Fair games and the martingale strategy I

I went back to my school a couple of weeks ago and gave a talk. I felt I’d given various incarnations of a talk on card-shuffling too many times, so it was time for a new topic. The following post (and time allowing, one or two more) is pretty much what I said.

The Martingale Strategy

Suppose we bet repeatedly on the outcome of tossing a fair coin. Since it’s November, my heart is set on buying an ice cream that costs £1, so my aim is to win this amount from our game. My strategy is this:

First, I bet £1. If I win, then that’s great, because I now have made exactly enough profit to buy the ice cream. If I lose, then I play again, and this time I bet £2. Again, if I win, then my total profit is £2-£1 = £1, so I stop playing and buy the ice cream. If I lose, then I play a third time, again doubling my stake. So if I win for the first time on the seventh go, my overall profit will be

£64 – (£1+£2+£4+£8+£16+£32) = £1,

and it’s clear that this can be continued and I will eventually win a round, and at this point my total profit will be £1. So I will always eventually be able to buy my ice cream.

But, there’s nothing special about the value £1, so I could replace the words ‘ice cream’ with ‘private tropical island’, so why am I still here in the UK on a wet Monday when I could be on my beach lounger?

There are some fairly obvious reasons why the strategy I’ve described is not actually a fail-safe way to make a profit. For a start, although with probability one a head will come up eventually, there is a small positive chance that the first 200 rolls will all be tails. At this point, I would have accrued a debt of roughly 2^{200} pounds, and this is slightly more than the number of atoms in the universe. All this for an ice cream?

So there are major problems carrying out this strategy in a finite world. And of course, it’s no good if we stop after a very large but finite number of turns, because then there’s always this very small chance that we’ve made a very large loss, which is bad, partly because we can’t have the ice cream, but also because it exactly cancels out the chance of making our £1 profit, and so our overall average profit is exactly zero.

Though I’ve set this up in an intentionally glib fashion, as so often is the case, we might have stumbled across an interesting mathematical idea. That is, if we play a fair game a finite number of times, we have a fair game overall, meaning our overall average profit is zero. But if we are allowed to play a potentially infinite number of times, then it’s not clear how to define our overall ‘average’ profit, since we feel it ought to be zero, as an extension of the finite case, but also might be positive, because it ends up being £1 with probability one.

It’s tempting at this stage to start writing statements like

1 \times 1 + (-\infty) \times 0=0 ,

to justify why this might have come about, where we consider the infinitely unlikely event that is infinitely costly. But this is only convincing at the most superficial level, and so it makes more sense to think a bit more carefully about under exactly what circumstances we can extend our observation about the overall fairness of a finite sequence of individual fair games.

A second example

The previous example was based upon a series of coin tosses, and we can use exactly the same source of randomness to produce a simple random walk. This is a process that goes up or down by 1 in each time step, where each option happens with probability ½, independently of the history.

We could avoid the requirement to deal with very large bets by always staking £1, and then cashing in the first time we have a profit of £1. Then, if we start the random walk at zero, it models our profit, and we stop the first time it gets to 1. It’s not obvious whether we hit 1 with probability one. Let’s show this.

In order to hit some positive value k, the random walk must pass through 1, 2, and so on, up to (k-1) and then finally k. So \mathbb{P}(\text{hit k}) = [\mathbb{P}(\text{hit 1})]^k. And similarly for negative values. Also, the probability that we return to zero is the same as the probability that we ever hit 1, since after one time-step they are literally the same problem (after symmetry). So, if the probability of hitting 1 is p<1, then the number of visits to zero is geometric (supported on 1,2,3,…) with parameter p, and so

\mathbb{E}[\text{visits to k}] = \mathbb{E}[\text{visits to zero}] \times \mathbb{P}(\text{hit k})=(1+1/p) \times p^{|k|} = (p+1)p^{|k|-1}.

Thus, when we sum over all values of k, we are summing a pair of geometric series with exponent <1, and so we get a finite answer. But if the expected number of visits to anywhere (ie the sum across all places) is finite, this is clearly ridiculous, since we are running the process for an infinite time, and at each time-step we must be somewhere! So we must in fact have p=1, and thus another potential counter-example to the claim that a sequence of fair games can sometimes be unfair.

We might have exactly the same set of practical objections, such as this method requiring arbitrarily large liquidity (even though it doesn’t grow exponentially fast so doesn’t seem so bad).

What will actually turn out to be useful is that although the bets are now small, the average time until we hit 1 is actually infinite. Remember that, even though most things we see in real life don’t have this property, it is completely possible for a random variable to take finite values yet have infinite expectation.

Notes on the Martingale Strategy

There’s no reason why the originally proposed strategy had to be based upon fair coin tosses. This strategy might work in a more general setting, where the chance of winning on a given turn is not ½, or is not even constant. So long as at each stage you bet exactly enough that, if you win, you recoup all your losses so far, and one extra pound, this has the same overall effect.

Of course, we need to check that we do eventually win a round, which is not guaranteed if the probability of winning (conditional on not having yet won) decays sufficiently fast. If we let p_k be the probability of winning on turn k, given that we haven’t previously won, then we require that the probability of never winning \prod_{k\ge 1}(1-p_k)=0. By taking logs and taking care of the approximations, it can be seen that the divergence or otherwise of \sum p_k determines which way this falls.

In the next post, we’ll talk about how the two problems encountered here, namely allowing large increments, and considering a stopping time with infinite expectation are exactly the two cases where something can go wrong. We’ll also talk about a slightly different setting, where the choice of when to stop playing becomes a bit more dynamic and complicated.

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Azuma-Hoeffding Inequality

It’s (probably) my last Michaelmas term in Oxford, at least for the time being, and so also the last time giving tutorials on either of the probability courses that students take in their first two years. This time, I’m teaching the second years, and as usual the aim of the majority of the first half of the course is to acquire as sophisticated an understanding as possible of the Central Limit Theorem. I feel a key step is appreciating that CLT tells you about the correct scaling for the deviations from the mean of these partial sums of IID random variables. The fact that these deviations on this correct scaling converge in law to a normal distribution, irrespective (apart from mild conditions) on the underlying distribution, is interesting, but should be viewed as a secondary, bonus, property.

Emphasising the scaling of deviations in CLT motivates the next sections of this (or any) course. We develop tools like Markov’s inequality to control the probability that a random variable is much larger than its expectation, and experiment with applying this to various functions of the random variable to get stronger bounds. When the moment generating function exists, this is an excellent choice for this analysis. We end up with a so-called Chernoff bound. For example, we might consider the probability that when we toss N coins, at least a proportion ¾ are Heads. A Chernoff bound says that this probability decays exponentially in N.

One direction to take is to ask how to control precisely the parameter of this exponential decay, which leads to Cramer’s theorem and the basis of the theory of Large Deviations. An alternative direction is to observe that the signed difference between the partial sums of independent random variables and their means is an example of a martingale, albeit not a very interesting one, since in general the increments of a martingale are not independent. So we might ask: under what circumstances can we show exponential tail bounds on the deviation of a martingale from its mean (that is, its initial value) at a fixed (perhaps large) time?

Azuma-Hoeffding inequality

The following result was derived and used by various authors in the 60s, including Azuma and Hoeffding (separately), but also others.

Let X_0,X_1,X_2,\ldots be a martingale with respect to some filtration, and we assume that the absolute value of each increment |X_i-X_{i-1}| is bounded almost surely by some c_i<\infty. Then, recalling that \mathbb{E}[X_n|\mathcal{F}_0]=X_0, we have

\mathbb{P}(X_n \ge X_0+t) \le \exp\left( -\frac{t^2}{2\sum_{i=1}^n c_i^2}\right).

