# Skorohod embedding

Background

Suppose we are given a standard Brownian motion $(B_t)$, and a stopping time T. Then, so long as T satisfies one of the regularity conditions under which the Optional Stopping Theorem applies, we know that $\mathbb{E}[B_T]=0$. (See here for a less formal introduction to OST.) Furthermore, since $B_t^2-t$ is a martingale, $\mathbb{E}[B_T^2]=\mathbb{E}[T]$, so if the latter is finite, so is the former.

Now, using the strong Markov property of Brownian motion, we can come up with a sequence of stopping times $0=T_0, T_1, T_2,\ldots$ such that the increments $T_k-T_{k-1}$ are IID with the same distribution as T. Then $0,B_{T_1},B_{T_2},\ldots$ is a centered random walk. By taking T to be the hitting time of $\{-1,+1\}$, it is easy to see that we can embed simple random walk in a Brownian motion using this approach.

Embedding simple random walk in Brownian motion.

The Skorohod embedding question asks: can all centered random walks be constructed in this fashion, by stopping Brownian motion at a sequence of stopping time? With the strong Markov property, it immediately reduces the question of whether all centered finite-variance distributions X can be expressed as $B_T$ for some integrable stopping time T.

The answer to this question is yes, and much of what follows is drawn from, or at least prompted by Obloj’s survey paper which details the problem and rich history of the many approaches to its solution over the past seventy years.

Applications and related things

The relationship between random walks and Brownian motion is a rich one. Donsker’s invariance principle asserts that Brownian motion appears as the scaling limit of a random walk. Indeed, one can construct Brownian motion itself as the limit of a sequence of consistent random walks with normal increments on an increasingly dense set of times. Furthermore, random walks are martingales, and we know that continuous, local martingales can be expressed as a (stochastically) time-changed Brownian motion, from the Dubins-Schwarz theorem.

The Skorohod embedding theorem can be used to prove results about random walks with general distribution by proving the corresponding result for Brownian motion, and checking that the construction of the sequence of stopping times has the right properties to allow the result to be carried back to the original setting. It obviously also gives a coupling between a individual random walk and a Brownian motion which may be useful in some contexts, as well as a coupling between any pair of random walks. This is useful in proving results for random walks which are much easier for special cases of the distribution. For example, when the increments are Gaussian, or when there are combinatorial approaches to a problem about simple random walk. At the moment no aspect of this blog schedule is guaranteed, but I plan to talk about the law of the iterated logarithm shortly, whose proof is approachable in both of these settings, as well as for Brownian motion, and Skorohod embedding provides the route to the general proof.

At the end, we will briefly compare some other ways to couple a random walk and a Brownian motion.

One thing we could do is sample a copy of X independently from the Brownian motion, then declare $T= \tau_{X}:= \inf\{t\ge 0: B_t=X\}$, the hitting time of (random value) X. But recall that unfortunately $\tau_x$ has infinite expectation for all non-zero x, so this doesn’t fit the conditions required to use OST.

Skorohod’s original method is described in Section 3.1 of Obloj’s notes linked above. The method is roughly to pair up positive values taken by X appropriately with negative values taken by X in a clever way. If we have a positive value b and a negative value a, then $\tau_{a,b}$, the first hitting time of $\mathbb{R}\backslash (a,b)$ is integrable. Then we choose one of these positive-negative pairs according to the projection of the distribution of X onto the pairings, and let T be the hitting time of this pair of values. The probability of hitting b conditional on hitting {a,b} is easy to compute (it’s $\frac{-a}{b-a}$) so we need to have chosen our pairs so that the ‘probability’ of hitting b (ie the density) comes out right. In particular, this method has to start from continuous distributions X, and treat atoms in the distribution of X separately.

The case where the distribution X is symmetric (that is $X\stackrel{d}=-X$) is particularly clear, as then the pairs should be $(-x,x)$.

However, it feels like there is enough randomness in Brownian motion already, and subsequent authors showed that indeed it wasn’t necessary to introduce extra randomness to provide a solution.

One might ask whether it’s possible to generate the distribution on the set of pairs (as above) out of the Brownian motion itself, but independently from all the hitting times. It feels like it might be possible to make the distribution on the pairs measurable with respect to

$\mathcal{F}_{0+} = \bigcap\limits_{t>0} \mathcal{F}_t,$

the sigma-algebra of events determined by limiting behaviour as $t\rightarrow 0$ (which is independent of hitting times). But of course, unfortunately $\mathcal{F}_{0+}$ has a zero-one law, so it’s not possible to embed non-trivial distributions there.

Dubins solution

The exemplar for solutions without extra randomness is due to Dubins, shortly after Skorohod’s original argument. The idea is to express the distribution X as the almost sure limit of a martingale. We first use the hitting time of a pair of points to ‘decide’ whether we will end up positive or negative, and then given this information look at the hitting time (after this first time) of two subsequent points to ‘decide’ which of four regions of the real interval we end up in.

I’m going to use different notation to Obloj, corresponding more closely with how I ended up thinking about this method. We let

$a_+:= \mathbb{E}[X \,|\, X>0], \quad a_- := \mathbb{E}[X\,|\, X<0],$ (*)

and take $T_1 = \tau_{\{a_-,a_+\}}$. We need to check that

$\mathbb{P}\left( B_{T_1}=a_+\right) = \mathbb{P}\left(X>0\right),$

for this to have a chance of working. But we know that

$\mathbb{P}\left( B_{T_1}=a_+\right) = \frac{a_+}{a_+-a_-},$

and we can also attack the other side using (*) and the fact that $\mathbb{E}[X]=0$, using the law of total expectation:

$0=\mathbb{E}[X]=\mathbb{E}[X\,|\, X>0] \mathbb{P}(X>0) + \mathbb{E}[X\,|\,X<0]\mathbb{P}(X<0) = a_+ \mathbb{P}(X>0) + a_- \left(1-\mathbb{P}(X>0) \right),$

$\Rightarrow\quad \mathbb{P}(X>0)=\frac{a_+}{a_+-a_-}.$

Now we define

$a_{++}=\mathbb{E}[X \,|\, X>a_+],\quad a_{+-}=\mathbb{E}[X\,|\, 0

and similarly $a_{-+},a_{--}$. So then, conditional on $B_{T_1}=a_+$, we take

$T_2:= \inf_{t\ge T_1}\left\{ B_t\not\in (a_{+-},a_{++}) \right\},$

and similarly conditional on $B_{T_1}=a_-$. By an identical argument to the one we have just deployed, we have $\mathbb{E}\left[B_{T_2} \,|\,\mathcal{F}_{T_1} \right] = B_{T_1}$ almost surely. So, although the $a_{+-+}$ notation now starts to get very unwieldy, it’s clear we can keep going in this way to get a sequence of stopping times $0=T_0,T_1,T_2,\ldots$ where $B_{T_n}$ determines which of the $2^n$ regions of the real line any limit $\lim_{m\rightarrow\infty} B_{T_m}$ should lie in.

A bit of work is required to check that the almost sure limit $T_n\rightarrow T$ is almost surely finite, but once we have this, it is clear that $B_{T_n}\rightarrow B_T$ almost surely, and $B_T$ has the distribution required.

