LAWS OF LARGE NUMBERS AND MODERATE DEVIATIONS FOR STOCHASTIC-PROCESSES WITH STATIONARY AND INDEPENDENT INCREMENTS

Let {X(t); t greater-than-or-equal-to 0} be a stochastic process with stationary and independent increments which has no Gaussian component. Assume that X(1) has a finite moment generating function. Let P(lambda) be the probability measure of the process {Z(lambda)(1); 0 less-than-or-equal-to t less-than-or-equal-to 1}, where Z(lambda)(t) = (1/lambda(q))X(lambda(alpha)[0, t]), alpha is a probability measure on [0, 1] and 1 < q < 2. We may regard P(lambda) as a probability measure on BV[0, 1], the space of functions of bounded variation on [0, 1]. In this paper, we establish some results on moderate deviations for {P(lambda); lambda > 0). We also present the Marcinkiewicz-Zygmund type Strong Law of Large Numbers for {X(t); t greater-than-or-equal-to 0}.