Allan Gut

Almost-Sure Results for a Class of Dependent Random Variables

The aim of this note is to establish almost-sure Marcinkiewicz-Zygmund type results for a class of random variables indexed by ℤd+ —the positive d-dimensional lattice points—and having maximal coefficient of correlation strictly smaller than 1. The class of applications include filters of certain Gaussian sequences and Markov processes.

Complete convergence for arrays

Let {(Xnk, 1≤k≤n),n≥1}, be an array of rowwise independent random variables. We extend and generalize some recent results due to Hu, Móricz and Taylor concerning complete convergence, in the sense of Hsu and Robbins, of the sequence of rowwise arithmetic means.

Extreme Shock Models

The standard assumptions in shock models are that the failure (of the system) is related either to the cumulative effect of a (large) number of shocks or that failure is caused by a shock which is larger than a certain critical level. The present paper is devoted to the second kind. Here the standard setting is that the shocks Xk, k ≥ 1, and the times between the shocks Yk, k ≥ 1, are independent, identically distributed random vectors (Xk, Yk), k ≥ 1. In particular, Xk and Yk may well be dependent (the typical case). The main object of interest is the time to failure, Tτ(t), where Tn = ∑k≤n Yk and τ(t) is the first exceedance time, viz. the first time that Xk > t. We derive moment relations and asymptotic distributions of Tτ(t) as t increases in such a way that P{X1} > t} tends to 0. A final section discusses some extensions; more general events of failure, the non-i.i.d. case, and point process convergence for a particular case.

Mixed shock models

Traditionally, shock models are of two kinds. The failure (of a system) is related either to the cumulative effect of a (large) number of shocks or it is caused by a shock which is larger than some critical level. The present paper is devoted to a mixed m

The weak law of large numbers for arrays

The laws of large numbers for sums of i.i.d. random variables can be generalized in various ways. The purpose of this note is to collect some domination conditions and to provide a fairly general weak law for arrays. AMS 1980 Subject Classifications: Primary: 60F05, 60F25, 60G42, 60G50

Complete convergence and Cesàro summation for i.i.d. random variables

SummaryVarious results generalizing summation methods for divergent series of real numbers to analogous results for independent, identically distributed random variables have appeared during the last two decades. The main result of this paper provides necessary and sufficient conditions for the complete convergence of the Cesàro means of i.i.d random variables.

Equivalences in strong limit theorems for renewal counting processes

A number of strong limit theorems for renewal counting processes, e.g. the strong law of large numbers, the Marcinkiewicz-Zygmund law of large numbers or the law of the iterated logarithm, can be derived from their corresponding counterparts for the underlying partial sums. In this paper, it is proved that these strong laws indeed hold simultaneously for both processes. As a byproduct it follows (i) that certain (moment) conditions are necessary and sufficient, and (ii) that the results, in fact, hold for (almost) arbitrary nonnegative summation processes. Renewal processes constructed from random walks with infinite expectation are studied, too, but results are essentially different from the case with linear drift.

Moments of the maximum of normed partial sums of random variables with multidimensional indices

SummaryFor a set of i.i.d. random variables indexed by the positive integer d-dimensional lattice points we give conditions for the existence of moments of the supremum of normed partial sums, thereby obtaining results related to the Kolmogorov-Marcinkiewicz strong law of large numbers and the law of the iterated logarithm.

Truncated Sequential Change‐point Detection based on Renewal Counting Processes

The typical approach in change-point theory is to perform the statistical analysis based on a sample of fixed size. Alternatively, one observes some random phenomenon sequentially and takes action as soon as one observes some statistically significant deviation from the “normal” behaviour. Based on the, perhaps, more realistic situation that the process can only be partially observed, we consider the counting process related to the original process observed at equidistant time points, after which action is taken or not depending on the number of observations between those time points. In order for the procedure to stop also when everything is in order, we introduce a fixed time horizon n at which we stop declaring “no change” if the observed data did not suggest any action until then. We propose some stopping rules and consider their asymptotics under the null hypothesis as well as under alternatives. The main basis for the proofs are strong invariance principles for renewal processes and extreme value asymptotics for Gaussian processes.

Anscombe’s theorem 60 years later

Abstract The point of departure of the present article is Anscombes seminal 1952 paper on limit theorems for randomly indexed processes. We discuss the importance of this result and mention some of its impact, mainly on stopped random walks. The main aim of the article is to illustrate the beauty and efficiency of what will be called the stopped random walk (SRW) method.