We want to know how close we can make this coupling between a centered random walk with variance 1, and a standard Brownian motion. Here, ‘close’ means uniformly close in probability. For large times, the typical difference between one of the stopping times $0,T_1,T_2,\ldots$ in the Skorohod embedding and its expectation (recall $\mathbb{E}[T_k]=k$) is $\sqrt{n}$. So, constructing the random walk $S_0,S_1,S_2,\ldots$ from the Brownian motion via Skorohod embedding leads to

$\left |S_k - B_k \right| = \omega(n^{1/4}),$

for most values of $k\le n$. Strassen (1966) shows that the true scale of the maximum

$\max_{k\le n} \left| S_k - B_k \right|$

is slightly larger than this, with some extra powers of $\log n$ and $\log\log n$ as one would expect.

The Komlos-Major-Tusnady coupling is a way to do a lot better than this, in the setting where the distribution of the increments has a finite MGF near 0. Then, there exists a coupling of the random walk and the Brownian motion such that

$\max_{k\le n}\left|S_k- B_k\right| = O(\log n).$

That is, there exists C such that

$\left[\max_{k\le n} \left |S_k-B_k\right| - C\log n\right] \vee 0$

is a tight family of distributions, indeed with uniform exponential tail. To avoid digressing infinitely far from my original plan to discuss the proof of the law of iterated logarithm for general distributions, I’ll stop here. I found it hard to find much coverage of the KMT result apart from the challenging original paper, and many versions expressed in the language of empirical processes, which are similar to random walks in many ways relevant to convergence and this coupling, but not for Skorohod embedding. So, here is a link to some slides from a talk by Chatterjee which I found helpful in getting a sense of the history, and some of the modern approaches to this type of normal approximation problem.

# DGFF 2 – Boundary conditions and Gibbs-Markov property

In the previous post, we defined the Discrete Gaussian Free Field, and offered some motivation via the discrete random walk bridge. In particular, when the increments of the random walk are chosen to be Gaussian, many natural calculations are straightforward, since Gaussian processes are well-behaved under conditioning and under linear transformations.

Non-zero boundary conditions

In the definition of the DFGG given last time, we demanded that $h\equiv 0$ on $\partial D$. But the model is perfectly well-defined under more general boundary conditions.

It’s helpful to recall again the situation with random walk and Brownian bridge. If we want a Brownian motion which passes through (0,0) and (1,s), we could repeat one construction for Brownian bridge, by taking a standard Brownian motion and conditioning (modulo probability zero technicalities) on passing through level s at time 1. But alternatively, we could set

$B^{\mathrm{drift-br}}(t) = B(t)+ t(s-B(1)),\quad t\in[0,1],$

or equivalently

$B^{\mathrm{drift-br}}(t)=B^{\mathrm{br}}(t)+ st, \quad t\in[0,1].$

That is, a Brownian bridge with drift can be obtain from a centered Brownian bridge by a linear transformation, and so certainly remains a Gaussian process. And exactly the same holds for a discrete Gaussian bridge: if we want non-zero values at the endpoints, we can obtain this distribution by taking the standard centred bridge and applying a linear transformation.

We can see how this works directly at the level of density functions. If we take $0=Z_0,Z_1,\ldots,Z_{N-1},Z_N=0$ a centred Gaussian bridge, then the density of $Z=\mathbf{z}\in \mathbb{R}^{N+1}$ is proportional to

$\mathbf{1}\{z_0=z_N=0\}\exp\left( -\frac12 \sum_{i=1}^N (z_i-z_{i-1})^2 \right).$ (3)

So rewriting $z_i= y_i- ki$ (where we might want $k=s/N$ to fit the previous example), the sum within the exponent rearranges as

$-\frac12 \sum_{i=1}^N (y_i-y_{i-1} - k)^2 = -\frac12 \sum_{i=1}^N (y_i-y_{i-1})^2 - 2k(y_N-y_0)+ Nk^2.$

So when the values at the endpoints $z_0,z_n,y_0,y_N$ are fixed, this middle term is a constant, as is the final term, and thus the density of the linearly transformed bridge has exactly the same form as the original one.

In two or more dimensions, the analogue of adding a linear function is to add a harmonic function. First, some notation. Let $\varphi$ be any function on $\partial D$. Then there is a unique harmonic extension of $\varphi$, for which $\nabla \varphi=0$ everywhere on D, the interior of the domain. Recall that $\nabla$ is the discrete graph Laplacian defined up to a constant by

$(\nabla \varphi) _x = \sum\limits_{x\sim y} \varphi_x - \varphi_y.$

If we want $h^D$ instead to have boundary values $\varphi$, it’s enough to replace $h^D$ with $h^D+\varphi$. Then, in the density for the DGFF ( (1) in the previous post), the term in the exponential becomes (ignoring the $\frac{1}{4d}$ )

$-\sum\limits_{x\sim y} \left[ (h^D_x-h^D_y)^2 + (\varphi_x-\varphi_y)^2 +2(h^D_x - h^D_y)(\varphi_x-\varphi_y)\right].$

For each $x\in D$, on taking this sum over its neighbours $y\in \bar D$, the final term vanishes (since $\varphi$ is harmonic), while the second term is just a constant. So the density of the transformed field, which we’ll call $h^{D,\varphi}$ is proportional to (after removing the constant arising from the second term above)

$\mathbf{1}\left\{h^{D,\varphi}_x = \varphi_x,\, x\in\partial D\right\} \exp\left( -\frac{1}{4d} \sum\limits_{x\sim y} \left( h^{D,\varphi}_x - h^{D,\varphi}_y \right)^2 \right).$

So $h^{D,\varphi}:= h^D + \varphi$ satisfies the conditions for the DGFF on D with non-zero boundary conditions $\varphi$.

Harmonic functions and RW – a quick review

Like the covariances in DGFF, harmonic functions on D are related to simple random walk on D stopped on $\partial D$. (I’m not claiming a direct connection right now.) We can define the harmonic extension $\varphi$ to an interior point x by taking $\mathbb{P}_x$ to be the law of SRW $x=Z_0,Z_1,Z_2,\ldots$ started from x, and then setting

$\varphi(x):= \mathbb{E}\left[ \varphi_{\tau_{\partial d}} \right],$

where $\tau_{\partial D}$ is the first time that the random walk hits the boundary.

Inverse temperature – a quick remark

In the original definition of the density of the DGFF, there is the option to add a constant $\beta>0$ within the exponential term so the density is proportional to

$\exp\left(-\beta \sum\limits_{x\sim y} (h_x-h_y)^2 \right).$

With zero boundary conditions, the effect of this is straightforward, as varying $\beta$ just rescales the values taken by the field. But with non-zero boundary conditions, the effect is instead to vary the magnitude of the fluctuations of the values of the field around the (unique) harmonic function on the domain with those BCs. In particular, when $\beta\rightarrow \infty$, the field is ‘reluctant to be far from harmonic’, and so $h^D \Rightarrow \varphi$.

This parameter $\beta$ is called inverse temperature. So low temperature corresponds to high $\beta$, and high stability, which fits some physical intuition.

