Searcharxiv⌕ Search

arXiv subjects

Rafał Kulik

Publications and source records attributed to Rafał Kulik.

12 recordsLinked to original sources
arXiv2024

A remarkable example on clustering of extremes for regularly-varying stochastic processes

The stable-regenerative multiple-stable model has been shown recently to have distinct candidate extremal index and extremal index. To understand further this rare phenomenon, two more results are established here for the double-stable model. The first is the convergence of point processes for the clusters of extremes, enhancing the previous result on the weak convergence of random sup-measures. Most interestingly, the second result reveals a new phase transition at the mesoscopic level when computing the asymptotic exceedance probability over a block, $\mathbb P(\max_{k=1,\dots,d_n} X_k>b_n)$, as $n\to\infty$. Here, the mesoscopic level is referred to the fact that the block size $d_n$ is allowed to grow at the rate $n^ρ$ with $ρ\in[0,1]$, while the threshold $b_n$ is such that $\mathbb P(X_1>b_n)\sim 1/n$. The recently discovered discrepancy between the candidate extremal index and the extremal index is shown to be just a reflection of this phase transition that is prohibited by the anticlustering condition.

math.PR↗
arXiv2023

Asymptotic expansions for blocks estimators: PoT framework

We consider disjoint and sliding blocks estimators of cluster indices for multivariate, regularly varying time series in the Peak-over-Threshold framework. We aim to provide a complete description of the limiting behaviour of these estimators. This is achieved by a precise expansion for the difference between the sliding and the disjoint blocks statistics. The rates in the expansion stem from internal clusters and boundary clusters. To obtain these rates we need to extend the existing results on vague convergence of cluster measures. We reveal dichotomous behaviour between small blocks and large blocks scenario.

math.ST↗
arXiv2020

Change-point tests for the tail parameter of Long Memory Stochastic Volatility time series

We consider a change-point test based on the Hill estimator to test for structural changes in the tail index of Long Memory Stochastic Volatility time series. In order to determine the asymptotic distribution of the corresponding test statistic, we prove a uniform reduction principle for the tail empirical process in a two-parameter Skorohod space. It is shown that such a process displays a dichotomous behavior according to an interplay between the Hurst parameter, i.e., a parameter characterizing the dependence in the data, and the tail index. Our theoretical results are accompanied by simulation studies and the analysis of financial time series with regard to structural changes in the tail index.

math.ST↗
arXiv2017

Testing for Change in Stochastic Volatility with Long Range Dependence

In this paper, change-point problems for long memory stochastic volatility models are considered. A general testing problem which includes various alternative hypotheses is discussed. Under the hypothesis of stationarity the limiting behavior of CUSUM- and Wilcoxon-type test statistics is derived. In this context, a limit theorem for the two-parameter empirical process of long memory stochastic volatility time series is proved. In particular, it is shown that the asymptotic distribution of CUSUM test statistics may not be affected by long memory, unlike Wilcoxon test statistics which are typically influenced by long range dependence. To avoid the estimation of nuisance parameters in applications, the usage of self-normalized test statistics is proposed. The theoretical results are accompanied by simulation studies which characterize the finite sample behavior of the considered testing procedures when testing for changes in mean, in variance, and in the tail index.

math.ST↗
arXiv2016

The accessibility of convex bodies and derandomization of the hit and run algorithm

We introduce the concept of accessibility and prove that any convex body $X$ in $\mathbb R^d$ is accessible with relevant constants depending on $d$ only. This property leads to a new algorithm which may be considered as a natural derandomization of the hit and run algorithm applied to generate a sequence of random points covering $X$ uniformly. We prove stability of the Markov chain generated by the proposed algorithm and provide its rate of convergence.

math.PR↗
arXiv2015

Multivariate Tail Estimation: Conditioning on an extreme event

We consider regularly varying random vectors. Our goal is to estimate in a non-parametric way some characteristics related to conditioning on an extreme event, like the tail dependence coefficient. We introduce a quasi-spectral decomposition that allow to improve efficiency of estimators. Asymptotic normality of estimators is based on weak convergence of tail empirical processes. Theoretical results are supported by simulation studies.

stat.ME↗
arXiv2012

Heavy tailed branching process with immigration

In this paper we analyze a branching process with immigration defined recursively by $X_t=θ_t\circ X_{t-1}+B_t$ for a sequence $(B_t)$ of i.i.d. random variables and random mappings $ θ_t\circ x:=θ_t(x)=\sum_{i=1}^xA_i^{(t)}, $ with $(A_i^{(t)})_{i\in \mathbb{N}_0}$ being a sequence of $\mathbb{N}_0$-valued i.i.d. random variables independent of $B_t$. We assume that one of generic variables $A$ and $B$ has a regularly varying tail distribution. We identify the tail behaviour of the distribution of the stationary solution $X_t$. We also prove CLT for the partial sums that could be further generalized to FCLT. Finally, we also show that partial maxima have a Fréchet limiting distribution.

math.PR↗
arXiv2011

Estimation of limiting conditional distributions for the heavy tailed long memory stochastic volatility process

We consider Stochastic Volatility processes with heavy tails and possible long memory in volatility. We study the limiting conditional distribution of future events given that some present or past event was extreme (i.e. above a level which tends to infinity). Even though extremes of stochastic volatility processes are asymptotically independent (in the sense of extreme value theory), these limiting conditional distributions differ from the i.i.d. case. We introduce estimators of these limiting conditional distributions and study their asymptotic properties. If volatility has long memory, then the rate of convergence and the limiting distribution of the centered estimators can depend on the long memory parameter (Hurst index).

math.ST↗
arXiv2009

Wavelet regression in random design with heteroscedastic dependent errors

We investigate function estimation in nonparametric regression models with random design and heteroscedastic correlated noise. Adaptive properties of warped wavelet nonlinear approximations are studied over a wide range of Besov scales, $f\in\mathcal{B}^s_{π,r}$, and for a variety of $L^p$ error measures. We consider error distributions with Long-Range-Dependence parameter $α,0<α\leq1$; heteroscedasticity is modeled with a design dependent function $σ$. We prescribe a tuning paradigm, under which warped wavelet estimation achieves partial or full adaptivity results with the rates that are shown to be the minimax rates of convergence. For $p>2$, it is seen that there are three rate phases, namely the dense, sparse and long range dependence phase, depending on the relative values of $s,p,π$ and $α$. Furthermore, we show that long range dependence does not come into play for shape estimation $f-\int f$. The theory is illustrated with some numerical examples.

math.ST↗
arXiv2008

Nonparametric deconvolution problem for dependent sequences

We consider the nonparametric estimation of the density function of weakly and strongly dependent processes with noisy observations. We show that in the ordinary smooth case the optimal bandwidth choice can be influenced by long range dependence, as opposite to the standard case, when no noise is present. In particular, if the dependence is moderate the bandwidth, the rates of mean-square convergence and, additionally, central limit theorem are the same as in the i.i.d. case. If the dependence is strong enough, then the bandwidth choice is influenced by the strength of dependence, which is different when compared to the non-noisy case. Also, central limit theorem are influenced by the strength of dependence. On the other hand, if the density is supersmooth, then long range dependence has no effect at all on the optimal bandwidth choice.

math.ST↗