The functional iterated logarithm law for stochastic processes represented by multiple Wiener integrals

Abstract

In a previous paper the authors obtained a functional law of the iterated logarithm for a class of self-similar processes X with stationary increments, which are represented by multiple Wiener integrals. This result is extended to a certain class of processes represented by multiple Wiener integrals which converge to X with an appropriate normalization. As an application a functional log log law for nonlinear functionals of some stationary Gaussian processes is given.

Approximation of Multiple Stratonovich Fractional Integrals

“…If the kernel h t t∈ a b is so called adapted, that is, h t = h · 1 0 t p · for some h ∈ L 2 s 0 1 p , then it is well known that the multiple integral process t −→ I We need the Hölder case when we have the following corresponding result: [15]). If the kernel h t t∈ a b is -Hölder continuous, h 0 = 0 then the multiple integral process t −→ I …”

“…Since the multiple stochastic fractional process it is not a martingale, we can not apply maximal martingale inequalities, but instead we can use maximal inequalities of exponentially type in terms of Hölder norm as obtained in [15].…”

Weak convergence of the sequential empirical copula processes under long-range dependence

“…Note that H r1,••• ,rp (η t ) = p j=1 H rj (η t,j ). Using the results of Mori & Oodaira (1987), we have…”

The law of iterated logarithm for subordinated Gaussian sequences: uniform Wasserstein bounds

“…The question of the iterated logarithm in this setup was partially solved by Taqqu in [30]. Later on, Lai and Stout [19] gave criteria for upper bounds, whereas the complete law of the iterated logarithm was proved by Mori and Oodaira in [22].…”

“…Note that, so far, there has been no attempt to prove a LIL in this delicate context. We believe indeed that it would be not possible (or, at least, technically very demanding) to extend the approaches by Arcones [2] and Mori and Oodaira in [22] to deal with this case. One plausible explanation for this impasse is that, in both cases (a) and (b), the convergence in distribution takes place at an algebraic speed in n (with respect e.g.…”