Operator self-similar processes and functional central limit theorems

Abstract

Let $\{X_k:k\ge1\}$ be a linear process with values in the separable Hilbert space $L_2(\mu)$ given by $X_k=\sum_{j=0}^\infty(j+1)^{-D}\varepsilon_{k-j}$ for each $k\ge1$, where $D$ is defined by $Df=\{d(s)f(s):s\in\mathbb S\}$ for each $f\in L_2(\mu)$ with $d:\mathbb S\to\mathbb R$ and $\{\varepsilon_k:k\in\mathbb Z\}$ are independent and identically distributed $L_2(\mu)$-valued random elements with $\operatorname E\varepsilon_0=0$ and $\operatorname E\|\varepsilon_0\|^2<\infty$. We establish sufficient conditions for the functional central limit theorem for $\{X_k:k\ge1\}$ when the series of operator norms $\sum_{j=0}^\infty\|(j+1)^{-D}\|$ diverges and show that the limit process generates an operator self-similar process.Comment: 22 page

Long-Range Dependent Curve Time Series

“…is a positive slowly varying function (c.f., Bingham et al 1987) which depends on i. Our limit distribution theory is comparable to theorems in Characiejus & Rauckauskas (2014) and Düker (2018) which consider Hilbert space-valued long-range dependent linear processes with derivation heavily relying on the theory of multiplication operator. However, it seems challenging to directly apply the methodology in Section 3 below to their model framework, making it difficult to achieve dimension reduction via FPCA.…”

“…This assumption is restrictive, but, motivated by Characiejus & Rauckauskas (2014) and Düker (2018), using the multiplication operator, it may be possible to extend (4.1) to…”

Limit theorems for Hilbert space-valued linear processes under long range dependence

“…, if a > 4, and the first assumption of the above corollary is fulfilled. and Račkauskas (2013and Račkauskas ( , 2014 investigate a central and a functional central limit theorem for a linear process (X k ) k∈Z in form of (1) with values in the real Hilbert space L 2 (µ) of square-integrable real-valued functions. We extend their result to the complex Hilbert space of square-integrable complex-valued functions denoted by L 2 (µ, C) with inner product…”

“…As in Characiejus and Račkauskas (2014) we get an operator self-similar process. Such processes were first introduced by Lamperti (1962) and play an important role in the context of long memory.…”

Local Whittle estimation of long‐range dependence for functional time series

“…The bulk of the literature studies functional data which are either independent or stationary SRD (e.g., Bosq, 2000;Ramsay and Silverman, 2005;Ferraty and Vieu, 2006;Hörmann and Kokoszka, 2010;Horváth and Kokoszka, 2012;Berkes et al, 2013;Hsing and Eubank, 2015). Li et al (2020) is among the first to extend the functional framework from SRD to LRD (see also Characiejus and Rackauskas, 2014;Düker, 2018). They not only establish the central limit theorem for a temporal sum of LRD functional observations, but also develop functional principal component analysis (FPCA) and estimate the memory parameter for the projected process via semiparametric R/S.…”