Proof

We apply a Chernoff argument to each increment. First, observe that for Y a distribution supported on [-1,1] with mean zero, by convexity \mathbb{E}[e^{tY}] is maximised by taking Y equal to +1 and -1 each with probability ½. Thus

\mathbb{E}[e^{tY}]\le \frac12 e^t + \frac 12 e^{-t}=\cosh(t) \le e^{-t^2/2},

where the final inequality follows by directly comparing the Taylor series.

We’ll use this shortly. Before that, we start the usual argument for a Chernoff bound on X_n-X_0.

\mathbb{P}(X_n-X_0\ge t) = \mathbb{P}(e^{\theta(X_n-X_0)}\ge e^{\theta t})\le e^{-\theta t} \mathbb{E}[e^{\theta(X_n-X_0)}]

= e^{-\theta t} \mathbb{E}[\mathbb{E}[e^{\theta((X_n-X_{n-1}) +X_{n-1}-X_0)} | \mathcal{F}_{n-1}]]

= e^{-\theta t} \mathbb{E}[e^{\theta(X_{n-1}-X_0)} \mathbb{E}[e^{\theta(X_n-X_{n-1})}|\mathcal{F}_{n-1}] ],

and our preliminary result allows us to control this inner expectation

\le e^{-\theta t} e^{\theta^2c_n^2/2} \mathbb{E}[e^{\theta(X_{n-1}-X_0)}].

So now we can apply this inductively to obtain

\mathbb{P}(X_n-X_0\ge t) \le e^{-\theta t+ \theta^2 \sum_{i=1}^n c_i^2}.

Finally, as usual in such an argument, we need to choose a sensible value of the free parameter \theta, and naturally we want to choose it to make this RHS as small as possible, which is achieved when \theta = \frac{t}{\sum_{i=1}^n c_i^2}, and leads exactly to the statement of the inequality.

Applications

Unsurprisingly, we can easily apply this to the process of partial sums of IID random variables with mean zero and bounded support, to recover a Chernoff bound.

A more interesting example involves revealing the state (ie open or closed) of the edges of an Erdos-Renyi graph one at a time. We need to examine some quantitative property of the graph which can’t ever be heavily influenced by the presence or non-presence of a single given edge. The size of the largest clique, or the largest cut, are good examples. Adding or removing an edge can change these quantities by at most one.

So if we order the edges, and let the filtration \mathcal{F}_k be generated by the state of the first k edges in this ordering, then X_k=\mathbb{E}[\text{max cut}| \mathcal{F}_k] is a martingale. (A martingale constructed backwards in this fashion by conditioning a final state on a filtration is sometimes called a Doob martingale.) Using A-H on this shows that the deviations from the mean are of order \sqrt{N}, where N is the size of the graph. In the sparse case, it can be justified fairly easily that the maximum cut has size \Theta(N), since for example there will always be some positive proportion of isolated vertices. However, accurate asymptotics for the mean of this quantity seem (at least after a brief search of the literature – please do correct me if this is wrong!) to be unknown. So this might be an example of the curious situation where we can control the deviations around the mean better than the mean itself!

Beyond bounded increments

One observation we might make about the proof is that it is tight only if all the increments X_i-X_{i-1} are supported on \{-c_i,+c_i\}, which is stronger than demanding that the absolute value is bounded. If in fact we have X_i-X_{i-1}\in[-d_i,c_i] almost surely, then, with a more detailed preliminary lemma, we can have instead a bound of \exp\left( -\frac{2t^2}{\sum_{i=1}^n (c_i+d_i)^2} \right).

While it isn’t a problem in these examples, in many settings the restriction to bounded increments is likely to be the obstacle to applying A-H. Indeed, in the technical corner of my current research problem, this is exactly the challenge I faced. Fortunately, at least in principle, all is not necessarily lost. We might, for example, be able to establish bounds (c_i) as described, such that the probability that any |X_i-X_{i-1}| exceeds its c_i is very small. You could then construct a coupled process (Y_i), that is equal to X_i whenever the increments are within the given range, and something else otherwise. For Y to fit the conditions of A-H, the challenge is to ensure we can do this such that the increments remain bounded (ie the ‘something else’ also has to be within [-c_i,c_i] ) and also that Y remains a martingale. This total probability of a deviation is bounded above by the probability of Y experiencing that deviation, plus the probability of Y and X decoupling. To comment on the latter probability is hard in general without saying a bit more about the dependence structure in X itself.

Multitype Branching Processes

One of the fundamental objects in classical probability theory is the Galton-Watson branching process. This is defined to be a model for the growth of a population, where each individual in a generation gives birth to some number (possibly zero) of offspring, who form the next generation. Crucially, the numbers of offspring of the individuals are IID, with the same distribution both within generations and between generations.

There are several ways one might generalise this, such as non-IID offspring distributions, or pairs of individuals producing some number of offspring, but here we consider the situation where each individual has some type, and different types have different offspring distributions. Note that if there are K types, say, then the offspring distributions should now be supported on \mathbb{Z}_{\ge 0}^K. Let’s say the offspring distribution from a parent of type i is \mu^{(i)}.

The first question to address is one of survival. Recall that if we want to know whether a standard Galton-Watson process has positive probability of having infinite size, that is never going extinct, we only need to know the expectation of the offspring distribution. If this is less than 1, then the process is subcritical and is almost surely finite. If it is greater than 1, then it is supercritical and survives with positive probability. If the expectation is exactly 1 (and the variance is finite) then the process is critical and although it is still almost surely finite, the overall population size has a power-law tail, and hence (or otherwise) the expected population size is infinite.

We would like a similar result for the multitype process, saying that we do not need to know everything about the distribution to decide what the survival probability should be.

The first thing to address is why we can’t just reduce the multitype change to the monotype setting. It’s easiest to assume that we know the type of the root in the multitype tree. The case where the type of the root is random can be reconstructed later. Anyway, suppose now that we want to know the offspring distribution of a vertex in the m-th generation. To decide this, we need to know the probability that this vertex has a given type, say type j. To calculate this, we need to work out all the type possibilities for the first m generations, and their probabilities, which may well include lots of complicated size-biasing. Certainly it is not easy, and there’s no reason why these offspring distributions should be IID. The best we can say is that they should probably be exchangeable within each generation.

Obviously if the offspring distribution does not depend on the parent’s type, then we have a standard Galton-Watson tree with types assigned in an IID manner to the realisation. If the types are symmetric (for example if M, to be defined, is invariant under permuting the indices) then life gets much easier. In general, however, it will be more complicated than this.

We can however think about how to decide on survival probability. We consider the expected number of offspring, allowing both the type of the parent and the type of the child to vary. So define m_{ij} to be the expected number of type j children born to a type i parent. Then write these in a matrix M=(m_{ij}).

One generalisation is to consider a Galton-Watson forest started from some positive number of roots of various types. Suppose we have a vector \nu=(\nu_i) listing the number of roots of each type. Then the expected number of descendents of each type at generation n is given by the vector \nu M^n.

Let \lambda be the largest eigenvalue of M. As for the transition matrices of Markov chains, the Perron-Frobenius theorem applies here, which confirms that, because the entries of M are positive, the eigenvalue with largest modulus is simple and real, and the associated eigenvector has entirely positive entries. [In fact we need a couple of extra conditions on M, including that it is possible to get from any type to any other type – we say irreducible – but that isn’t worth going into now.]

