Stochastic Processes and Functional Analysis

A Volume of Recent Advances in Honor of M. M. Rao

Alan C. Krinik, Randall J. Swift, Alan C. Krinik, Randall J. Swift

• Publisher: CRC Press
• Year: 2004
• ISBN:
  9781135536497
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

This extraordinary compilation is an expansion of the recent American Mathematical Society Special Session celebrating M. M. Rao's distinguished career and includes most of the presented papers as well as ancillary contributions from session invitees. This book shows the effectiveness of abstract analysis for solving fundamental problems of stochas

Stochastic Analysis and Function Spaces

M. M. Rao

Department of Mathematics,
University of California
Riverside, CA 92521

Abstract

In this paper some interesting and nontrivial relations between certain key areas of stochastic processes and some classical and other function spaces connected with exponential Orlicz spaces are shown. The intimate relationship between these two areas, and several resulting problems for investigation in both areas are pointed out. The connection between the theory of large deviations and exponential as well as vector Orlicz, Fenchel-Orlicz, and Besov-Orlicz spaces are presented. These lead to new problems for solution. Relations between certain Hölder spaces and the range of stochastic flows as well as stochastic Sobolev spaces for SPDEs are also pointed out.

1. Introduction

To motivate the problem, consider a real random variable X on (Ω, Σ, P), a probability triple, with its Laplace transform Mx(·), or its moment generating function, existing so that

is finite. Since Mx(t) ≥ 0, consider its (natural) logarithm also called the cumulant (or semi-invariant) function Λ : t → log M_X(t). Then Λ (0) = 0 and has the remarkable property that it is convex. In fact, if 0 < α = 1 — β < 1, then one has

So as t ↑ ∞, 0 = Λ (0) ≤ Λ (t) ↑ ∞, and the convexity of Λ (·) plays a fundamental role in connecting the probabilistic behavior of X and the continuouity properties of Λ. First let us note that by the well-known integral representation, one has

where Λ′ (·) is the left derivative of Λ which exists everywhere and is nondecreasing. Taking a = 0, consider the (generalized) inverse of Λ′, say

′. It is given by

which, if Λ′ is strictly increasing, is the usual inverse function

′ = (Λ′)^-1. Then

′ is also nondecreasing and left continuous. Let

be its indefinite integral:

A problem of fundamental importance in Probability Theory is the rate of convergence in a limit theorem for its application in practical situations. It will be very desirable if the decay to the limit is exponentially fast. The class of problems for which this occurs constitute a central part of the large deviation analysis. Its relation with Orlicz spaces and related function spaces is of interest here. Let us illustrate this with a simple, but nontrivial, problem which also serves as a suitable motivation for the subject to follow.

Consider a sequence of independent random variables X₁, X₂, ... on a probability space (Ω, Σ, P) with a common distribution F for which the Laplace transform (or the moment generating function) exists. Then the classical Kolmogorov law of large numbers states that the averages converge with probability 1 to their mean, i.e.,

Expressed differently, one has for each ε > 0, h_n(ε) → 0 as n→ ∞ in:

and later it was found that hn(ε) = e-

(ε) with

as the Leqendre transform of Λ, the latter being the cumulant function of F, or Λ (t) = log M_F(t), t

IR, and

is given by

The function

defined differently by (4) and (6) can be shown to be the same so that there is no conflict in notation. The following example illustrates and leads to further work. [

of (6) is also termed the complementary or conjugate function of Λ.]

Let the X_n above be Bernoulli variables so that P[XM_n = 1] = p and P[X_n = 0] = q(= 1 — p), 0 < p < 1. Then the cumulant function Λ is given by Λ (t) = log(q + pe^t) and hence

One finds its complementary function to be, since m = E(X₁) = p and

(m) = 0,

and for other values

(t) = ∞. If the X_n describe a fair coin, so that p = q = ½, one gets

otherwise. The complementary function, written in a more symmetric form can be expressed as:

A dire...