The large deviation principle for stochastic processes. I

Random fractals generated by a local Gaussian process indexed by a class of functions

“…It follows readily from Theorem 5.2 in Arcones [1]. (This same method is used for the proof of Prop.…”

“…Lévy [8] For each Λ > 0 E Λ collects the exceptional points in [0,1] where the law of the iterated logarithm (1.2) fails. Orey and Taylor [11] showed that, with probability 1, E Λ is a random fractal with Hausdorff dimension, given by…”

Moderate Deviations for I.I.D. Random Variables

“…Hence, following the ideas already indicated in [10] and [2], we symmetrize the X i : introduce an i.i.d. sequence (X i ) i∈N which is also independent of (X i ) i∈N and such that L(X 1 ) = L(X 1 ).…”

“…Here -for all such sequences (b n ) n -the rate functions are given by I(t) = 1 2 Var X1 t 2 and the speed is a n = b 2 n /n. In a recent paper [2] Arcones states that assumption (2.3) below is necessary and sufficient for 1 bn n i=1 (X i − EX i ) to satisfy a moderate deviations principle. As a matter of fact it seems that the search for such an equivalent condition is almost as old as the investigation of moderate deviations for means of i.i.d.…”

“…Indeed, already Ledoux [9] discusses the connection of our assumption (2.3) below with the moderate deviations upper bound for a restricted range of normalizations (b n ). Unfortunately, Arcones proof in [2] of this interesting equivalence is spread over several theorems and articles and technically very involved, since it relies on MDPs for Banach space valued stochastic processes. Moreover it is most unfortunately spoiled by several flaws and therefore hardly penetrable.…”