So in fact the total number of descendents at generation n grows like \lambda^n in expectation, and so we have the same description of subcriticality and supercriticality. We can also make a sensible comment about the left-\lambda-eigenvector of M. This is the limiting proportion of the different types of vertices.

It’s a result (eg. [3]) that the height profile of a depth-first search on a standard Galton-Watson tree converges to Brownian Motion. Another way to phrase this is that a GW tree conditioned to have some size N has the Brownian Continuum Random Tree as a scaling limit as N grows to infinity. Miermont [4] proves that this result holds for the multitype tree as well. In the remainder of this post I want to discuss one idea along the way to the proof, and one application.

I said initially that there wasn’t a trivial reduction of a multitype process to a monotype process. There is however a non-trivial embedding of a monotype process in a multitype process. Consider all the vertices of type 1, and all the paths between such vertices. Then draw a new tree consisting of just the type 1 vertices. Two of these are joined by an edge if there is no other type 1 vertex on the unique path between them in the original tree. If that definition is confusing, think of the most sensible way to construct a tree on the type 1 vertices from the original, and you’ve probably chosen this definition.

There are two important things about this new tree. 1) It is a Galton-Watson tree, and 2) if the original tree is critical, then this reduced tree is also critical. Proving 1) is heavily dependent on exactly what definitions one takes for both the multitype branching mechanism and the standard G-W mechanism. Essentially, at a type 1 vertex, the number of type 1 descendents is not dependent on anything that happened at previous generations, nor in other branches of the original tree. This gives IID offspring distributions once it is formalised. As for criticality, we note that by the matrix argument given before, under the irreducibility condition discussed, the expectation of the total population size is infinite iff the expected number of type 1 vertices is also infinite. Since the proportion of type 1 vertices is given by the first element of the left eigenvector, which is positive, we can make a further argument that the number of type 1 vertices has a power-law tail iff the total population size also has a power-law tail.

I want to end by explaining why I was thinking about this model at all. In many previous posts I’ve discussed the forest fire model, where occasionally all the edges in some large component are deleted, and the component becomes a set of singletons again. We are interested in the local limit. That is, what do the large components look like from the point of view of a single vertex in the component? If we were able to prove that the large components have BCRT as the scaling limit, this would answer this question.

This holds for the original random graph process. There are two sensible ways to motivate this. Firstly, given that a component is a tree (which it is with high probability if its size is O(1) ), its distribution is that of the uniform tree, and it is known that this has BCRT as a scaling limit [1]. Alternatively, we know that the components have a Poisson Galton-Watson process as a local limit by the same argument used to calculate the increments of the exploration process. So we have an alternative description of the BCRT appearing: the scaling limit of G-W trees conditioned on their size.

Regarding the forest fires, if we stop the process at some time T>1, we know that some vertices have been burned several times and some vertices have never received an edge. What is clear though is that if we specify the age of each vertex, that is, how long has elapsed since it was last burned; conditional on this, we have an inhomogeneous random graph. Note that if we have two vertices of ages s and t, then the probability that there is an edge between them is 1-e^{-\frac{s\wedge t}{n}}, ie approximately \frac{s\wedge t}{n}. The function giving the probabilities of edges between different types of vertices is called the kernel, and here it is sufficiently well-behaved (in particular, it is bounded) that we are able to use the results of Bollobas et al in [2], where they discuss general sparse inhomogeneous random graphs. They show, among many other things, that in this setting as well the local limit is a multitype branching process.

So in conclusion, we have almost all the ingredients towards proving the result we want, that forest fire components have BCRT scaling limit. The only outstanding matter is that the Miermont result deals with a finite number of types, whereas obviously in the setting where we parameterise by age, the set of types is continuous. In other words, I’m working hard!

References

[1] Aldous – The Continuum Random Tree III

[2] Bollobas, Janson, Riordan – The phase transition in inhomogeneous random graphs

[3] Le Gall – Random Trees and Applications

[4] Miermont – Invariance principles for spatial multitype Galton-Watson trees

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Generating Functions for Dice

So last week I was writing an article for Betting Expert about laws of large numbers, and I was trying to produce some representations of distributions to illustrate the Weak LLN and the Central Limit Theorem. Because tossing a coin feels too simplistic, and also because the natural state space for this random variable, at least verbally, is not a subset of the reals, I decided to go for dice instead. So it’s clear what the distribution of the outcome of a single dice roll is, and with a bit of thought or a 6×6 grid, we can work out the distribution of the average of two dice rolls. But what about 100 rolls? Obviously, we need large samples to illustrate the laws of large numbers! In this post, we discuss how to calculate the distribution of the sample mean of n dice rolls.

First we observe that the total set of outcomes of n dice rolls is 6^n. The sum of the outcomes must lie between n and 6n inclusive. The distribution of the sum and the distribution of the sample mean are equivalent up to dividing by n. The final observation is that because the total number of outcomes has a nice form, we shouldn’t expect it to make any difference to the method if we calculate the probability of a given sum, or the number of configurations giving rise to that sum.

Indeed, tying in nicely with the first year probability course, we are going to use generating functions, and there is no difference in practice between the probability generating function, and the combinatorial generating function, if the underlying mechanism is a uniform choice. Well, in practice, there is a small difference, namely a factor of 6 here. The motivation for using generating functions is clear: we are considering the distribution of a sum of independent random variables. This is pretty much exactly why we bother to set up the machinery for PGFs.

Anyway, since each of {1,2,…,6} is equally likely, the GF of a single dice roll is

x+x^2+\ldots+x^6=x\cdot \frac{1-x^6}{1-x}.

So, if we want the generating function of the sum of n independent dice rolls, we can obtain this by raising the above function to the power n. We obtain

x^n(1-x^6)^n(1-x)^{-n}.

Note the factor of x^n at the beginning arises because the minimum value of the sum is n. So to work out the number of configurations giving rise to sum k, we need to evaluate the coefficient of x^k. We can deal with (1-x^6)^n fairly straightforwardly, but some thought it required regarding whether it’s possible to do similar job on (1-x)^{-n}.

We have to engage briefly with what is meant by a binomial coefficient. Note that

\binom{x}{k}=\frac{x(x-1)\ldots(x-k+1)}{1\cdot\ldots\cdot k}

is a valid definition even when x is not a positive integer, as it is simply a degree k polynomial in x. This works if x is a general positive real, and indeed if x is a general negative real. At this stage, we do need to keep k a positive integer, but that’s not a problem for our applications.

So we need to engage with how the binomial theorem works for exponents that are not positive integers. The tricky part with the standard expression as

(a+b)^n=\binom{n}{0}a^n+\ldots + \binom{n}{n}b^n,

is that the attraction of this symmetry in a and b prompts us to work in more generality than is entirely necessary to state the result. Note if we instead write

(1+x)^n=1+\binom{n}{1}x+\binom{n}{2}x^2+\ldots,

we have unwittingly described this finite sum as an infinite series. It just happens that all the binomial coefficients apart from the first (n+1) are zero. The nice thing about this definition is that it might plausibly generalise to non-integer or negative values of n. And indeed it does. I don’t want to go into the details here, but it’s just a Taylor series really, and the binomial coefficients are set up with factorials in the right places to look like a Taylor series, so it all works out.

It is also worth remarking that it follows straight from the definition of a negative binomial coefficient, that

\binom{-n}{j}=(-1)^j \binom{n+j-1}{j}.

