MODERATELY LARGE DEVIATION PRINCIPLES FOR THE TRAJECTORIES OF RANDOM WALKS AND PROCESSES WITH INDEPENDENT INCREMENTS

Let xi,xi(1),xi(2), ... be a sequence of independent identically distributed d-dimensional random vectors with zero mean and unit covariance matrix. Let S-0 := 0, S-n := Sigma(n)(i=1)xi(i), n >= 1, and let s(n) = s(n)(t), 0 <= t <= 1, be a random polygon with nodes at the points (k/n, Sk/x), 0 <= k <= n, where x = x(n) >> root n, x = o(n) as n -> infinity We establish a moderately large deviation principle for s(n) which asserts that, for a broad class of measurable sets B of continuous functions, one has logP(s(n) epsilon B) similar to -x(2)/n inf (f epsilon B) I(f), where I(f) := {1/2 integral(1)(0)vertical bar f'(t)vertical bar(2)dt if f(0) = 0, f is absolutely continuous, infinity otherwise. We establish the moderately large deviation principle for homogeneous processes with independent increments as well.