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# Stochastic Analysis and Function Spaces

DOI link for Stochastic Analysis and Function Spaces

Stochastic Analysis and Function Spaces book

# Stochastic Analysis and Function Spaces

DOI link for Stochastic Analysis and Function Spaces

Stochastic Analysis and Function Spaces book

## ABSTRACT

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

Mx(t) = etx dP, t IR, (1)

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

log(( esX dP) ( etX dP) ), by Hölder's inequality, = (s)+ (t). (2)

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

(t) = (a) + t

'(u) du,

where '(.) is the left derivative of which exists everywhere and is nondecreasing. Taking a = 0, consider the (generalized) inverse of ', say '. It is given by '

: t inf{t > a = 0 : '(u) > t}, which, if ' is strictly increas-

ing, is the usual inverse function ' = ( ') 1. rhen ' 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 ( , , 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.,

1 n

i=l Xi E(X1) = m. (a.e.)

Expressed differently, one has for each > 0, hn( ) 0 as n —> in:

X(t). ( a –

dP)

(0) (t) , •

•

.