In any case, we can rewrite our expression for the generating function of the IID sum as

x^n\left[\sum_{k=0}^n \binom{n}{k}(-1)^k x^{6k}\right]\left[\sum_{j\ge 0} \binom{-n}{j}(-1)^j x^j\right]

By accounting for where we can gather exponents from each bracket, we can evaluate the coefficient of x^m as

\sum_{6k+j=m+n}\binom{n}{k}\binom{n+j-1}{j}(-1)^k.

Ie, k in the sum takes values in \{0,1,\ldots, \lfloor \frac{m+n}{6}\rfloor\}. At least in theory, this now gives us an explicit way to calculate the distribution of the average of multiple dice rolls. We have to be wary, however, that many compilers will not be happy dealing with large binomial coefficients, as the large factorials grow extremely rapidly. An approximation using logs is likely to be more tractable for larger settings.

Anyway, I leave you with the fruits of my labours.

dice30Related articles

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The Contour Process

As I explained in my previous post, I haven’t been reading around as much as I would generally like to recently. A few days in London staying with my parents and catching up with some friends has therefore been a good chance to get back into the habit of leafing through papers and Pitman’s book among other things.

This morning’s post should be a relatively short one. I’m going to define the contour process, a function of a (random or deterministic) tree, related to the exploration process which I have mentioned a few times previously. I will then use this to prove a simple but cute result equating in distribution the sizes of two different branching processes via a direct bijection.

The Contour Process

To start with, we have to have a root, and from that root we label the tree with a depth-first labelling. An example of this is given below. It is helpful at this stage to conceive this process as an explorer walking on the tree, and turning back on themselves only when there is no option to visit a vertex they haven’t already seen. So in the example tree shown, the depth-first exploration visits vertex V_2 exactly four times. Note that with this description, it is clear that the exploration traverses every edge exactly twice, and so the length of the sequence is 2n-1, where n is the number of vertices in the tree since obviously, we start and end at the root.

Another common interpretation of this depth-first exploration is to take some planar realisation of the tree. (Note trees are always planar – proof via induction after removing a leaf.) Then if you treat the tree as a hedge and starting at the root walk along, following the outer boundary with your right hand, this exactly recreates the process.

The height of a tree at a particular vertex is simply the graph distance between that vertex and the root. So when we move from one vertex to an adjacent vertex, the height must increase or decrease by 1.

The contour process is the sequence of heights seen along the depth-first exploration. It is therefore a sequence:

0=h_0,h_1,\ldots,h_{2n-1}=0,\quad h_i\geq 0,

and such that |h_{i+1}-h_i|=1.

Note that though the contour process uniquely determines the tree structure, the choice of depth-first labelling is a priori non-canonical. For example, in the display above, V_3 might have been explored before V_2. Normally this is resolved by taking the suitable vertex with the smallest label in the original tree to be next. It makes little difference to any analysis to choose the ordering of descendents of some vertex in a depth-first labelling randomly. Note that this explains why it is rather hard to recover Cayley’s theorem about the number of rooted trees on n vertices from this characterisation. Although the number of suitable contour functions is possible to calculate, we would require a complicated multiplicative correction for labelling if we wanted to recover the number of trees.

The only real observation about the uses of the contour process at this stage is that it is not in general a random walk with IID increments for a Galton-Watson branching process. This equivalence is what made the exploration process so useful. In particular, it made it straightforward, at least heuristically, to see why large trees might have a limit interpretation through Brownian excursions. If for example, the offspring distribution is bounded above, say by M, then the contour process certainly cannot be a random walk, as if we have visited a particular vertex exactly M+1 times, then it cannot have another descendent, and so we must return closer to the root at the next step.

I want to mention that in fact Aldous showed his results on scaling limits towards the Continuum Random Tree through the contour process rather than the exploration process. However, I don’t want to say any more about that right now.

A Neat Equivalence

What I do want to talk about is the following distribution on the positive integers. This comes up in Balazs Rath and Balint Toth’s work on forest-fires on the complete graph that I have been reading about recently. The role of this distribution is a conjectured equilibrium distribution for component size in a version of the Erdos-Renyi process where components are deleted (or ‘struck by lightning’) at a rate tuned so that giant components ‘just’ never emerge.

This distribution has the possibly useful property that it is the distribution of the total population size in a Galton-Watson process with Geom(1/2) offspring distribution. It is also the distribution of the total number of leaves in a critical binary branching process, where every vertex has either two descendents or zero descendents, each with probability 1/2. Note that both of these tree processes are critical, as the expected number of offspring is 1 in each case. This is a good start, as it suggests that the relevant equilibrium distribution should also have the power-law tail that is found in these critical branching processes. This would confirm that the forest-fire model exhibits self-organised criticality.

Anyway, as a sanity check, I tried to find a reason why, ignoring the forest-fires for now, these two distributions should be the same. One can argue using generating functions, but there is also the following nice bijective argument.

We focus first on the critical Geometric branching process. We examine its contour function. As explained above, the contour process is not in general a random walk with IID increments. However, for this particular case, it is. The geometric distribution should be viewed as the family of discrete memoryless distributions.

This is useful for the contour process. Note that if we are at vertex V for the (m+1)th time, that is we have already explored m of the edges out of V, then the probability that there is at least one further edge is 1/2, independently of the history of the exploration, as the offspring distribution is Geometric(1/2), which we can easily think of as adding edges one at a time based on independent fair coin tosses until we see a tail for example. The contour process for this random tree is therefore a simple symmetric random walk on Z. Note that this will hit -1 at some point, and the associated contour process is the RW up to the final time it hits 0 before hitting -1. We can check that this obeys the clear rule that with probability 1/2 the tree is a single vertex.

Now we consider the other model, the Galton-Watson process with critical binary branching mechanism. We should consider the exploration process. Recall that the increments in this process are given by the offspring distribution minus one. So this random sequence also behaves as a simple symmetric random walk on Z, again stopped when we hit -1.

To complete the bijective argument, we have to relate leaves in the binary process to vertices in the geometric one. A vertex is a leaf if it has no offspring, so the number of leaves is the number of times before the hitting time of -1 that the exploration process decreases by 1. (*)

Similarly for the contour process. Note that there is bijection between the set of vertices that aren’t the root and the set of edges. The contour process explores every edge exactly twice, once giving an increase of 1 and once giving a decrease of 1. So there is a bijection between the times that the contour process decreases by 1 and the non-root vertices. But the contour process was defined only up to the time we return to the root. This is fine if we know in advance how large the tree is, but we don’t know which return to the root is the final return to the root. So if we extend the random walk to the first time it hits -1, the portion up until the last increment is the contour process, and the final increment must be a decrease by 1, hence there is a bijection between the number of vertices in the Geom(1/2) G-W tree and the number of times that the contour process decreases by 1 before the hitting time of -1. Comparing with (*) gives the result.

Minimum Spanning Trees

In my last post, I discussed the Uniform Spanning Tree. To summarise very briefly, given a connected graph on n vertices, a tree is a subgraph, that is a subset of the edges, which is connected, but which contains no cycles. It turns out this requires the tree to have n-1 edges.

We are interested in natural mechanisms for generating randomly chosen spanning trees of a given graph. One way we can always do this is to choose uniformly at random from the set of possible trees. This UST is in some sense canonical, but it is worth knowing about some other measures on trees that might be of interest.

A family of natural problems in operations research concerns an arbitrary complex network, with some weight or cost associated to each connection. The question is how to perform some operation on the network so as to minimise the resulting cost. Perhaps the most famous such problem is that of the Travelling Salesman. The story is that a salesman needs to visit n locations and wants to do the trip as efficiently as possible. This might be thought of as some sort of financial or time cost, but proably the easiest way to set it up is to imagine he is trying to minimise the distance he has to travel. It is not hard to see why this problem might genuinely arise in plenty of real-world situations, where a organisation or agent is trying to be as efficient as possible.

