Rate of convergence of the joint distribution of order statistics to the normal law

1983 ◽  
Vol 23 (3) ◽  
pp. 2275-2280 ◽  
Author(s):  
V. A. Egorov ◽  
V. B. Nevzorov
Keyword(s):  
Rate Of Convergence ◽  
Order Statistics ◽  
Joint Distribution ◽  
Normal Law

The rate of convergence of extremes of stationary normal sequences

1983 ◽  
Vol 15 (01) ◽  
pp. 54-80 ◽  
Author(s):  
Holger Rootzén
Keyword(s):  
Rate Of Convergence ◽  
Joint Distribution ◽  
Point Processes ◽  
Rates Of Convergence ◽  
Joint Distribution Function ◽  
Maximal Correlation ◽  
Standard Normal ◽  
Image Position ◽  
The Right ◽  
Right Order

Let {ξ; t = 1, 2, …} be a stationary normal sequence with zero means, unit variances, and covariances let be independent and standard normal, and write . In this paper we find bounds on which are roughly of the order where ρ is the maximal correlation, ρ =sup {0, r 1 , r 2, …}. It is further shown that, at least for m-dependent sequences, the bounds are of the right order and, in a simple example, the errors are evaluated numerically. Bounds of the same order on the rate of convergence of the point processes of exceedances of one or several levels are obtained using a ‘representation' approach (which seems to be of rather wide applicability). As corollaries we obtain rates of convergence of several functionals of the point processes, including the joint distribution function of the k largest values amongst ξ1, …, ξn.

Representations and limit theorems for extreme value distributions

1978 ◽  
Vol 15 (03) ◽  
pp. 639-644 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Stochastic Process ◽  
Order Statistics ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Canonical Representation ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Extreme Value Distributions ◽  
Independent Identically Distributed

LetXn1≦Xn2≦ ··· ≦Xnndenote the order statistics from a sample ofnindependent, identically distributed random variables, and suppose that the variablesXnn, Xn,n–1, ···, when suitably normalized, have a non-trivial limiting joint distributionξ1,ξ2, ···, asn → ∞. It is well known that the limiting distribution must be one of just three types. We provide a canonical representation of the stochastic process {ξn,n≧ 1} in terms of exponential variables, and use this representation to obtain limit theorems forξnasn →∞.

Joint exceedances of high levels under a local dependence condition

1993 ◽  
Vol 30 (01) ◽  
pp. 112-120
Author(s):  
Helena Ferreira
Keyword(s):  
Order Statistics ◽  
Long Range ◽  
Joint Distribution ◽  
Long Range Dependence ◽  
Local Dependence ◽  
Stationary Sequences ◽  
Compound Poisson ◽  
Dependence Conditions

Under appropriate long-range dependence conditions, it is well known that the joint distribution of the number of exceedances of several high levels is asymptotically compound Poisson. Here we investigate the structure of a cluster of exceedances for stationary sequences satisfying a suitable local dependence condition, under which it is only necessary to get certain limiting probabilities, easy to compute, in order to obtain limiting results for the highest order statistics, exceedance counts and upcrossing counts.

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