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1

Weighted Laws of Large Numbers for A Class of Independent Summands

Pakes Anthony G.

Probability and Mathematical Statistics

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2021

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Vol. 41, Fasc. 1

55--75

EN

This paper obtains a necessary and sufficient condition for a weak law of large numbers for weighted averages of positive-valued independent random variables whose distributions belong to a class which includes the Fα-scheme of record theory. Additional general conditions are found under which the weak law extends to a strong law with the same norming. Examples show these conditions can be fulfilled, and that if they are not, then the weighted averages exhibit multiple growth rates.

2

J1 convergence of partial sum processes with a reduced number of jumps

Krizmanić D.

Probability and Mathematical Statistics

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2015

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Vol. 35, Fasc. 1

107--128

EN

Various functional limit theorems for partial sum processes of strictly stationary sequences of regularly varying random variables in the space of càdlàg functions D[0, 1] with one of the Skorokhod topologies have already been obtained. The mostly used Skorokhod J1 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, Skorokhod M1 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 allows us to obtain the functional J1 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.

3

Extremes of moving averages and moving maxima on a regular lattice

Basrak B., Tafro A.

Probability and Mathematical Statistics

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2014

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Vol. 34, Fasc. 1

61--79

EN

We study the extremal behaviour of spatial moving averages and moving maxima on a regular discrete grid. Our main assumption is that these random fields are stationary and regularly varying with the tail index α > 0. Using the asymptotic theory for point processes we characterise the limiting behaviour of their extremes over an increasing grid. Our approach builds on the results of Davis and Resnick concerning linear processes. By analogy to the analysis of time series data, an appropriate Hill estimator of the tail index can be defined.We exhibit a sufficient condition for the consistency of this estimator in a certain class of spatial lattice models. Finally, we show that this condition holds for the models in our title.

4

Stable probability distributions and their domains of attraction : a direct approach

Geluk J. L., de Haan L.

Probability and Mathematical Statistics

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2000

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Vol. 20, Fasc. 1

169--188

EN

The theory of stable probability distributions and their domains of attraction is derived in a direct way (avoiding the usual route via infinitely divisible distributions) using Fourier transforms. Regularly varying functions play an important role in the exposition.

5

Multivariate large deviations with stable limit laws

Zaigraev A.

Probability and Mathematical Statistics

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1999

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Vol. 19, Fasc. 2

323--335

EN

The large deviation problem for sums of i.i.d. random vectors is considered. It is assumed that the underlying distribution is absolutely continuous and its density is of regular variation. An asymptotic expression for the probability of large deviations is established in the case of a non-normal stable limit law. The role of the maximal summand is also emphasized.

6

Multivariable regular variation of functions and measures

Meerschaert M. M., Scheffler H.-P.

Journal of Applied Analysis

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1999

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Vol. 5, nr 1

125-146

EN

Regular variation is an asymptotic property of functions and measures. The one variable theory is well-established, and has found numerous applications in both pure and applied mathematics. In this paper we present several new results on mul-tivariable regular variation for functions and measures.