It might be the case that it is not possible to travel between every pair of locations, but we needn’t assume that for now. So if he knows the distance between any pair of cities, he wants to know which of the possible routes gives the shortest overall distance. The problem is that there are n! routes, and this grows roughly like n^n, which is faster than exponential, so for as few as 20 cities it has turned into a comparison which is too large to compute.

There are various algorithms which reduce the number of routes that must be checked, and some approximation methods. But if you want the exact answer, it is not currently possible to calculate this in polynomial time.

Minimal Spanning Trees and Uniqueness

For the travelling salesman, we were looking for the minimal cost spanning path. In the case of the complete graph, this is the same as the minimal cost non-repeating path of length n-1. Such paths are a subset of the set of spanning trees on the underlying graph. So what if we look instead for the minimal cost spanning tree? This exists as after all, there are only finitely many spanning trees.

So far, this has been deterministic, but we were looking for a random spanning tree. We can achieve this by choosing the weights at random. Anything other than assigning the weights as an IID sequence seems likely to be complicated, but there isn’t a canonical choice of the distribution of the weights. Our first question will be whether the distribution of the weights affects the distribution of the induced MST. In fact it will turn out that so long as the distribution is continuous, it has no effect on the distribution of the MST. The continuous condition might seem odd, but it is present only to ensure that the weights almost certainly end up generating a unique MST.

It turns out that there is a straightforward greedy algorithm to find the MST once the weights are known. We will examine some consequences of this algorithm in the random setting. First we check uniqueness. The condition required for uniqueness is that the weights be distinct. Note that this is slightly weaker than the statement that all of sums of (n-1)-tuples be distinct, which immediately implies a unique MST.

We now prove this condition. Suppose we have distinct weights, and an associated MST. If the underlying graph is a tree, then the result is clear. Otherwise, add some extra edge e, with weight w(e). By the definition of a tree, this generates exctly one cycle. Consider the other edges, say e_1,\ldots,e_k in this cycle. If any of w(e_i)>w(e) then we can replace e_i with e to get a spanning tree with smaller weight, a contradiction of the claim that we started with an MST. So by distinctness of weights, we conclude that w(e)>w(e_i) for all i.

Conversely, suppose we remove some edge e which IS in the MST. We end up with exactly two connected components. Consider all the edges in the underlying graph between the two components, and suppose that one of these f satisfies w(f)<w(e). Then if we add in edge f, which is by construction not in the original MST, we end up with a smaller total weight than we started with, a further contradiction.

We can summarise this in a neat form. Given an edge e between x and y, consider the set of all edges in the underlying graph with weight LESS THAN w(e). Then if x and y are in different components, the edge e must be in the MST. Since we have an explicit description of which edges are present, it follows that the MST is unique. The problem is that working out the component structure of the graph with higher weights removed is computationally rather intensive. We want a slightly faster algorithm.

Kruskal’s Algorithm

Several rather similar algorithms were developed roughly simultaneously. Prim’s algorithm is a slight generalisation of what we will discuss. Anyway, for now we consider Kruskal’s algorithm which has the advantage that it can be described without really needing to draw a diagram.

We start by ordering the weights. Without loss of generality, we might as well relabel the edges so that

w(e_1)< w(e_2)<\ldots< w(e_{|E|}).

Now, by the condition derived in the argument for uniqueness, we must have e_1 and e_2 in any MST. Now consider e_3. Unless doing so would create a cycle, add e_3. Then, unless doing so would create a cycle, add e_4. Continue. It is clear that the result of this procedure is acyclic. To check it is actually a spanning tree, we show that it is also connected. Suppose not, and two of the components are A and B. Let e be the edge between A and B with minimal weight. According to the algorithm, we should have included e in our MST because at no point would adding it possibly have created a cycle. So we have proved that this greedy algorithm does indeed give the (unique) MST.

A useful consequence of this is that we know the two edges with overall minimum weight are definitely in the MST. In the search for a random measure on spanning trees, what is most important is that we didn’t use the actual values of the weights in this construction, only the order. In other words, we might as well have assumed the weights were a random permutation from S_{|E|}. This now answers our original question about how the random weight MST depends in distribution on the underlying edge weight distribution. So long as with probability one the weights are distinct (which holds if the distribution is continuous), then the distribution of the resulting spanning tree is constant.

It’s not too hard to show this isn’t the same as UST: n=4 suffices as a counterexample. But the difference in asymptotic behaviour of properties such as the diameter is of interest, and will be explored in the next post.

Beyond Erdos-Renyi: more realistic models of networks

The claim is often made that the study of random graphs such as the Erdos-Renyi model is worthwhile because it gives us information about complex systems which exist in the real world. The internet or social networks provide the example du jour at the moment, but it’s equally plausible to think about traffic flows, electrical systems or interacting biological processes too.

If this were entirely true, it would be great for two reasons. Firstly, in my opinion at least, it is a beautiful subject in its own right, and to have a concrete applicable reason to continue studying it would make it even better. (Not to mention the dreaded competition for funding…) Secondly, Erdos-Renyi is so simple. After all, it involves little more than adding some simple topology to a collection of IID Bernoulli random variables, and so it would surely be possible to draw some significant conclusions about how complicated real-world objects interact without too much mathematical effort.

Unfortunately, but unsurprising, this simplicity is a drawback as far as applications go. It is fairly clear that most real-world systems cannot offer any property even approaching the niceness of the independent, same probability edges condition. But rather than consign E-R to the ‘pretty but useless’ category of mathematical structures, we should think carefully about exactly why it fails to be a good model for real-world networks, and see whether there are any small adjustments that could be made to improve it.

This is something I’ve been meaning to read up about for ages and ages. What follows is based heavily on the Albert and Barabasi 2002 review paper. I suspect that many of the open problems and intuitive calculations have since been finished and formalised, but for an overview I hope that doesn’t matter hugely. I’ve also leafed through the relevant chapters of Remco Van der Hofstad’s notes, but am setting the details and the exercises aside for the holidays when I have a bit more time!

Problems with Erdos-Renyi

Recall that G(n,p) takes n vertices, and adds edges between any pair of vertices independently with probability p.

One property shared by most real-world networks is the scale-free phenomenon, which says that the degree distribution has a power law tail. The Albert-Barabasi papers gives a comprehensive survey of data verifying this claim. By contrast, G(n,p) has degree distribution which is approximately Poisson as n grows. This is concentrated near the average degree with a thin exponential tail, so does not satisfy this requirement. I was and still am a bit confused by the term ‘scale-free’. The idea is certainly that the local structure is independent of the size of the system, which seems to be true for the degree distributions in sparse ER, that is where p = O(1/n). But I think the correct heuristic is that it doesn’t matter how far zoomed in you are – the macroscopic structure looks similar for n vertices as for n^2 vertices. This certainly fails to be true for ER, where no vertex has O(n) neighbours, whereas with a power law tail, this does hold.

The main consequence of this is that there are a few vertices with very high degree. These are often called ‘hubs’ and parallels are drawn to the internet, where key websites and servers connect lots of traffic and pages from different areas. The idea is that the hubs are almost certainly well-connected to each other, and this offers a step towards a small-world phenomenon, where the shortest path between any two vertices is very small relative to the size of the system. This notion was introduced to mainstream culture by Stanley Milgram’s ‘Six degrees of separation’ experiment in the 60s, where it became clear that subjects were able to deliver a package to a complete stranger on the other side of the USA, using only personal contacts, in about six stages. The graph theoretic notion for this is the diameter, defined as the maximal graph distance between two points. Here, the graph distance means the length of the shortest path between the points. This definition, with the max-min formalism looks rather complicated, but isn’t really. The diameter of an Erdos-Renyi graph for fixed p, increases like log n, which is small relative to n, and so this property holds.