A Markov property

For a discrete (Gaussian) random walk, the Markov property says that conditional on a given value at a given time, the trajectory of the process before this time is independent of the trajectory afterwards. The discrete Gaussian bridge is similar. Suppose we have as before $0=Z_0,Z_1,\ldots, Z_N=0$ a centred Gaussian bridge, and condition that $Z_k=y$, for $k\in\{1,\ldots,N-1\}$, and $y\in\mathbb{R}$. With this conditioning, the density (3) splits as a product

$\mathbf{1}\{z_0=z_N=0, z_k=y\}\exp\left(-\frac12 \sum\limits_{i=1}^N (z_i-z_{i-1})^2 \right) =$

$\mathbf{1}\{z_0=0,z_k=y\} \exp\left(-\frac12 \sum\limits_{i=1}^k (z_i-z_{i-1})^2 \right) \cdot \mathbf{1}\{z_k=y,z_N=0\} \exp\left(-\frac12 \sum\limits_{i=k+1}^N (z_i-z_{i-1})^2 \right).$

Therefore, with this conditioning, the discrete Gaussian bridge splits into a pair of independent discrete Gaussian bridges with drift. (The same would hold if the original process had drift too.)

The situation for the DGFF is similar, though rather than focusing on the condition, it makes sense to start by focusing on the sub-domain of interest. Let $A\subset D$, and take $B=\bar D\backslash A$. So in particular $\partial A\subset B$.

Then we have that conditional on $h^D\big|_{\partial A}$, the restricted fields $h^D\big|_{B\backslash \partial A}$ and $h^D\big|_A$ are independent. Furthermore, $h^D\big|_A$ has the distribution of the DGFF on A, with boundary condition given by $h^D\big|_{\partial A}$. As in the discrete bridge, this follows just by splitting the density. Every gradient term corresponds to an edge in the underlying graph that lies either entirely inside $\bar A$ or entirely inside B. This holds for a general class of Gibbs models where the Hamiltonian depends only on the sum of some function of the heights (taken to be constant in this ‘free’ model) and the sum of some function of their nearest-neighbour gradients.

One additional and useful interpretation is that if we only care about the field on the restricted region A, the dependence of $h^D\big|_A$ on $h^D\big|_{D\backslash A}$ comes only through $h^D\big|_{\partial A}$. But more than that, it comes only through the (random) harmonic function which extends the (random) values taken on the boundary of A to the whole of A. So, if $h^A$ is an independent DGFF on A with zero boundary conditions, we can construct the DGFF $h^D$ from its value on $D\backslash A$ via

$h^D_x \stackrel{d}= h^A_x + \varphi^{h^D\big|_{\partial A}},$

where $\varphi^{h^D\big|_{\partial A}}$ is the unique harmonic extension of the (random) values taken by $h^D$ on $\partial A$ to $\bar A$.

This Markov property is crucial to much of the analysis to come. There are several choices of the restricted domain which come up repeatedly. In the next post we’ll look at how much one can deduce by taking A to be the even vertices in D (recalling that every integer lattice $\mathbb{Z}^d$ is bipartite), and then taking A to be a finer sublattice within D. We’ll use this to get some good bounds on the probability that the DGFF is positive on the whole of D. Perhaps later we’ll look at a ring decomposition of $\mathbb{Z}^d$ consisting of annuli spreading out from a fixed origin. Then the distribution of the field at this origin can be considered, via the final idea discussed above, as the limit of an infinite sequence of random harmonic functions given by the values taken by the field at increasingly large radius from the origin. Defining the DGFF on the whole lattice depends on the existence or otherwise of this local limit.

# Ornstein-Uhlenbeck Process

A large part of my summer has been spent proving some technical results pertaining to the convergence of some functionals of a critical Frozen Percolation process. This has been worthwhile, but hasn’t involved a large amount of reading around anything in particular, which has probably contributed to the lack of posts in recent months. Perhaps a mixture of that and general laziness?

Anyway, it turns out that the limit of the discrete processes under consideration is the Ornstein-Uhlenbeck process. The sense in which this limit holds (or at least, for now, is conjectured to hold) is something for another article. However, I thought it would be worth writing a bit about this particular process and why it is interesting.

The O-U process is described by the SDE

$dX_t=-\beta (X_t-\mu)dt+\sigma dW_t,$

where W is a standard Brownian motion. We think of $\mu$ as the ‘mean’. The extent to which this behaves as a mean will be discussed shortly. The process is then mean-reverting, in the sense that the drift is directed against deviations of the process away from this mean. The parameter $\beta$ measures the extent of this mean reversion, while as usual $\sigma$ controls the magnitude of the Brownian noise.

The motivation for considering mean-reverting processes is considerable. One measure of this is how many equations with articles on Wikipedia turn out to be precisely this Ornstein-Uhlenbeck process with different context or notation. In most cases, the motivation arises because Brownian motion is for some reason unsuitable to take as a canonical random process. We will see why the O-U process is somehow the next most canonical choice for a random process.

In physics, it is sometimes unsatisfactory to model the trajectory of a particle with Brownian motion (even though this motivated the name…) as the velocities are undefined (see this post from ages ago), or infinite, depending on your definition of velocity. Using the Ornstein-Uhlenbeck process to model the velocity of a particle is often a satisfactory alternative. It is not unreasonable that there should be a mean velocity, presumably zero. The mean reversion models a frictional force from the underlying medium, while the Brownian noise describes random collisions with similar particles.

In financial applications, the Ornstein-Uhlenbeck model has been applied, apparently under the title of the Vasicek model since the 70s to describe quantities such as interest rates where there is some underlying reason to ban indefinite growth, and require mean reversion. Another setting might be a commodity which, because of external driving factors, has over the relevant time-scale well-defined mean value, around which mean-reverting fluctuations on the observed time-scale can be described. As with other financial models, it is undesirable for a process to take negative values. This can be fixed by taking a positive mean, then setting the volatility to be state dependent, decaying to zero as the state tends to zero, so for small values, the positive drift dominates. I don’t fully understand why patching this aspect is significantly more important than patching any other non-realistic properties of the model, but the resulting SDE is, at least in one particular case where the volatility is $\sqrt{X_t}$, called the Cox-Ingersoll-Ross model.

Anyway, a mathematical reason to pay particular attention to this Ornstein-Uhlenbeck process is the following. It is the unique family of continuous Markov processes to have a stationary Gaussian distribution. It is the mean-reverting property that is key. There is no chance of Brownian motion having any stationary distribution, let alone a Gaussian one. If this isn’t clear, you can convince yourself by thinking of the stationary distribution of SRW on $\mathbb{Z}$. Since the process is space-homogeneous, the only stationary measure is the uniform measure.

I want to focus on one particular property of the O-U process, through which some other aspects will be illuminated. If we take $\sigma=\beta$ and let $\beta\rightarrow\infty$, then the stationary processes converge to white noise.

First though, we should note this is perhaps the easiest SDE to solve explicitly. We consider $X_t e^{\theta t}$, and applying Ito’s lemma rapidly gives

$X_t=\mu + (X_0-\mu)e^{-\beta t}+\sigma\int_0^t e^{-\beta(t-s)}dW_s.$

W is Gaussian so the distribution of $X_t$ conditional on $X_0=x_0$ is also Gaussian, and since W is centred we can read off the expectation. Applying the Ito isometry then gives the variance. In conclusion:

$X_t\stackrel{d}{=}\mathcal{N}(\mu+(x_0-\mu)e^{-\beta t}, \frac{\sigma^2}{\beta}(1-e^{-2\beta t})).$

In particular, note that the variation has no dependence on $x_0$. So as t grows to infinity, this converges to $\mathcal{N}(\mu, \frac{\sigma^2}{\beta})$. This is, unsurprisingly, the stationary distribution of the process.

