Rimas Norvaiša | Searcharxiv

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↗

arXiv2015

Weighted power variation of integrals with respect to a Gaussian process

We consider a stochastic process $Y$ defined by an integral in quadratic mean of a deterministic function $f$ with respect to a Gaussian process $X$, which need not have stationary increments. For a class of Gaussian processes $X$, it is proved that sums of properly weighted powers of increments of $Y$ over a sequence of partitions of a time interval converge almost surely. The conditions of this result are expressed in terms of the $p$-variation of the covariance function of $X$. In particular, the result holds when $X$ is a fractional Brownian motion, a subfractional Brownian motion and a bifractional Brownian motion.

math.PR↗