A functional limit theorem for dependent sequences with infinite variance stable limits

Abstract

Under an appropriate regular variation condition, the affinely normalized partial sums of a sequence of independent and identically distributed random variables converges weakly to a non-Gaussian stable random variable. A functional version of this is known to be true as well, the limit process being a stable Lévy process. The main result in the paper is that for a stationary, regularly varying sequence for which clusters of high-threshold excesses can be broken down into asymptotically independent blocks, the properly centered partial sum process still converges to a stable Lévy process. Due to clustering, the Lévy triple of the limit process can be different from the one in the independent case. The convergence takes place in the space of càdlàg functions endowed with Skorohod's M1 topology, the more usual J1 topology being inappropriate as the partial sum processes may exhibit rapid successions of jumps within temporal clusters of large values, collapsing in the limit to a single jump. The result rests on a new limit theorem for point processes which is of independent interest. The theory is applied to moving average processes, squared GARCH(1, 1) processes and stochastic volatility models. . This reprint differs from the original in pagination and typographic detail. 1 2 B. BASRAK, D. KRIZMANIĆ AND J. SEGERS tributional behavior of the D[0, 1] valued process V n (t) = a −1 n (S ⌊nt⌋ − ⌊nt⌋b n ), t ∈ [0, 1],

Some Remarks on Definitions of Memory for Stationary Random Processes and Fields

“…At the same time, it is necessary to note that there are many papers dealing with the convergence of partial sums of stationary sequences to an α-stable law, 0 < α < 2; see, for example, [3,6,16,17,18]. In [16], necessary and sufficient conditions are given, in [3], functional convergence in the space D[0, 1] with Skorokhod's M 1 topology is proved, but in all these mentioned papers (and in many others not mentioned here but dealing with stationary sequences with infinite variance), limit theorems are considered with normalization of the form A n = n 1/α L(n). This means that, according to Definition 5, only the case of zero memory is considered.…”

The cutoff phenomenon for random birth and death chains

“…Theorem 3.4 of [2] states that Theorem 13 (Functional Limit Theorem for Mixing Sequences). Let X n be a strictly stationary stochastic process which is jointly regularly varying with exponent 1.…”

“…To build Y (2) j step by step, begin by coupling Y (2) 0 to X 0 so that they are both at the same quantile in their respective distributions. That is, choose X 0 = x from its distribution, then set Y (2) 0 to the unique value a which satisfies…”

“…That is, choose X 0 = x from its distribution, then set Y (2) 0 to the unique value a which satisfies…”

Editorial: special issue on time series extremes

“…This example also indicates that serial and extremal dependence are two different concepts. The clustering of extremes appears also in the aforementioned point process methodology, where the typical limit for weakly dependent time series is a cluster Poisson process; see for example (Hsing et al 1988;Davis and Hsing 1995) and (Basrak et al 2012).…”