To address the white noise convergence, we need to consider $\text{Cov}(X_0,X_t)$ in stationarity. Let’s assume WLOG that $\mu=0$ so most of the expectations will vanish. We obtain

$\text{Cov}(X_0,X_t)=\mathbb{E}[X_0X_t]=\mathbb{E}_{x_0}\left[\mathbb{E}[X_t| X_0=x_0]\right]=\mathbb{E}[X_0^2 e^{-\beta t}]= \frac{sigma^2}{2\beta}2^{-\beta t}.$

If we want, the Chapman-Kolmogorov equations work particularly nicely here, and we are able to derive a PDE for the evolution of the density function, though obviously this is very related to the result above. This PDE is known as the Fokker-Planck equation.

So, in particular, when $\sigma=\beta\rightarrow \infty$, this covariance tends to 0. I’m not purporting that this constitutes a proof that the Ornstein-Uhlenbeck processes converge as processes to white noise. It’s not obvious how to define process convergence, not least because there’s flexibility about how to view white noise as a process. One doesn’t really want to define the value of white noise at a particular time, but you can consider the covariance of integrals of white noise over disjoint intervals as a limit, in similar way to convergence of finite dimensional distributions.

The fact that taking $\beta=0$ gives Brownian motion, and this case gives white noise, intermediate versions of the Ornstein-Uhlenbeck process are sometimes referred to as coloured noise.

Finally, the Ornstein-Uhlenbeck process emerges as the scaling limit of mean-reverting discrete Markov chains, analogous to Brownian motion as the scaling limit of simple random walk. One particularly nice example is the Ehrenfest Urn model. We have two urns, and 2N balls. In each time step one of the 2N balls is chosen uniformly at random, and it is moved to the other urn. So a ball is more likely to be removed from an urn with more than N balls. We can view this as a model for molecules in, say a room, with a slightly porous division between them, eg a small hole. More complicated interface models in higher dimensions lead to fascinating PDEs, such as the famous KPZ equation, which are the subject of much ongoing interest in this area.

This result can be an application of the theory of convergence of Markov chains to SDEs pioneered by Stroock and Varadhan, about which more may follow very soon. In any case, it turns out that the fluctuations in the Ehrenfest Urn model are on the scale of $\sqrt{n}$, unsurprisingly, and are given by a centred Ornstein-Uhlenbeck process.

Investigating this has reminded me how much I’ve forgotten, or perhaps how little I ever knew, about the technicalities of stochastic processes are their convergence results, so next up will probably be a summary of all the useful definitions and properties for this sort of analysis.

# Reflected Brownian Motion

A standard Brownian motion is space-homogeneous, meaning that the behaviour of $B_{T+t}-B_T$ does not depend on the value of $B_T$. By Donsker’s Theorem, such a Brownian motion is also the limit in a process space of any homogeneous random walk with zero-drift and constant variance, after suitable rescaling.

In many applications, however, we are interested in real-valued continuous-time Markov processes that are defined not on the whole of the real line, but on the half-line $\mathbb{R}_{\ge 0}$. So as BM is the fundamental real-valued continuous-time Markov process, we should ask how we might adjust it so that it stays non-negative. In particular, we want to clarify uniqueness, or at least be sure we have found all the sensible ways to make this adjustment, and also to consider how Donsker’s Theorem might work in this setting.

We should consider what properties we want this non-negative BM to have. Obviously, it should be non-negative, but it is also reasonable to demand that it looks exactly like BM everywhere except near 0. But since BM has a scale-invariance property, it is essentially meaningful to say ‘near 0’, so we instead demand that it looks exactly like BM everywhere except at 0. Apart from this, the only properties we want are that it is Markov and has continuous sample paths.

A starting point is so-called reflected Brownian motion, defined by $X_t:=|B_t|$. This is very natural and very convenient for analysis, but there are some problems. Firstly, this has the property that it looks like Brownian motion everywhere except 0 only because BM is space-homogeneous but also symmetric, in the sense that $B_t\stackrel{d}{=}-B_t$. This will be untrue for essentially any other process, so as a general method for how to keep stochastic processes positive, this will be useless. My second objection is a bit more subtle. If we consider this as an SDE, we get

$dX_t=\text{sign}(B_t)dB_t.$

This is a perfectly reasonable SDE but it is undesirable, because we have a function of B as coefficient on the RHS. Ideally, increments of X would be a function of X, and the increments of B, rather than the values of B. That is, we would expect $X_{t+\delta t}-X_t$ to depend on $X_t$ and on $(B_{t+s}-B_t, 0\le s\le \delta t)$, but not on $B_t$ itself, as that means we have to keep track of extra information while constructing X.

So we need an alternative method. One idea might be to add some non-negative process to the BM so that the sum stays non-negative. If this process is deterministic and finite, there there is some positive probability that the sum will eventually be negative, so this won’t do. We are looking therefore so a process which depends on the BM. Obviously we could take $\max(-B_t,0)$, but this sum would then spend macroscopic intervals of time at 0, and these intervals would have the Raleigh distribution (for Brownian excursions) rather than the exponential distribution, hence the process given by the sum would not be memoryless and Markov.

The natural alternative is to look for an increasing process $A_t$, and then it makes sense to talk about the minimal increasing process that has the desired property. A moment’s thought suggests that $A_t=-min_{s\le t}B_t$ satisfies this. So we have the decomposition

$B_t=-A_t+S_t,$

where $S_t$ is the height of B above its running minimum. So S is an ideal alternative definition of reflecting BM. In particular, when B is away from its minimum, $dB_t=dS_t$, so this has the property that it evolves exactly as the driving Brownian motion.

What we have done is to decompose a general continuous process into the sum of a decreasing continuous process and a non-negative process. This is known as the Skorohod problem, and was the subject of much interest, even in the deterministic case. Note that process A has the property that it is locally constant almost everywhere, and is continuous, yet non-constant. Unsurprisingly, since A only changes when the underlying BM is 0, A is continuous with respect to the local time process at 0. In fact, A is the local time process of the underlying Brownian motion, by comparison with the construction by direct reflection.

One alternative approach is to look instead at the generator of the process. Recall that the generator of a process is an operator on some space of functions, with $\mathcal{L}f$ giving the infinitissimal drift of $f(X_t)$. In the case of Brownian motion, the generator $(\mathcal{L}f)(x)=\frac12 f''(x)$ for bounded smooth functions f. This is equivalent to saying that

$f(X_t)-f(X_0)-\int_0^t \frac12 f''(X_s)ds$ (*)

is a martingale. This must hold also for reflected Brownian motion, whenever x is greater than 0. Alternatively, if the function f is zero in a small neighbourhood of 0, it should have the same generator with respect to reflected BM. Indeed, for a general smooth bounded function f, we can still consider the expression (*) with respect to reflected BM. We know this expression behaves as a martingale except when X is zero. If f'(0)>0, and T is some hitting time of 0, then $f(X_{T+\delta T})-f(X_T)\ge 0$, hence the expression (*) is a submartingale. So if we restrict attention to functions with f'(0)=0, the generator remains the same. Indeed, by patching together all such intervals, it can be argued that even if f'(0) is not zero,

$f(X_t)-f(X_0)-\int_0^t \frac12 f''(X_s)ds - f'(0)A_t$

is a martingale, where A is the local time process at zero.

