On asymptotics of discretized functionals of long-range dependent functional data

Abstract

The paper studies the asymptotic behaviour of weighted functionals of long-range dependent data over increasing observation windows. Various important statistics, including sample means, high order moments, occupation measures can be given by these functionals. It is shown that in the discrete sampling case additive functionals have the same asymptotic distribution as the corresponding integral functionals for the continuous functional data case. These results are applied to obtain non-central limit theorems for weighted additive functionals of random fields. As the majority of known results concern the discrete sampling case the developed methodology helps in translating these results to functional data without deriving them again. Numerical studies suggest that the theoretical findings are valid for wider classes of long-range dependent data.

Long range dependence of heavy-tailed random functions

“…Abel's uniform convergence test allows us to interchange the sum and the integral over Im (G) 2 . Since b k ≥ 0, we get…”

“…Proof. Consider the random variable According to [23, Theorem 4] and [2, Theorem 4.3], the random variables have the same limiting distributions as . Furthermore, if we have by [23, Theorem 5] that converges in distribution to random variable R .…”

Reduction principle for functionals of strong-weak dependent vector random fields

“…Obtaining such analogous of Theorems 4 and 5, it requires an extension of Arcones-Major results, see Arcones (1994); Major (2019), to continuous settings. While the direct proof may need substantial efforts, see Major (2019), one can try the simpler strategy proposed in Alodat and Olenko (2019). Namely, to prove that discrete and continuous functionals have same limits and then to apply the known discrete result from Arcones (1994) and Major (2019); (4) cyclically dependent components, i.e.…”

Strong law of large numbers for functionals of random fields with unboundedly increasing covariances

“…It follows from results in Alodat and Olenko (2020); Ivanov and Leonenko (2012) that for the field X(s) in the examples above one can obtain not only SLLN, but also limit theorems about the convergence of distributions. Namely, the following result holds true.…”

“…Theorem 3. Alodat and Olenko (2020) Let a function gðsÞ, s 2 R d , satisfy the condition l 2dÀb 0 k g 2 ðl Á 1 d ÞL k ðlÞ ! 1, when l !…”