A quick glance at your list of Facebook friends will confirm that the independent edges condition in an Erdos-Renyi random graph is not a plausible model for social networks. How many friends do you have? Let’s say about 1000, more to make the calculation easier than because you’re necessarily very popular. How many does your friend Tom have? Let’s say 1000 again. As was in the news a few months ago, there are now over a billion people on Facebook. Let’s say exactly a billion (that is 10^9 for these purposes). So both you and Tom are friends with 1/10^6 of the total membership of the network. So how large would you expect the overlap of your friendships to be, if they were all chosen independently at random? Well, the probability that you are both friends with Alice is 10^-12, and so the expected number of your mutual friends is 10^-12 x 10^9 = 10^-3 which is substantially less than 1. Yet I imagine if you substituted names suitably, you and Tom might well have over 50 mutual friends if you were, say, in the same year at school or niversity and haven’t yet purged your list.

We want a statistic that records this idea quantitatively. There are various candidates for such a clustering coefficient. The underlying notion that we might expect there to be greater connectivity between neighbours of some fixed point v than in the graph as a whole gives an intuition for a possible definition. Compare the proportion of triangles in the graph to the cube of the proportion of edges. When this ratio is large, then there is a lot of clustering. In the E-R case, we would expect these to be equal, as the probability of forming a triangle is equal to the cube of the probability of the presence of each of the three independent edges that make up the triangle.

So we have three properties of real networks that we would like to incorporate into a model: small diameter, power-law degree distribution, and high clustering. To avoid this turning into a book, I’m going to write a paragraph about each of the possibilities discussed by Albert and Barabasi.

Generalised Random Graph

The degree distribution will typically emerge as a consequence of the construction of a given model. The general idea here is to condition on the degree distribution having the form we want, and see what this does to the structure. Of course, the choice of how to do this conditioning is absolutely key. It certainly isn’t obvious what it means to ‘condition G(n,p) to have power-law distribution’, since the very idea of a power-law vs exponential tail requires the number of vertices to be large.

The first idea for achieving this gives the vertices ‘stubs’, which join up in pairs to form edges. We decide on the distribution of stubs according to this power law, then pair them up uniformly at random. Obviously, there is a possibility of getting some loops, but this is not going to happen so often as to be a genuine problem in the limit. This construction is similarly open to the branching process exploration ideas well covered for the E-R random graph, though we have to be careful to size-bias the degree distributions when necessary. There is still an underlying independence in the location of edges though, so it is reasonably clear that the amount of clustering may be closer to E-R than to the real examples cited.

The other possibility suggested is to retain the independent edge property, but give the vertices weights, and let the probability of an edge between two vertices be some sensible function of the weights. In the end it turns out to make little difference whether the weights are chosen deterministically or randomly, but by taking the weights i.i.d. with infinite mean, we can generate a so-called generalised random graph where the degree distribution has a power law.

Watts-Strogatz

In the WS model, the idea is to interpolate between a graph with maximal clustering and a random graph. A d-regular graph, say on a ring, where every vertex is connected to its d nearest neighbours has high clustering, but large diameter, as for example it takes roughly n/2d steps to get to the other side of the ring. Whereas in the standard E-R model we add edges with some fixed probability p, here we replace edges with some fixed probability p. That is, we take an edge in the regular graph and with some small probability we remove it and instead add an edge between two vertices chosen uniformly at random. The theoretical motivation is that removing a few edges doesn’t destroy the high clustering evident in the regular graph, but even a sparse random graph has small diameter, so adding a few ‘long-range’ edges should be enough to decrease the diameter significantly.

It obviously needs to be checked that a substantial drop in diameter occurs before a substantial decrease in clustering, and there is a calculation and diagram to support this intuitive idea in the paper. The one drawback of this model is that it fails to provide the power-law degree distributions we want. After all, an E-R graph has a concentrated degree distribution, and a d-regular graph has all degrees the same, so we would expect some interpolation between the two to have a concentrated distribution as well. Nonetheless, this model accords well with an idea of how complex networks might form, particularly if there is some underlying geometry. It is reasonable to assume that an initial setup for a network would be that people are connected to those closest to them, and then slowly acquire distant contacts as time progresses.

Preferential Attachment – Barabasi-Albert model

Most of our intuition for networks can be extended to an intuition for the formation of networks. The idea of prescribing a degree distribution is neat, but it doesn’t give any account to the mechanism of formation. Complexity emerges over time, and a good model should be able to describe why this happens. The Barabasi-Albert model takes this as its starting point, with the aim of producing a highly clustered system dynamically. Recall that we can describe G(n,p) as a process by coupling, then increasing p from 0 to 1, and seeing edges emerge. The independence assumption can be lifted through the coupling, and so which edge appears next is independent of the current state of the system.

This is what we need to relax. Recall the motivating idea of ‘hubs’, where a small collection of vertices have very high connectivity across the whole system, as observed in several real situations. A consequence of this is that new edge is more likely to be attached to a hub, than to a pair of poorly connected vertex elsewhere. But it turns out that this idea of preferential attachment isn’t enough by itself. Because as a network forms, it is not just the connectivity that increases, but also the size of the system itself. So in fact it makes sense to add vertices rather than edges, and join the new vertices to existing vertices in proportion to the degrees of the existing vertices. This combination of growth and preferential attachment is key to the scale-free graphs that this Barabasi-Albert model generates. Relaxing either mechanism returns us to the case of exponential tails. However, there are methods in the literature for generating such graphs without the need for a dynamic model, but they are harder to understand and describe. None I have seen so far has a high clustering coefficient.

Hubs are effectively a way to reduce the diameter. Recall the description of Milgram’s experiment where he encouraged randomly chosen people to send a package to Harvard. For the purposes of this model, an undergraduate from Wyoming or a husband from Alabama moving in with his wife in Boston are clear hubs, as for very many people near their previous home, they represent a good connection to Harvard. So it is unsurprising that BA, which reinforces hubs, has a sub-logarithmic small diameter.

Conclusions

I’m not entirely what conclusions I should draw from my reading. Probably the main one is that I should read more as there is plenty of interesting stuff going on in this area. Intuitively, it seems unlikely that there is going to be a single model which unites the descriptions of all relevant real-world networks. As ever, it is pleasant to find structures that are both mathematically interesting in their own right and relevant to applied problems. So it is reassuring to observe how similar many of the models discussed above are to the standard random graph.

Poisson Tails

I’ve had plenty of ideas for potential probability posts recently, but have been a bit too busy to write any of them up. I guess that’s a good thing in some sense. Anyway, this is a quick remark based on an argument I was thinking about yesterday. It combines Large Deviation theory, which I have spent a lot of time learning about this year, and the Poisson process, which I have spent a bit of time teaching.

Question

Does the Poisson distribution have an exponential tail? I ended up asking this question for two completely independent reasons yesterday. Firstly, I’ve been reading up about some more complex models of random networks. Specifically, the Erdos-Renyi random graph is interesting mathematical structure in its own right, but the independent edge condition results in certain regularity properties which are not seen in many real-world networks. In particular, the degree sequence of real-world networks typically follows an approximate power law. That is, the tail is heavy. This corresponds to our intuition that most networks contain ‘hubs’ which are connected to a large region of the network. Think about key servers or websites like Wikipedia and Google which are linked to by millions of other pages, or the social butterfly who will introduce friends from completely different circles. In any case, this property is not observed in an Erdos-Renyi graph, where the degrees are binomial, and in the sparse situation, rescale in the limit to a Poisson distribution. So, to finalise this observation, we want to be able to prove formally that the Poisson distribution has an exponential (so faster than power-law) tail.