I was aware when I started reading about this that there was another family of processes called ‘Sticky Brownian Motion’ that shared properties with Reflected BM, in that it behaves like standard BM away from zero, but is also constrained to the non-negative reals. I think this will get too long if I also talk about that here, so that can be postponed, and for now we consider reflected BM as a limit of reflected (or other) random walks, bearing in mind that there is at least one other candidate to be the limit.

Unsurprisingly, if we have a family of random walks constrained to the non-negative reals, that are zero-drift unit-variance away from 0, then if they converge as processes, the limit is Brownian away from zero, and non-negative. Note that “away from 0” means after rescaling. So the key aspect is behaviour near zero.

What is the drift of reflected BM at 0? We might suspect it is infinite because of the form of the generator, but we can calculate it directly. Given $X_0=0$, we have:

$\frac{\mathbb{E}X_t}{t}=\frac{\mathbb{E}|B_t|}{t}=\frac{\sqrt{t}\mathbb{E}|B_1|}{t},$

so letting $t\rightarrow 0$, we see indeed that the drift is infinite at 0.

For convergence of discrete processes, we really need the generators to converge. Typically we index the discrete-time processes by the time unit h, which tends to 0, and $b_h(x),a_h(x)$ are the rescaled drift and square-drift from x. We assume that we don’t see macroscopic jumps in the limit. For the case of simple random walk reflected at 0, it doesn’t matter exactly how we construct the joint limit in h and x, as the drift is uniform on x>0, but in general this does matter. I don’t want to discuss sticky BM right now, so it’s probably easiest to be vague and say that the discrete Markov processes converge to reflected BM so long they don’t spend more time than expected near 0 in the limit, as the title ‘sticky’ might suggest.

The two ways in which this can happen is if the volatility term $a_h(x)$ is too small, in which case the process looks almost deterministic near 0, or if the drift doesn’t increase fast enough. And indeed, this leads to two conditions. The first is straightforward, if $a_h(x)$ is bounded below, in the sense that $\liminf_{h,x\rightarrow 0} a_h(x)\ge C>0$, then we have convergence to reflected BM. Alternatively, the only danger can arise down those subsequences where $a_h(x)\rightarrow 0$, so if we have that $b_h(x)\rightarrow +\infty$ whenever $h,x,a_h(x)\rightarrow 0$, then this convergence also holds.

Next time I’ll discuss what sticky BM means, what it doesn’t mean, why it isn’t easy to double the local time, and how to obtain sticky BM as a limit of discrete random walks in a similar way to the above.

REFERENCES

S. Varadhan – Chapter 16 from a Lecture Course at NYU can be found here.

# 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

# Critical Components in Erdos-Renyi

In various previous posts, I’ve talked about the phase transition in the Erdos-Renyi random graph process. Recall the definition of the process. Here we will use the Gilbert model G(n,p), where we have n vertices, and between any pair of vertices we add an edge, independently of other pairs with probability p. We are interested in the sparse scaling, where the typical vertex has degree O(1) in n, and so p=c/n for constant c>0, and we assume throughout that n is large. We could alternatively have considered the alternative Erdos-Renyi model where we choose uniformly at random from the set of graphs with n vertices and some fixed number of edges. Almost all the results present work equally well in this setting.

As proved by Erdos and Renyi, the typical component structure of such a graph changes noticeably around the threshold c=1. Below this, in the subcritical regime, all the components are small, meaning of size at most order O(log n). Above this, in the supercritical regime, there is a single giant component on some non-zero proportion of the vertices. The rest of the graph looks subcritical. The case c=1 exhibits a phase transition between these qualitatively different behaviours. They proved that here, the largest component is with high probability O(n^2/3). It seems that they thought this result held whenever c=1-o(1), but it turns out that this is not the case. In this post, I will discuss some aspects of behaviour around criticality, and the tools needed to treat them.

The first question to address is this: how many components of size n^{2/3} are there? It might be plausible that there is a single such component, like for the subsequent giant component. It might also be plausible that there are n^1/3 such components, so O(n) vertices are on such critical components. As then it is clear how we transition out of criticality into supercriticality – all the vertices on critical components coalesce to form the new giant component.

In fact neither of these are correct. The answer is that for all integers k>0, with high probability the k-th largest component is on a size scale of n^2/3. This is potentially a confusing statement. It looks like there are infinitely many such components, but of course for any particular value of n, this cannot be the case. We should think of there being w(1) components, but o(n^b) for any b>0.

The easiest way to see this is by a duality argument, as we have discussed previously for the supercritical phase. If we remove a component of size O(n^2/3), then what remains is a random graph with n-O(n^2/3) vertices, and edge probability the same as originally. It might make sense to rewrite this probability 1/n as

$\frac{1}{n-O(n^{2/3})}\cdot \frac{n-O(n^{2/3})}{n}=\frac{1-O(n^{-1/3})}{n-O(n^{2/3})}.$

The approximation in the final numerator is basically the same as

$1-o\left(n-O(n^{2/3})\right).$

Although we have no concrete reasoning, it seems at least plausible that this should look similar in structure to G(n,1/n). In particular, there should be another component of size

$O\left([n-O(n^{2/3})]^{2/3}\right)=O(n^{2/3})$.

In fact, the formal proof of this proceeds by an identical argument, only using the exploration process. Because I’ve described this several times before, I’ll be brief. We track how far we have gone through each component in a depth-first walk. In both the supercritical and subcritical cases, when we scale correctly we get a random path which is basically deterministic in the limit (in n). For exactly the same reasons as visible CLT fluctuations for partial sums of RVs with expectation zero, we start seeing interesting effects at criticality.

The important question is the order of rescaling to choose. At each stage of the exploration process, the number of vertices added to the stack is binomial. We want to distinguish between components of size $O(n^{2/3})$ so we should look at the exploration process at time $sn^{2/3}$. The drift of the exploration process is given by the expectation of a binomial random variable minus one (since we remove the current vertex from the stack as we finish exploring it). This is given by

$\mathbb{E}=\left[n-sn^{2/3}\right]\cdot \frac{1}{n}-1=-sn^{-1/3}.$

Note that this is the drift in one time-step. The drift in $n^{2/3}$ time-steps will accordingly by $sn^{1/3}$. So, if we rescale time by $n^{2/3}$ and space by $n^{1/3}$, we should get a nice stochastic process. Specifically, if Z is the exploration process, then we obtain:

$\frac{1}{n^{1/3}}Z^{(n)}_{sn^{2/3}} \rightarrow_d W_s,$

where W is a Brownian motion with inhomogeneous drift -s at time s. The net effect of such a drift at a fixed positive time is given by integrating up to that time, and hence we might say the process has quadratic drift, or is parabolic.

We should remark that our binomial expectation is not entirely correct. We have discounted those $sn^{2/3}$ vertices that have already been explored, but we have not accounted for the vertices currently in the stack. We should also be avoiding considering these. However, we now have a heuristic for the approximate number of these. The number of vertices in the stack should be $O(n^{1/3})$ at all times, and so in particular will always be an order of magnitude smaller than the number of vertices already considered. Therefore, they won’t affect this drift term, though this must be accounted for in any formal proof of convergence. On the subject of which, the mode of convergence is, unsurprisingly, weak convergence uniformly on compact sets. That is, for any fixed S, the convergence holds weakly on the random functions up to time $sn^{2/3}$.

