arXiv subjects
Publications and source records attributed to Danijel Krizmanic.
For a stationary sequence of random variables we derive a self-normalized functional limit theorem under joint regular variation with index $α\in (0,2)$ and weak dependence conditions. The convergence takes place in the space of real-valued cadlag functions on $[0,1]$ with the Skorokhod $M_{1}$ topology.
In this article we derive a self-normalized functional limit theorem for strictly stationary linear processes with i.i.d. heavy-tailed innovations and random coefficients under the condition that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series. The convergence takes part in the space of càdlàg functions on $[0,1]$ with the Skorokhod $M_{2}$ topology.
We derive functional convergence of the partial maxima stochastic processes of multivariate linear processes with weakly dependent heavy-tailed innovations and random coefficients. The convergence takes place in the space of $\mathbb{R}^{d}$--valued càdlàg functions on $[0,1]$ endowed with the weak Skorokhod $M_{1}$ topology. We also show that this topology in general can not be replaced by the standard (or strong) $M_{1}$ topology.
For moving average processes with random coefficients and heavy-tailed innovations that are weakly dependent in the sense of strong mixing and local dependence condition $D'$ we study joint functional convergence of partial sums and maxima. Under the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series we derive a functional limit theorem in the space of $\mathbb{R}^{2}$-valued càdlàg functions on $[0, 1]$ with the Skorokhod weak $M_{2}$ topology.
For a strictly stationary sequence of random variables we derive functional convergence of the joint partial sum and partial maxima process under joint regular variation with index $α\in (0,2)$ and weak dependence conditions. The limiting process consists of an $α$--stable Lévy process and an extremal process. We also describe the dependence between these two components of the limit. The convergence takes place in the space of $\mathbb{R}^{2}$--valued càdlàg functions on $[0,1]$, with the Skorohod weak $M_{1}$ topology. We further show that this topology in general can not be replaced by the stronger (standard) $M_{1}$ topology.
For linear processes with independent identically distributed innovations that are regularly varying with tail index $α\in (0, 2)$, we study functional convergence of the joint partial sum and partial maxima processes. We derive a functional limit theorem under certain assumptions on the coefficients of the linear processes which enable the functional convergence to hold in the space of $\mathbb{R}^{2}$--valued càdlàg functions on $[0, 1]$ with the Skorohod weak $M_{2}$ topology. Also a joint convergence in the $M_{2}$ topology on the first coordinate and in the $M_{1}$ topology on the second coordinate is obtained.
Recently, for the joint partial sum and partial maxima processes constructed from linear processes with independent identically distributed innovations that are regularly varying with tail index $α\in (0, 2)$, a functional limit theorem with the Skorohod weak $M_{2}$ topology has been obtained. In this paper we show that, if all the coefficients of the linear processes are of the same sign, the functional convergence holds in the stronger topology, i.e. in the Skorohod weak $M_{1}$ topology on the space of $\mathbb{R}^{2}$--valued càdlàg functions on $[0, 1]$.
Various functional limit theorems for partial sum processes of strictly stationary sequences of regularly varying random variables in the space of cadlag functions $D[0,1]$ with one of the Skorohod topologies have already been obtained. The mostly used Skorohod $J_{1}$ topology is inappropriate when clustering of large values of the partial sum processes occurs. When all extremes within each cluster of high-threshold excesses do not have the same sign, Skorohod $M_{1}$ topology also becomes inappropriate. In this paper we alter the definition of the partial sum process in order to shrink all extremes within each cluster to a single one, which allow us to obtain the functional $J_{1}$ convergence. We also show that this result can be applied to some standard time series models, including the GARCH(1,1) process and its squares, the stochastic volatility models and $m$-dependent sequences.