The second occurrence of this question concerns large deviations for the exploration process of a random graph. This is a topic I’ve mentioned elsewhere (here for the exploration process, here for LDs) so I won’t recap extensively now. Anyway, the results we are interested in give estimates for the rate of decay in probability for the event that the path defined by the exploration process differs substantially from the expected path as n grows. A major annoyance in this analysis is the possibility of jumps. A jump occurs if a set of o(n) adjacent underlying random variables (here, the increments in the exploration process) have O(n) sum. A starting point might be to consider whether O(1) adjacent RVs can have O(n) sum, or indeed whether a single Poisson random variable can have sum of order n. In practice, this asks whether the probability \mathbb{P}(X>\alpha n) decays faster than exponentially in n. If it does, then this is dominated on a large deviations scale. If it decays exactly exponentially in n, then we have to consider such jumps in the analysis.

Approach

We can give a precise statement of the probabilities that a Po(\lambda) random variable X returns a given integer value:

\mathbb{P}(X=k)=e^{-\lambda}\frac{\lambda^k}{k!}.

Note that these are the terms in the Taylor expansion of e^{\lambda} appropriately normalised. So, while it looks like it should be possible to evaluate

\mathbb{P}(X>\alpha n)=e^{-\lambda}\sum_{\alpha n}^\infty \frac{\lambda^k}{k!},

this seems impossible to do directly, and it isn’t even especially obvious what a sensible bounding strategy might be.

The problem of estimating the form of the limit in probability of increasing unlikely deviations from expected behaviour surely reminds us of Cramer’s theorem. But this and other LD theory is generally formulated in terms of n random variables displaying some collective deviation, rather than a single random variable, with the size of the deviation growing. But we can transform our problem into that form by appealing to the three equivalent definitions of the Poisson process.

Recall that the Poisson process is the canonical description of, say, an arrivals process, where events in disjoint intervals are independent, and the expected number of arrives in a fixed interval is proportional to the width of the interval, giving a well-defined notion of ‘rate’ as we would want. The two main ways to define the process are: 1) the times between arrivals are given by i.i.d. Exponential RVs with parameter \lambda equal to the rate; and 2) the number of arrivals in interval [s,t] is independent of all other times, and has distribution given by Po(\lambda(t-s)). The fact that this definition gives a well-defined process is not necessarily obvious, but let’s not discuss that further here.

So the key equivalence to be exploited is that the event X>n for X\sim \text{Po}(\lambda) is a statement that there are at least n arrivals by time 1. If we move to the exponential inter-arrival times definition, we can write this as:

\mathbb{P}(Z_1+\ldots+Z_n<1),

where the Z’s are the i.i.d. exponential random variables. But this is exactly what we are able to specify through Cramer’s theorem. Recall that the moment generating function of an exponential distribution is not finite everywhere, but that doesn’t matter as we construct our rate function by taking the supremum over some index t of:

I(x)=\sup_t (xt-\log \mathbb{E}e^{tZ_1})=\sup_t(xt-\log(\frac{\lambda}{\lambda-t})).

A simple calculation then gives

I(x)=\lambda x-1 - \log \lambda x.

\Rightarrow I(x)\uparrow \infty\text{ as }x\downarrow 0.

Note that I(1) is the same for both Exp(\lambda) and Po(\lambda), because of the PP equality of events:

\{Z_1+\ldots+Z_n\leq n\}=\{\text{Po}(\lambda n)=\text{Po}(\lambda)_1+\ldots+\text{Po}(\lambda)_n> n\},

similar to the previous argument. In particular, for all \epsilon>0,

\mathbb{P}(\text{Po}(\lambda)>n)=\mathbb{P}(\frac{Z_1+\ldots+Z_n}{n}<\frac{1}{n})<\mathbb{P}(\frac{Z_1+\ldots+Z_n}{n}<\epsilon),\text{ for large }n.

\mathbb{P}(\text{Po}(\lambda)>n)=O(e^{-nI(\epsilon)}),\text{ for all }\epsilon.

Since we can take I(\epsilon) as large as we want, we conclude that the probability decays faster than exponentially in n.

Extreme Value Theory

This is something interesting which came up on the first problem sheet for the Part A Statistics course. The second question introduced the Weibull distribution, defined in terms of parameters \alpha,\lambda>0 through the distribution function:

F(x)=\begin{cases}0 & x<0\\ 1-\exp(-(\frac{x}{\lambda})^\alpha) & x\geq 0.\end{cases}

As mentioned in the statement of the question, this distribution is “typically used in industrial reliability studies in situations where failure of a system comprising many similar components occurs when the weakest component fails”. Why could that be? Expressed more theoretically, the lifetimes of various components might reasonably be assumed to behave like i.i.d. random variables in many contexts. Then the failure time of the system is given by the minimum of the constituent random variables.

So this begs the question: what does the distribution of minimum of a collection of i.i.d. random variables look like? First, we need to think why there should be an answer at all. I mean, it would not be unreasonable to assume that this would depend rather strongly on the underlying distribution. But of course, we might say the same thing about sums of i.i.d. random variables, but there is the Central Limit Theorem. Phrased in a way that is deliberately vague, this says that subject to some fairly mild conditions on the underlying distribution (finite variance in this case), the sum of n i.i.d. RVs look like a normal distribution for large n. Here we know what ‘looks like’ means, since we have a notion of a family of normal distributions. Formally, though, we might say that ‘looks like’ means that the image of the distribution under some linear transformation, where the coefficients are possibly functions of n, converges to the distribution N(0,1) as n grows.

The technical term for this is to say that the underlying RV we are considering, which in this case would be X_1+\ldots +X_n) is in the domain of attraction of N(0,1). Note that other distributions in the family of normals are also in the domain of attraction of N(0,1), and vice versa, so this forms an equivalence relation on the space of distributions, though this is not hugely helpful since most interesting statements involve some sort of limit.

Anyway, with that perspective, it is perhaps more reasonable to imagine that the minimum of a collection of i.i.d. RVs might have some limit distribution. Because we typically feel more comfortable thinking about right-tails rather than left-tails of probability distributions, this problem is more often considered for the maximum of i.i.d. RVs. The Fisher-Tippett-Gnedenko theorem, proved in various forms in the first half of the 20th century, asserts that again under mild regularity assumptions, the maximum of such a collection does lie in the domain of attraction of one of a small set of distributions. The Weibull distribution as defined above is one of these. (Note that if we are considering domains of attraction, then scaling x by a constant is of no consequence, so we can drop the parameterisation by \lambda.)

This is considered the first main theorem of Extreme Value Theory, which addresses precisely this sort of problem. It is not hard to consider why this area is of interest. To decide how much liquidity they require, an insurance company needs to know the likely size of the maximum claim during the policy. Similarly, the designer of a sea-wall doesn’t care about the average wave-height – what matters is how strong the once-in-a-century storm which threatens the town might be. A good answer might also explain how to resolve the apparent contradiction that most human characteristics are distributed roughly normally across the population. Normal distributions are unbounded, yet physiological constraints enable us to state with certainty that there will never be twelve foot tall men (or women). In some sense, EVT is a cousin of Large Deviation theory, the difference being that unlikely events in a large family of i.i.d. RVs are considered on a local scale rather than globally. Note that large deviations for Cramer’s theorem in the case where the underlying distribution has a heavy tail are driven by a single very deviant value, rather than by lots of slightly deviant data, so in this case the theories are comparable, though generally analysed from different perspectives.