Note that this process will tend to minus infinity almost surely. Component sizes are given by excursions above the running minimum. The process given by the height of the original process above the running minimum is called reflected. Essentially, we construct the reflected process by having the same generator when the current value is positive, and forcing the process up when it is at zero. There are various ways to construct this more formally, including as the scaling limit of some simple random walks conditioned never to stay non-negative.

The cute part of the result is that it holds equally well in a so-called critical window either side of the critical probability 1/n. When the probability is $\frac{1+tn^{-1/3}}{n}$, for any $t\in \mathbb{R}$, the same argument holds. Now the drift at time s is t-s, though everything else still holds.

This result was established by Aldous in [1], and gives a mechanism for calculating distributions of component sizes and so on through this critical window.

In particular, we are now in a position to answer the original question regarding how many such components there were. The key idea is that because whenever we exhaust a component in the exploration process, we choose a new vertex uniformly at random, we are effectively choosing a component according to the size-biased distribution. Roughly speaking, the largest components will show up near the beginning. Note that a critical $O(n^{2/3})$ component will not necessarily be exactly the first component in the exploration process, but the components that are explored before this will take up sufficiently few vertices that they won’t show up in the scaling of the limit.

In any case, the reflected Brownian motion ‘goes on forever’, and the drift is eventually very negative, so there cannot be infinitely wide excursions, hence there are infinitely many such critical components.

If we care about the number of cycles, we can treat this also via the exploration process. Note that in any depth-first search we are necessarily only interested in a spanning tree of the host graph. Anyway, when we are exploring a vertex, there could be extra edges to other vertices in the stack, but not to vertices we’ve already finished exploring (otherwise the edge would have been exposed then). So the expected number of excess edges into a vertex is proportional to the height of the exploration process at that vertex. So the overall expected number of excess edges, conditional on the exploration process is the area under the curve. This carries over perfectly well into the stochastic process limit. It is then a calculation to verify that the area under the curve is almost surely infinite, and thus that we expect there to be infinitely many cycles in a critical random graph.

REFERENCES

[1] Aldous D. – Brownian excursions, critical random graphs and the multiplicative coalescent

# Large Deviations 5 – Stochastic Processes and Mogulskii’s Theorem

Motivation

In the previous posts about Large Deviations, most of the emphasis has been on the theory. To summarise briefly, we have a natural idea that for a family of measures supported on the same metric space, increasingly concentrated as some index grows, we might expect the probability of seeing values in a set not containing the limit in distribution to grow exponentially. The canonical example is the sample mean of a family of IID random variables, as treated by Cramer’s theorem.

It becomes apparent that it will not be enough to specify the exponent for a given large deviation event just by taking the infimum of the rate function, so we have to define an LDP topologically, with different behaviour on open and closed sets. Now we want to find some LDPs for more complicated measures, but which will have genuinely non-trivial applications. The key idea in all of this is that the infimum present in the definition of an LDP doesn’t just specify the rate function, it also might well give us some information about the configurations or events that lead to the LDP.

The slogan for the LDP as in Frank den Hollander’s excellent book is: “A large deviation event will happen in the least unlikely of all the unlikely ways.” This will be useful when our underlying space is a bit more complicated.

Setup

As a starting point, consider the set-up for Cramer’s theorem, with IID $X_1,\ldots,X_n$. But instead of investigating LD behaviour for the sample mean, we investigate LD behaviour for the whole set of RVs. There is a bijection between sequences and the partial sums process, so we investigate the partial sums process, rescaled appropriately. For the moment this is a sequence not a function or path (continuous or otherwise), but in the limit it will be, and furthermore it won’t make too much difference whether we interpolate linearly or step-wise.

Concretely, we consider the rescaled random walk:

$Z_n(t):=\tfrac{1}{n}\sum_{i=1}^{[nt]}X_i,\quad t\in[0,1],$

with laws $\mu_n$ supported on $L_\infty([0,1])$. Note that the expected behaviour is a straight line from (0,0) to (1,$\mathbb{E}X_1$). In fact we can say more than that. By Donsker’s theorem we have a functional version of a central limit theorem, which says that deviations from this expected behaviour are given by suitably scaled Brownian motion:

$\sqrt{n}\left(\frac{Z_n(t)-t\mathbb{E}X}{\sqrt{\text{Var}(X_1)}}\right)\quad\stackrel{d}{\rightarrow}\quad B(t),\quad t\in[0,1].$

This is what we expect ‘standard’ behaviour to look like:

The deviations from a straight line are on a scale of $\sqrt{n}$. Here are two examples of potential large deviation behaviour:

Or this:

Note that these are qualitatively different. In the first case, the first half of the random variables are in general much larger than the second half, which appear to have empirical mean roughly 0. In the second case, a large deviation in overall mean is driven by a single very large value. It is obviously of interest to find out what the probabilities of each of these possibilities are.

We can do this via an LDP for $(\mu_n)$. Now it is really useful to be working in a topological context with open and closed sets. It will turn out that the rate function is supported on absolutely continuous functions, whereas obviously for finite n, none of the sample paths are continuous!

We assume that $\Lambda(\lambda)$ is the logarithmic moment generating function of X_1 as before, with $\Lambda^*(x)$ the Fenchel-Legendre transform. Then the key result is:

Theorem (Mogulskii): The measures $(\mu_n)$ satisfy an LDP on $L_\infty([0,1])$ with good rate function:

$I(\phi)=\begin{cases}\int_0^1 \Lambda^*(\phi'(t))dt,&\quad \text{if }\phi\in\mathcal{AC}, \phi(0)=0,\\ \infty&\quad\text{otherwise,}\end{cases}$

where AC is the space of absolutely continuous functions on [0,1]. Note that AC is dense in $L_\infty([0,1])$, so any open set contains a $\phi$ for which $I(\phi)$ is at least in principle finite. (Obviously, if $\Lambda^*$ is not finite everywhere, then extra restrictions of $\phi'$ are required.)

The following picture may be helpful at providing some motivation:

So what is going on is that if we take a path and zoom in on some small interval around a point, note first that behaviour on this interval is independent of behaviour everywhere else. Then the gradient at the point is the local empirical mean of the random variables around this point in time. The probability that this differs from the actual mean is given by Cramer’s rate function applied to the empirical mean, so we obtain the rate function for the whole path by integrating.

More concretely, but still very informally, suppose there is some $\phi'(t)\neq \mathbb{E}X$, then this says that:

$Z_n(t+\delta t)-Z_n(t)=\phi'(t)\delta t+o(\delta t),$

$\Rightarrow\quad \mu_n\Big(\phi'(t)\delta t+o(\delta t)=\frac{1}{n}\sum_{i=nt+1}^{n(t+\delta t)}X_i\Big),$

$= \mu_n\Big( \phi'(t)+o(1)=\frac{1}{n\delta t}\sum_{i=1}^{n\delta t}X_i\Big)\sim e^{-n\delta t\Lambda^*(\phi'(t))},$

by Cramer. Now we can use independence:

$\mu_n(Z_n\approx \phi)=\prod_{\delta t}e^{-n\delta t \Lambda^*(\phi'(t))}=e^{-\sum_{\delta t}n\delta t \Lambda^*(\phi'(t))}\approx e^{-n\int_0^1 \Lambda^*(\phi'(t))dt},$

as in fact is given by Mogulskii.

