Skorohod Space

The following is a summary of a chapter from Billingsley’s Convergence of Probability Measures. The ideas are easy to explain heuristically, but this was the first text I could find which explained how to construct Skorohod space for functions on the whole of the non-negative reals in enough stages that it was easily digestible.

It is relatively straightforward to define a topology on C[0,1], as we can induce from the most sensible metric. In this topology, functions f and g are close together if

\sup_{t\in[0,1]} |f(t)-g(t)| is small.

For cadlag functions, things are a bit more complicated. Two functions might be very similar, but have a discontinuity of similar magnitude at slightly different places. The sup norm of the difference is therefore macroscopically large. So we want a metric that also allows uniformly small deformations of the time scale.

We define the Skorohod (or Skorokhod depending on your transliteration preferences) metric d on D[0,1] as follows. Let \Lambda be the family of continuous, strictly increasing functions from [0,1] to [0,1] which map 0 to 0 and 1 to 1. This will be our family of suitable reparameterisations of the time scale (or abscissa – a new word I learned today. The other axis in a co-ordinate pair is called the ordinate). Anyway, we now say that

d(x,y)<\epsilon\quad\text{if }\exists \lambda\in\Lambda\text{ s.t. }

||\lambda - id||_\infty<\epsilon\quad\text{and}\quad ||f-\lambda\circ g||_\infty<\epsilon.

In other words, after reparameterising the time scale for g, without moving any time by more than epsilon, the functions are within epsilon in the sup metric.

Weak Convergence

We have the condition: if \{P_n\} is a tight sequence of probability measures and we have

\text{If }P_n\pi_{t_1,\ldots,t_k}^{-1}\Rightarrow P\pi_{t_1,\ldots,t_k}^{-1}\quad\forall t_1,\ldots,t_k\in[0,1],\quad\text{then }P_n\Rightarrow P,

where \pi_{t_1,\ldots,t_k} is the projection onto a finite-dimensional set. This is a suitable condition for C[0,1]. For D[0,1], we have the additional complication that these projections might not be continuous everywhere.

We can get over this problem. For a measure P, set T_P to be the set of t\in[0,1] such that \pi_t is continuous P-almost everywhere (ie for all f\in D apart from a collection with P-measure = 0). Then, for all P, it is not hard to check that 0,1\in T_P and [0,1]\backslash T_P is countable.

The tightness condition requires two properties:

1) \lim_{K\rightarrow\infty} \limsup_{n}P_n[f:||f||\geq K]=0.

2) \forall \epsilon>0:\,\lim_\delta\limsup_n P_n[f:w_f'(\delta)\geq\epsilon]=0.

These say, respectively, that the measure of ||f|| doesn’t escape to \infty, and there is no mass given in the limit to functions which ‘wiggle with infinite frequency on an epsilon scale of amplitude’.

D_\infty=D[0,\infty)

Our earlier definition of the Skorohod metric could have been written:

d(f,g)=\inf_{\lambda\in\Lambda}\{||\lambda-\text{id}||\vee||f-\lambda\circ g||\}.

From a topological convergence point of view, there’s no need to use the sup norm on \lambda - \text{id}. We want to regulate smoothness of the reparameterisation, so we could use the norm:

||\lambda||^\circ=\sup_{s<t}|\log\frac{\lambda(t)-\lambda(s)}{t-s}|,

that is, the slope is uniformly close to 1 if ||\lambda||^\circ is small. The advantage of this choice of norm is that an extension to D[0,\infty) is immediate. Also, the induced product norm

d^\circ(f,g)=\inf_{\lambda\in\Lambda} \{||\lambda - \text{id}||^\circ \vee||x-\lambda\circ y||\}

is complete. This gives us a few problems, as for example

d_\circ(1_{[0,1)},1_{[0,1-\frac{1}{n})})=1,

as you can’t reparameterise over the jump in a way that ensures the log of the gradient is relatively small. (In particular, to keep the sup norm less than 1, we would need \lambda to send [1-\frac{1}{n}]\mapsto 1, and so ||\lambda||^\circ=\infty by definition.)

So we can’t immediately define Skorohod convergence on D_\infty by demanding convergence on any restriction to [0,t]. We overcome this in a similar way to convergence of distribution functions.

Lemma: If d_t^\circ (f_n,f)\rightarrow_n 0 then for any s<t with f cts at s, then d_s^\circ(f_n,f)\rightarrow_n 0.

So this says that the functions converge in Skorohod space if for arbitrarily large times T where the limit function is continuous, the restrictions to [0,T] converge. (Note that cadlag functions have at most countably many discontinuities, so this is fine.)

A metric for D_\infty

If we want to specify an actual metric d_\infty^\circ, the usual tools for specifying a countable product metric will do here:

d_\infty^\circ(f,g)=\sum_{m\geq 1}2^{-m}[1\wedge d_m^\circ(f^m,g^m)],

where f^m is the restriction of f to [0,m], with the potential discontinuity at m smoothed out:

f^m(t)=\begin{cases}t&t\leq m-1\\ (m-t)f(t)&t\in[m-1,m]\\ 0&t\geq m.\end{cases}

In particular, d_\infty^\circ(f,g)=0\Rightarrow f^m=g^m\,\forall m.

It can be checked that:

Theorem: d_\infty^\circ(f_n,f)\rightarrow 0 in D_\infty if and only iff

\exists \lambda_n\in\Lambda_\infty\text{ s.t. }||\lambda_n-\text{id}||\rightarrow 0

\text{and }\sup_{t\leq m}|\lambda_n\circ f_n-f|\rightarrow_n 0,\,\forall m,

and that d_\infty^\circ (f_n,f)\rightarrow 0 \Rightarrow d_t^\circ(f_n,f)\rightarrow 0 for every point of continuity t of f.

Similarly weak convergence and tightness properties are available, roughly as you might expect. It is probably better to reference Billingsley’s book or similar sources rather than further attempting to summarise them here.

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2 thoughts on “Skorohod Space

  1. Interesting that you should say “abscissa” is a new word. Perhaps it’s a bit quaint and old-fashioned. I remember “abscissa” and “ordinate” from middle and high school almost 20 years ago, when we were making basic graphs of functions in math class.

    The distinction became more critical in high school and college economics when we would plot supply and demand curves as price vs. quantity. In that case price was the abscissa and quantity was the ordinate, even though price is always plotted on the vertical axis and quantity on the horizontal axis.

  2. Billingsley is good, but I’ve recently discovered notes by Kersting (Konvergenz und Martingalprobleme) which present this topic in a very compact way (e.g. going to D[0,\infty) directly and a few other tweaks). They are easy to find online on his website (just type the title to google and it is probably the first hit). The only downside is that they are German. However, I’m not a native speaker and my German is not even that good but you learn the “maths phrases” quite quickly.

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