Limit theorems for extremes with random sample size

1998 ◽  
Vol 30 (3) ◽  
pp. 777-806 ◽  
Author(s):  
Dmitrii S. Silvestrov ◽  
Jozef L. Teugels
Keyword(s):  
Weak Convergence ◽  
Sample Size ◽  
Random Sample ◽  
Limit Theorems ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Random Sample Size ◽  
Extremal Processes ◽  
Convergence Type

This paper is devoted to the investigation of limit theorems for extremes with random sample size under general dependence-independence conditions for samples and random sample size indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems for extremal processes with random sample size indexes.

Limit theorems for extremes with random sample size

1998 ◽  
Vol 30 (03) ◽  
pp. 777-806 ◽  
Author(s):  
Dmitrii S. Silvestrov ◽  
Jozef L. Teugels
Keyword(s):  
Weak Convergence ◽  
Sample Size ◽  
Random Sample ◽  
Limit Theorems ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Random Sample Size ◽  
Extremal Processes ◽  
Convergence Type

This paper is devoted to the investigation of limit theorems for extremes with random sample size under general dependence-independence conditions for samples and random sample size indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems for extremal processes with random sample size indexes.

scholarly journals On weak convergence of empirical processes for random number of independent stochastic vectors

1973 ◽  
Vol 73 (1) ◽  
pp. 139-144 ◽  
Author(s):  
Pranab Kumar Sen
Keyword(s):  
Weak Convergence ◽  
Sample Size ◽  
Random Sample ◽  
Random Number ◽  
Empirical Process ◽  
Empirical Processes ◽  
Random Sample Size ◽  
Martingale Property

AbstractBy the use of a semi-martingale property of the Kolmogorov supremum, the results of Pyke (6) on the weak convergence of the empirical process with random sample size are simplified and extended to the case of p(≥1)-dimensional stochastic vectors.

On maximum family size in branching processes

1999 ◽  
Vol 36 (3) ◽  
pp. 632-643 ◽  
Author(s):  
Ibrahim Rahimov ◽  
George P. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Sample Size ◽  
Random Sample ◽  
Family Size ◽  
Limit Theorems ◽  
Branching Processes ◽  
Random Sample Size ◽  
Convergence Results ◽  
Watson Process ◽  
Supercritical Branching Processes

The number Yn of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Yn as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Yn and EYn provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

On maximum family size in branching processes

1999 ◽  
Vol 36 (03) ◽  
pp. 632-643 ◽  
Author(s):  
Ibrahim Rahimov ◽  
George P. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Sample Size ◽  
Random Sample ◽  
Family Size ◽  
Limit Theorems ◽  
Branching Processes ◽  
Random Sample Size ◽  
Convergence Results ◽  
Watson Process ◽  
Supercritical Branching Processes

The number Y n of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Y n as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Y n and EY n provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

Weak convergence in an appointment system

1975 ◽  
Vol 12 (1) ◽  
pp. 188-194 ◽  
Author(s):  
C. W. Anderson
Keyword(s):  
Weak Convergence ◽  
Convergence Theorem ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Service Facility ◽  
Appointment System ◽  
Weak Convergence Theorem ◽  
Carry Over

It is assumed that customers at a service facility have appointments at times 0,1,2, … for which they may be unpunctual by random amounts or may never arrive at all. A weak convergence theorem is proved for the process which counts the number of arrivals. This makes it possible to carry over the results of Iglehart and Whitt (1970a) to obtain heavy traffic functional limit theorems for queues with arrivals by appointment.

Functional limit theorems for the number of busy servers in a G/G/∞ queue

2018 ◽  
Vol 55 (1) ◽  
pp. 15-29 ◽  
Author(s):  
Alexander Iksanov ◽  
Wissem Jedidi ◽  
Fethi Bouzeffour
Keyword(s):  
Weak Convergence ◽  
Integral Representation ◽  
Gaussian Process ◽  
Limit Theorems ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Open Problems ◽  
Skorokhod Space

Abstract We discuss weak convergence of the number of busy servers in a G/G/∞ queue in the J1-topology on the Skorokhod space. We prove two functional limit theorems with random and nonrandom centering, thereby solving two open problems stated in Mikosch and Resnick (2006). A new integral representation for the limit Gaussian process is given.

Multi-server assembly queues

1974 ◽  
Vol 11 (03) ◽  
pp. 629-632
Author(s):  
Michael A. Crane
Keyword(s):  
Weak Convergence ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Queue Size ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Random Functions ◽  
Wiener Processes ◽  
Multi Server

A model is studied in which each of several servers assembles finished products consisting of N different input items. Items of each type arrive independently at the assembly station and are grouped into N-tuples consisting of one item of each type. N-tuples are assembled into finished products by the servers on a first come-first-served basis. The model is analyzed by means of the theory of weak convergence, and functional limit theorems are obtained for appropriately normalized random functions induced by the queue size processes. The limits are expressed as functionals of multi-dimensional Wiener processes, with ordinary central limit theorems obtained as corollaries.

Multi-server assembly queues

1974 ◽  
Vol 11 (3) ◽  
pp. 629-632 ◽  
Author(s):  
Michael A. Crane
Keyword(s):  
Weak Convergence ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Queue Size ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Random Functions ◽  
Wiener Processes ◽  
Multi Server

A model is studied in which each of several servers assembles finished products consisting of N different input items. Items of each type arrive independently at the assembly station and are grouped into N-tuples consisting of one item of each type. N-tuples are assembled into finished products by the servers on a first come-first-served basis. The model is analyzed by means of the theory of weak convergence, and functional limit theorems are obtained for appropriately normalized random functions induced by the queue size processes. The limits are expressed as functionals of multi-dimensional Wiener processes, with ordinary central limit theorems obtained as corollaries.

Weak convergence in an appointment system

1975 ◽  
Vol 12 (01) ◽  
pp. 188-194
Author(s):  
C. W. Anderson
Keyword(s):  
Weak Convergence ◽  
Convergence Theorem ◽  
Limit Theorems ◽  
Heavy Traffic ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Service Facility ◽  
Appointment System ◽  
Weak Convergence Theorem ◽  
Carry Over

It is assumed that customers at a service facility have appointments at times 0,1,2, … for which they may be unpunctual by random amounts or may never arrive at all. A weak convergence theorem is proved for the process which counts the number of arrivals. This makes it possible to carry over the results of Iglehart and Whitt (1970a) to obtain heavy traffic functional limit theorems for queues with arrivals by appointment.

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