Remarks

1) The absolutely continuous requirement is useful. We really wouldn’t want to be examining carefully the tail of the underlying distribution to see whether it is possible on an exponential scale that o(n) consecutive RVs would have sum O(n).

2) In general $\Lambda^*(x)$ will be convex, which has applications as well as playing a useful role in the proof. Recalling den Hollander’s mantra, we are interested to see where infima hold for LD sets in the host space. So for the event that the empirical mean is greater than some threshold larger than the expectation, Cramer’s theorem told us that this is exponentially the same as same the empirical mean is roughly equal to the threshold. Now Mogulskii’s theorem says more. By convexity, we know that the integral functional for the rate function is minimised by straight lines. So we learn that the contributions to the large deviation are spread roughly equally through the sample. Note that this is NOT saying that all the random variables will have the same higher than expected value. The LDP takes no account of fluctuations in the path on a scale smaller than n. It does however rule out both of the situations pictured a long way up the page. We should expect to see roughly a straight line, with unexpectedly steep gradient.

3) The proof as given in Dembo and Zeitouni is quite involved. There are a few stages, the first and simplest of which is to show that it doesn’t matter on an exponential scale whether we interpolate linearly or step-wise. Later in the proof we will switch back and forth at will. The next step is to show the LDP for the finite-dimensional problem given by evaluating the path at finitely many points in [0,1]. A careful argument via the Dawson-Gartner theorem allows lifting of the finite-dimensional projections back to the space of general functions with the topology of pointwise convergence. It remains to prove that the rate function is indeed the supremum of the rate functions achieved on projections. Convexity of $\Lambda^*(x)$ is very useful here for the upper bound, and this is where it comes through that the rate function is infinite when the comparison path is not absolutely continuous. To lift to the finer topology of $L_\infty([0,1])$ requires only a check of exponential tightness in the finer space, which follows from Arzela-Ascoli after some work.

In conclusion, it is fairly tricky to prove even this most straightforward case, so unsurprisingly it is hard to extend to the natural case where the distributions of the underlying RVs (X) change continuously in time, as we will want for the analysis of more combinatorial objects. Next time I will consider why it is hard but potentially interesting to consider with adaptations of these techniques an LDP for the size of the largest component in a sparse random graph near criticality.

# Supremum of Brownian Motion

We define the supremum process of Brownian Motion by:

$S_t:=\sup_{0\leq s\leq t}B_s.$

Here are two facts about Brownian Motion. Firstly, the Reflection Principle:

$\mathbb{P}(S_t\geq b,B_t\leq a)=\mathbb{P}(B_t\geq 2b-a),$

which we motivate by ‘stopping’ at time $S_t$, and using the SMP for Brownian Motion, even though it isn’t a stopping time. By setting a=b, we get:

$\mathbb{P}(S_t\geq b)=\mathbb{P}(S_t\geq b,B_t\leq b)+\mathbb{P}(B_t\geq b)=2\mathbb{P}(B_t\geq b)=\mathbb{P}(|B|\geq b),$

and conclude that

$S_t\stackrel{d}{=}|B_t|\quad\text{for each }t\geq 0.$

The second fact comes from the decomposition of BM into local times and excursions:

$(S_t,S_t-B_t)_{t\geq 0}\stackrel{d}{=}(L_t,|B_t|)_{t\geq 0},$

where L is the local time process at 0, and this equality in distribution holds for the processes. See the previous post on excursion theory for explanation of what local times mean.

In particular, combining these two facts gives:

$S_t\stackrel{d}{=}S_t-B_t\quad\text{for every }t\geq 0.$

I thought that was rather surprising, and wanted to think of a straightforward reason why this should be true. I think the following works:

Brownian motion is time-reversible. In particular, as processes, we have

$(B_s)_{s\geq 0}\stackrel{d}{=}(B_{t-s}-B_t)_{s\geq 0}$

$\Rightarrow \sup_{0\leq r\leq t}B_r\stackrel{d}{=}\sup_{0\leq r\leq t}(B_{t-r}-B_t)$

$\Rightarrow S_t\stackrel{d}{=}S_t-B_t.$

# Subordinators and the Arcsine rule

After the general discussion of Levy processes in the previous post, we now discuss a particular class of such processes. The majority of content and notation below is taken from chapters 1-3 of Jean Bertoin’s Saint-Flour notes.

We say $X_t$ is a subordinator if:

• It is a right-continuous adapted stochastic process, started from 0.
• It has stationary, independent increments.
• It is increasing.

Note that the first two conditions are precisely those required for a Levy process. We could also allow the process to take the value $\infty$, where the hitting time of infinity represents ‘killing’ the subordinator in some sense. If this hitting time is almost surely infinite, we say it is a strict subordinator. There is little to be gained right now from considering anything other than strict subordinators.

Examples

• A compound Poisson process, with finite jump measure supported on $[0,\infty)$. Hereafter we exclude this case, as it is better dealt with in other languages.
• A so-called stable Levy process, where $\Phi(\lambda)=\lambda^\alpha$, for some $\alpha\in(0,1)$. (I’ll define $\Phi$ very soon.) Note that checking that the sample paths are increasing requires only that $X_1\geq 0$ almost surely.
• The hitting time process for Brownian Motion. Note that this does indeed have jumps as we would need. (This has $\Phi(\lambda)=\sqrt{2\lambda}$.)

Properties

• In general, we describe Levy processes by their characteristic exponent. As a subordinator takes values in $[0,\infty)$, we can use the Laplace exponent instead:

$\mathbb{E}\exp(-\lambda X_t)=:\exp(-t\Phi(\lambda)).$

• We can refine the Levy-Khintchine formula;

$\Phi(\lambda)=k+d\lambda+\int_{[0,\infty)}(1-e^{-\lambda x})\Pi(dx),$

• where k is the kill rate (in the non-strict case). Because the process is increasing, it must have bounded variation, and so the quadratic part vanishes, and we have a stronger condition on the Levy measure: $\int(1\wedge x)\Pi(dx)<\infty$.
• The expression $\bar{\Pi}(x):=k+\Pi((x,\infty))$ for the tail of the Levy measure is often more useful in this setting.
• We can think of this decomposition as the sum of a drift, and a PPP with characteristic measure $\Pi+k\delta_\infty$. As we said above, we do not want to consider the case that X is a step process, so either d>0 or $\Pi((0,\infty))=\infty$ is enough to ensure this.

Analytic Methods

We give a snapshot of a couple of observations which make these nice to work with. Define the renewal measure U(dx) by:

$\int_{[0,\infty)}f(x)U(dx)=\mathbb{E}\left(\int_0^\infty f(X_t)dt\right).$

If we want to know the distribution function of this U, it will suffice to consider the indicator function $f(x)=1_{X_t\leq x}$ in the above.

The reason to exclude step processes specifically is to ensure that X has a continuous inverse:

$L_x=\sup\{t\geq 0:X_t\leq x\}$ so $U(x)=\mathbb{E}L_x$ is continuous.

