arXiv subjects

Alfredas Račkauskas

Publications and source records attributed to Alfredas Račkauskas.

7 recordsLinked to original sources

arXiv2019

Convergence of U-Processes in Hölder Spaces with Application to Robust Detection of a Changed Segment

To detect a changed segment (so called epidemic changes) in a time series, variants of the CUSUM statistic are frequently used. However, they are sensitive to outliers in the data and do not perform well for heavy tailed data, especially when short segments get a high weight in the test statistic. We will present a robust test statistic for epidemic changes based on the Wilcoxon statistic. To study their asymptotic behavior, we prove functional limit theorems for U-processes in Hölder spaces. We also study the finite sample behavior via simulations and apply the statistic to a real data example.

math.ST↗

arXiv2019

Uniform asymptotic normality of weighted sums of short-memory linear processes

Let $X_1, X_2,\dots$ be a short-memory linear process of random variables. For $1\leq q<2$, let $\cF$ be a bounded set of real-valued functions on $[0,1]$ with finite $q$-variation. It is proved that $\{n^{-1/2}\sum_{i=1}^nX_if(i/n)\colon\,f\in\cF\}$ converges in outer distribution in the Banach space of bounded functions on $\cF$ as $n\to\infty$. Several applications to a regression model and a multiple change point model are given.

math.PR↗

arXiv2016

Operator self-similar processes and functional central limit theorems

Let $\{X_k:k\ge1\}$ be a linear process with values in the separable Hilbert space $L_2(μ)$ 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 $f\in L_2(μ)$ with $d:\mathbb S\to\mathbb R$ and $\{\varepsilon_k:k\in\mathbb Z\}$ are independent and identically distributed $L_2(μ)$-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.

math.PR↗

arXiv2016

Weak law of large numbers for linear processes

We establish sufficient conditions for the Marcinkiewicz-Zygmund type weak law of large numbers for a linear process $\{X_k:k\in\mathbb Z\}$ defined by $X_k=\sum_{j=0}^\inftyψ_j\varepsilon_{k-j}$ for $k\in\mathbb Z$, where $\{ψ_j:j\in\mathbb Z\}\subset\mathbb R$ and $\{\varepsilon_k:k\in\mathbb Z\}$ are independent and identically distributed random variables such that $x^p\Pr\{|\varepsilon_0|>x\}\to0$ as $x\to\infty$ with $1<p<2$ and $\operatorname E\varepsilon_0=0$. We use an abstract norming sequence that does not grow faster than $n^{1/p}$ if $\sum|ψ_j|<\infty$. If $\sum|ψ_j|=\infty$, the abstract norming sequence might grow faster than $n^{1/p}$ as we illustrate with an example. Also, we investigate the rate of convergence in the Marcinkiewicz-Zygmund type weak law of large numbers for the linear process.

math.PR↗

arXiv2015

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

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.

math.PR↗

arXiv2014

Testing epidemic change in nearly nonstationary process with statistics based on residuals

We study an epidemic type change in innovations of a first order autoregressive process $ y_{n,k} = φ_n y_{n,k-1} + ε_{k} + a_{n,k}$, where $ϕ_n$ is either a constant in $(-1,1)$ or a sequence in $(0,1)$, converging to 1. For $k$ inside some unknown interval $\mathbb{I}_n^\ast=(k^\ast,k^\ast+\ell^\ast]$, $a_{n,k}=a_n$ while $a_{n,k}=0$ for $k$ outside $\mathbb{I}_n^\ast$. When $a_n\neq 0$, we have an epidemic deviation from the usual (zero) mean of innovations. Since innovations are not observed, we build uniform increments statistics on residuals $(\widehatε_k)$ of the process $y_{n,k}$. We assume that innovations $(ε_k)$ are regularly varying with index $p \ge 2$ or satisfies integrability condition $\lim_{t \to \infty} t^p P(|ε_1| > t) = 0$ for $p > 2$ and $Eε_k^2 < \infty$ for $p=2$. We find the limit distributions of the tests under no change and prove consistency under short epidemics that is $\ell^\ast=O(n^β)$ for some $0<β\le 1/2$.

math.ST↗

arXiv2010

The limit distribution of the maximum increment of a random walk with regularly varying jump size distribution

In this paper, we deal with the asymptotic distribution of the maximum increment of a random walk with a regularly varying jump size distribution. This problem is motivated by a long-standing problem on change point detection for epidemic alternatives. It turns out that the limit distribution of the maximum increment of the random walk is one of the classical extreme value distributions, the Fréchet distribution. We prove the results in the general framework of point processes and for jump sizes taking values in a separable Banach space.

math.ST↗