The extremal index and clustering of high values for derived stationary sequences

1995 ◽  
Vol 32 (4) ◽  
pp. 972-981 ◽  
Author(s):  
Ishay Weissman ◽  
Uri Cohen
Keyword(s):  
Random Variables ◽  
Extremal Index ◽  
Stationary Sequence ◽  
Stationary Sequences ◽  
Limiting Behaviour ◽  
Maximum Sequence ◽  
Independent Identically Distributed ◽  
Pointwise Maximum

Given a sequence of independent identically distributed random variables, we derive a moving-maximum sequence (with random translations). The extremal index of the derived sequence is computed and the limiting behaviour of clusters of high values is studied. We are then given two or more independent stationary sequences whose extremal indices are known. We derive a new stationary sequence by taking either a pointwise maximum or by a mixture of the original sequences. In each case, we compute the extremal index of the derived sequence.

The extremal index and clustering of high values for derived stationary sequences

1995 ◽  
Vol 32 (04) ◽  
pp. 972-981 ◽  
Author(s):  
Ishay Weissman ◽  
Uri Cohen
Keyword(s):  
Random Variables ◽  
Extremal Index ◽  
Stationary Sequence ◽  
Stationary Sequences ◽  
Limiting Behaviour ◽  
Maximum Sequence ◽  
Independent Identically Distributed ◽  
Pointwise Maximum

Given a sequence of independent identically distributed random variables, we derive a moving-maximum sequence (with random translations). The extremal index of the derived sequence is computed and the limiting behaviour of clusters of high values is studied. We are then given two or more independent stationary sequences whose extremal indices are known. We derive a new stationary sequence by taking either a pointwise maximum or by a mixture of the original sequences. In each case, we compute the extremal index of the derived sequence.

scholarly journals Asymptotic Conditional Distribution of Exceedance Counts

2012 ◽  
Vol 44 (01) ◽  
pp. 270-291 ◽  
Author(s):  
Michael Falk ◽  
Diana Tichy
Keyword(s):  
Asymptotic Distribution ◽  
Conditional Distribution ◽  
Random Variables ◽  
Extremal Index ◽  
Stationary Sequence ◽  
Independent Random Variables ◽  
Fragility Index ◽  
Expected Number

We investigate the asymptotic distribution of the number of exceedances among d identically distributed but not necessarily independent random variables (RVs) above a sequence of increasing thresholds, conditional on the assumption that there is at least one exceedance. Our results enable the computation of the fragility index, which represents the expected number of exceedances, given that there is at least one exceedance. Computed from the first d RVs of a strictly stationary sequence, we show that, under appropriate conditions, the reciprocal of the fragility index converges to the extremal index corresponding to the stationary sequence as d increases.

scholarly journals Asymptotic Conditional Distribution of Exceedance Counts

2012 ◽  
Vol 44 (1) ◽  
pp. 270-291 ◽  
Author(s):  
Michael Falk ◽  
Diana Tichy
Keyword(s):  
Asymptotic Distribution ◽  
Conditional Distribution ◽  
Random Variables ◽  
Extremal Index ◽  
Stationary Sequence ◽  
Independent Random Variables ◽  
Fragility Index ◽  
Expected Number

We investigate the asymptotic distribution of the number of exceedances amongdidentically distributed but not necessarily independent random variables (RVs) above a sequence of increasing thresholds, conditional on the assumption that there is at least one exceedance. Our results enable the computation of thefragility index, which represents the expected number of exceedances, given that there is at least one exceedance. Computed from the firstdRVs of a strictly stationary sequence, we show that, under appropriate conditions, the reciprocal of the fragility index converges to the extremal index corresponding to the stationary sequence asdincreases.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

scholarly journals A counterexample to the existence of a general central limit theorem for pairwise independent identically distributed random variables

2021 ◽  
Vol 499 (1) ◽  
pp. 124982
Author(s):  
Benjamin Avanzi ◽  
Guillaume Boglioni Beaulieu ◽  
Pierre Lafaye de Micheaux ◽  
Frédéric Ouimet ◽  
Bernard Wong
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Independent Identically Distributed
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