In fact, this renewal measure characterises the subordinator uniquely, as we see by taking the Laplace transform:

$\mathcal{L}U(\lambda)=\int_{[0,\infty)}e^{-\lambda x}U(dx)=\mathbb{E}\int e^{-\lambda X_t}dt$

$=\int \mathbb{E}e^{-\lambda X_t}dt=\int\exp(-t\Phi(\lambda))dt=\frac{1}{\Phi(\lambda)}.$

The Arcsine Law

X is Markov, which induces a so-called regenerative property on the range of X, $\mathcal{R}$. Formally, given s, we do not always have $s\in\mathcal{R}$ (as the process might jump over s), but we can define $D_s=\inf\{t>s:t\in\mathcal{R}\}$. Then

$\{v\geq 0:v+D_s\in\mathcal{R}\}\stackrel{d}{=}\mathcal{R}.$

In fact, the converse holds as well. Any random set with this regenerative property is the range of some subordinator. Note that $D_s$ is some kind of dual to X, since it is increasing, and the regenerative property induces some Markovian properties.

In particular, we consider the last passage time $g_t=\sup\{s, in the case of a stable subordinator with $\Phi(\lambda)=\lambda^\alpha$. Here, $\mathcal{R}$ is self-similar with scaling exponent $\alpha$. The distribution of $\frac{g_t}{t}$ is thus independent of t. In this situation, we can derive the generalised arcsine rule for the distribution of $g_1$:

$\mathbb{R}(g_1\in ds)=\frac{\sin \alpha\pi}{\pi}s^{\alpha-1}(1-s)^{-\alpha}ds.$

The most natural application of this is to the hitting time process of Brownian Motion, which is stable with $\alpha=\frac12$. Then $g_1=S_1-B_1$, in the usual notation for the supremum process. Furthermore, we have equality in distribution of the processes (see previous posts on excursion theory and the short aside which follows):

$(S_t-B_t)_{t\geq 0}\stackrel{d}{=}(|B_t|)_{t\geq 0}.$

So $g_1$ gives the time of the last zero of BM before time 1, and the arcsine law shows that its distribution is given by:

$\mathbb{P}(g_1\leq t)=\frac{2}{\pi}\text{arcsin}\sqrt{t}.$

# The Levy-Khintchine Formula

Because of a string of coincidences involving my choice of courses for Part III and various lecturers’ choices about course content, I didn’t learn what a Levy process until a few weeks’ ago. Trying to get my head around the Levy-Khintchine formula took a little while, so the following is what I would have liked to have been able to find back then.

A Levy process is an adapted stochastic process started from 0 at time zero, and with stationary, independent increments. This is reminiscent, indeed a generalisation, of the definition of Brownian motion. In that case, we were able to give a concrete description of the distribution of $X_1$. For a general Levy process, we have

$X_1=X_{1/n}+(X_{2/n}-X_{1/n})+\ldots+(X_1-X_{1-1/n}).$

So the distribution of $X_1$ is infinitely divisible, that is, can be expressed as the distribution of the sum n iid random variables for all n. Viewing this definition in terms of convolutions of distributions may be more helpful, especially as we will subsequently consider characteristic functions. If this is the first time you have seen this property, note that it is not a universal property. For example, it is not clear how to write a U[0,1] random variable as a convolution of two iid RVs. Note that exactly the same argument suffices to show that the distribution of $X_t$ is infinitely divisible.

It will be most convenient to work with the characteristic functions

$\mathbb{E}\exp(i\langle \lambda,X_t\rangle).$

By stationarity of increments, we can show that this is equal to

$\exp(-\Psi(\lambda)t)\quad\text{where}\quad \mathbb{E}\exp(i\langle \lambda,X_1\rangle)=:\exp(-\Psi(\lambda)).$

This function $\Psi(\lambda)$ is called the characteristic exponent. The argument resembles that used for Cauchy’s functional equations, by dealing first with the rationals using stationarity of increments, then lifting to the reals by the (right-)continuity of

$t\mapsto \mathbb{E}\exp(i\langle \lambda,X_t\rangle).$

As ever, $\Psi(\lambda)$ uniquely determines the distribution of $X_1$, and so it also uniquely determines the distribution of Levy process. The only condition on $\Psi$ is that it be the characteristic function of an infinitely divisible distribution. This condition is given explicitly by the Levy-Khintchine formula.

Levy-Khintchine

$\Psi(\lambda)$ is the characteristic function of an infinitely divisible distribution iff

$\Psi(\lambda)=i\langle a,\lambda\rangle +\frac12 Q(\lambda)+\int_{\mathbb{R}^d}(1-e^{i\langle \lambda,x\rangle}+i\langle \lambda,x\rangle 1_{|x|<1})\Pi(dx).$

for $a\in\mathbb{R}^d$, Q a quadratic form on $\mathbb{R}^d$, and $\Pi$ a so-called Levy measure satisfying $\int (1\wedge |x|^2)\Pi(dx)<\infty$.

This looks a bit arbitrary, so first let’s explain what each of these terms ‘means’.

• $i\langle a,\lambda\rangle$ comes from a drift of $-a$. Note that a deterministic linear function is a (not especially interesting) Levy process.
• $\frac12Q(\lambda)$ comes from a Brownian part $\sqrt{Q}B_t$.

The rest corresponds to the jump part of the process. Note that a Poisson process is an example of a Levy process, hence why we might consider thinking about jumps in the first place. The reason why there is an indicator function floating around is that we have to think about two regimes separately, namely large and small jumps. Jumps of size bounded below cannot happen too often as otherwise the process might explode off to infinity in finite time with positive probability. On the other hand, infinitesimally small jumps can happen very often (say on a dense set) so long as everything is controlled to prevent an explosion on the macroscopic scale.

There is no canonical choice for where the divide between these regimes happens, but conventionally this is taken to be at $|x|=1$. The restriction on the Levy measure near 0 ensures that the sum of the squares all jumps up some finite time converges absolutely.

• $\Pi\cdot 1_{|x|\geq 1}$ gives the intensity of a standard compound Poisson process. The jumps are well-spaced, and so it is a relatively simple calculation to see that the characteristic function is

$\int_{\mathbb{R}^d}(1-e^{i\langle \lambda,x\rangle})1_{|x|\geq 1}\Pi(dx).$

The intensity $\Pi\cdot 1_{|x|<1}$ gives infinitely many hits in finite time, so if the expectation of this measure is not 0, we explode immediately. We compensate by drifting away from this at rate

$\int_{\mathbb{R}^d}x1_{|x|<1}\Pi(dx).$

To make this more rigorous, we should really consider $1_{\epsilon<|x|<1}$ then take a limit, but this at least explains where all the terms come from. Linearity allows us to interchange integrals and inner products, to get the term

$\int_{\mathbb{R}^d}(1-e^{-i\langle \lambda,x\rangle}+i\langle\lambda,x\rangle 1_{|x|<1})\Pi(dx).$

If the process has bounded variation, then we must have Q=0, and also

$\int (1\wedge |x|)\Pi(dx)<\infty,$

that is, not too many jumps on an |x| scale. In this case, then this drift component is well-defined and linear $\lambda$, so can be incorporated with the drift term at the beginning of the Levy-Khintchine expression. If not, then there are some $\lambda$ for which it does not exist.

There are some other things to be said about Levy processes, including

• Stable Levy processes, where $\Psi(k\lambda)=k^\alpha \Psi(\lambda)$, which induces the rescaling-invariance property: $k^{-1/\alpha}X_{kt}\stackrel{d}{=}X$. The distribution of each $X_t$ is then also a stable distribution.
• Resolvents, where instead of working with the process itself, we work with the distribution of the process at a random exponential time.