14 - Appendix

Summary

Kolmogorov's Theorem for path continuity

Kolmogorov's Theorem gives a simple condition for Hölder continn a complete separable metric space. Since we only use it in Chapter 2 for processes on R^d, we simplify matters and consider only processes on [0, 1]^d. This result is interesting from a historical perspective since it contains the germs of the much deeper continuity conditions obtained in Chapter 5 (actually, in Chapter 5, we only consider Gaussian processes, but the methods developed have a far larger scope, as is shown in Ledoux and Talagrand (1991)).

Let D_m be the set of d-dimensional vectors in [0, 1]^d with components of the form i/2^m for some integer i ∈ [0, 2^m]. Let D denote the set of dyadic numbers in [0, 1]^d, that is,.

Theorem 14.1.1Let X = {X_t, t ∈ [0, 1]^d} be a stochastic process satisfying

for constants c, r > 0 and h ≥ 1. Then, for any α < r/h,

for some random variable C(ω) < ∞ almost surely.

Proof Let N_m be the set of nearest neighbors in D_m. This is the set of pairs s, t ∈ D_m with |s − t| = 2^−m. Using (14.1) and the fact that |N_m| ≥ 2d2^dm, we have

For any s ∈ D, let s_m be the vector in D_m with s_m ≤ s that is closest to s. Then, for any m, we have that either s_m = s_m+1 or else the pair {s_m, s_m+1} ∈ N_m+1. Now let s, t ∈ D with |s − t| ≤ 2^−m.