The limiting behaviour of the last exit time for sequences of independent, identically distributed random variables

J�rg H�sler

doi:10.1007/bf00533637

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

Journal of Applied Probability ◽

10.2307/3215211 ◽

1995 ◽

Vol 32 (4) ◽

pp. 972-981 ◽

Cited By ~ 9

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.

Download Full-text

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

Journal of Applied Probability ◽

10.1017/s0021900200103456 ◽

1995 ◽

Vol 32 (04) ◽

pp. 972-981 ◽

Cited By ~ 1

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.

Download Full-text

A Note on the Accuracy of Normal Approximation of Random Quantities

Calcutta Statistical Association Bulletin ◽

10.1177/00080683211013510 ◽

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.

Download Full-text