In fact, we have to consider the reversed Weibull distribution for a maximum, which is supported on (-\infty,0]. This is one of three possibly distribution families for the limit of a maximum. The other two are the Gumbel distribution

F(x)=e^{-e^{-x}},

and the Frechet distribution

F(x)=\exp(-x^{-\alpha}),\quad x>0.

Note that \alpha is a positive parameter in both the Frechet and Gumbel distributions. These three distributions can be expressed as a single one parameter family, the Generalised Extreme Value distribution.

The differences between them lie in the tail behaviour. The reversed Weibull distribution has an actual upper bound, the Gumbel an exponential, fast-decaying tail, and the Frechet a polynomial ‘fat’ tail. It is not completely obvious that these properties are inherited from the original distribution. After all, to get from the original distribution to extreme value distribution, we are taking the maximum, then rescaling and translating in a potentially quite complicated way. However, it is perhaps reasonable to see that the property of the underlying distribution having an upper bound is preserved through this process. Obviously, the bound itself is not preserved – after all, we are free to apply arbitrary linear transformations to the distributions!

In any case, it does turn out to be the case that the U[0,1] distribution has maximum converging to a reversed Weibull; the exponential tails of the Exp(1) and N(0,1) distributions lead to a Gumbel limit; and the fat-tailed Pareto distribution gives the Frechet limit. The calculations are reasonably straightforward, especially once the correct rescaling is known. See this article from Duke for an excellent overview and the details for these examples I have just cited. These notes discuss further properties of these limiting distributions, including the unsurprising fact that their form is preserved under taking the maximum of i.i.d. copies. This is analogous to the fact that the family of normal distributions is preserved under taking arbitrary finite sums.

From a statistical point of view, devising a good statistical test for what class of extreme value distribution a particular set of data obeys is of great interest. Why? Well mainly because of the applications, some of which were suggested above. But also because of the general statistical principle that it is unwise to extrapolate beyond the range of the available data. But that is precisely what we need to do if we are considering extreme values. After all, the designer of that sea-wall can’t necessarily rely on the largest storm in the future being roughly the same as the biggest storm in the past. So because the EVT theorem gives a clear description of the distribution, to find the limiting properties, which is where the truly large extremes might occur, it suffices to find a good test for the form of the limit distribution – that is, which of the three possibilities is relevant, and what the parameter should be. This seems to be fairly hard in general. I didn’t understand much of it, but this paper provided an interesting review.

Anyway, that was something interesting I didn’t know about (for the record, I also now know how to construct a sensible Q-Q plot for the Weibull distribution!), though I am assured that EVT was a core element of the mainstream undergraduate mathematics syllabus forty years ago.

Large Deviations 4 – Sanov’s Theorem

Although we could have defined things for a more general topological space, most of our thoughts about Cramer’s theorem, and the Gartner-Ellis theorem which generalises it, are based on means of real-valued random variables. For Cramer’s theorem, we genuinely are interested only in means of i.i.d. random variables. In Gartner-Ellis, one might say that we are able to relax the condition on independence and perhaps identical distribution too, in a controlled way. But this is somewhat underselling the theorem: using G-E, we can deal with a much broader category of measures than just means of collections of variables. The key is that convergence of the log moment generating function is exactly enough to give a LDP with some rate, and we have a general method for finding the rate function.

So, Gartner-Ellis provides a fairly substantial generalisation to Cramer’s theorem, but is still similar in flavour. But what about if we look for additional properties of a collection of i.i.d. random variables (X_n). After all, the mean is not the only interesting property. One thing we could look at is the actual values taken by the X_ns. If the underlying distribution is continuous, this is not going to give much more information than what we started with. With probability, \{X_1,\ldots,X_n\} is a set of size n, with distribution given by the product of the underlying measure. However, if the random variables take values in a discrete set, or better still a finite set, then (X_1,\ldots,X_n) gives a so-called empirical distribution.

As n grows towards infinity, we expect this empirical distribution to approximate the real underlying distribution fairly well. This isn’t necessarily quite as easy as it sounds. By the strong law of large numbers applied to indicator functions 1(X_i\leq t), the empirical cdf at t converges almost surely to the true cdf at t. To guarantee that this convergence is uniform in t is tricky in general (for reference, see the Glivenko-Cantelli theorem), but is clear for random variables defined on finite sets, and it seems reasonable that an extension to discrete sets should be possible.

So such empirical distributions might well admit an LDP. Note that in the case of Bernoulli random variables, the empirical distribution is in fact exactly equivalent to the empirical mean, so Cramer’s theorem applies. But, in fact we have a general LDP for empirical distributions. I claim that the main point of interest here is the nature of the rate function – I will discuss why the existence of an LDP is not too surprising at the end.

The rate function is going to be interesting whatever form it ends up taking. After all, it is effectively going to some sort of metric on measures, as it records how far a possible empirical measure is from the true distribution. Apart from total variation distance, we don’t currently have many standard examples for metrics on a space of measures. Anyway, the rate function is the main content of Sanov’s theorem. This has various forms, depending on how fiddly you are prepared for the proof to be.

Define L_n:=\sum_{i=1}^n \delta_{X_i}\in\mathcal{M}_1(E) to be the empirical measure generated by X_1,\ldots,X_n. Then L_n satisfies an LDP on \mathcal{M}_1(E) with rate n and rate function given by H(\cdot,\mu), where \mu is the underlying distribution.

The function H is the relative entropy, defined by:

H(\nu|\mu):=\int_E \log\frac{\nu(x)}{\mu(x)}d\nu(v),

whenever \nu<<\mu, and \infty otherwise. We can see why this absolute continuity condition is required from the statement of the LDP. If the underlying distribution \mu has measure zero on some set A, then the observed values will not be in A with probability 1, and so the empirical measure will be zero on A also.

Note that an alternative form is:

H(\nu|\mu)=\int_E \frac{\nu(x)}{\mu(x)}\log\frac{\nu(x)}{\mu(x)}d\mu(v)=\mathbb{E}_\nu\frac{\nu(x)}{\mu(x)}\log\frac{\nu(x)}{\mu(x)}.

Perhaps it is more clear why this expectation is something we would want to minimise.

In particular, if we want to know the most likely asymptotic empirical distribution inducing a large deviation empirical mean (as in Cramer), then we find the distribution with suitable mean, and smallest entropy relative to the true underlying distribution.

A remark on the proof. If the underlying set of values is finite, then a proof of this result is essentially combinatorial. The empirical distribution is some multinomial distribution, and we can obtain exact forms for everything and then proceed with asymptotic approximations.

I said earlier that I would comment on why the LDP is not too surprising even in general, once we know Gartner-Ellis. Instead of letting X_i take values in whatever space we were considering previously, say the reals, consider instead the point mass function \delta_{X_i} which is effectively exactly the same random variable, only now defined on the space of probability measures. The empirical measure is then exactly:

\frac{1}{n}\sum_{i=1}^n \delta_{X_i}.

If the support K of the (X_i)s is finite, then in fact this space of measures is a convex subspace of \mathbb{R}^K, and so the multi-dimensional version of Cramer’s theorem applies. In general, we can work in the possibly infinite-dimensional space [0,1]^K, and our relevant subset is compact, as a closed subset of a compact space (by Tychonoff). So the LDP in this case follows from our previous work.