The central limit theorem for a sequence of random processes with space-varying long memory*

Abstract

In this paper we investigate a sequence of square integrable random processes with space varying memory. We establish sufficient conditions for the central limit theorem in the space L 2 (µ) for the partial sums of the sequence of random processes with space varying long memory. Of particular interest is a non-standard normalization of the partial sums in the central limit theorem.

Operator self-similar processes and functional central limit theorems

“…The following two asymptotic relations are proved in Characiejus and Račkauskas [2]: if 1/2 < d(r) < 1 and 1/2 < d(s) < 1, then…”

“…The first approach is to define {X k } as stochastic processes with space varying memory and square µ-integrable sample paths. The second approach is to define L 2 (µ) valued random variable X k for each k ≥ 1 as series (1) with u j given by (2) and to investigate the convergence of such series. We present both of these two approaches.…”

“…Now we establish a necessary and sufficient condition for the mean square convergence of series (1) with {u j } given by (2). Recall that (j + 1) −D f = {(j + 1) −d(s) f (s) : s ∈ S} for each j ≥ 0 and f ∈ L 2 (µ) since e T = ∞ j=0 T j /j!…”

“…Roughly speaking, if the series ∞ j=0 u j converges, the asymptotic behaviour of {S n } and {ζ n } is inherited from {ε k } (see Merlevède, Peligrad, and Utev [17], Račkauskas and Suquet [18] for more details). However, this is not the case when ∞ j=0 u j = ∞ (see Račkauskas and Suquet [19] and Characiejus and Račkauskas [2]).…”

Limit theorems in the context of multivariate long-range dependence

“…For further procedure we use a proof idea like Characiejus and Račkauskas (2013), first introduced in Račkauskas and Suquet (2011). It will make it possible to prove the theorem under the assumption of a Gaussian white noise process.…”