On Functional Limit Theorems for a Class of Stochastic Processes Indexed by Pseudo-Metric Parameter Spaces (with applications to empirical processes)

1990 ◽  
pp. 82-100
Author(s):  
Peter Gaenssler ◽  
Wilhelm Schneemeier
Keyword(s):  
Stochastic Processes ◽  
Limit Theorems ◽  
Empirical Processes ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Parameter Spaces

Functional limit theorems for stochastic processes based on embedded processes

1975 ◽  
Vol 7 (01) ◽  
pp. 123-139 ◽  
Author(s):  
Richard F. Serfozo
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Limit Theorems ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Limit Laws ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Subordinated Processes ◽  
Analogous Functional

The techniques used by Doeblin and Chung to obtain ordinary limit laws (central limit laws, weak and strong laws of large numbers, and laws of the iterated logarithm) for Markov chains, are extended to obtain analogous functional limit laws for stochastic processes which have embedded processes satisfying these laws. More generally, it is shown how functional limit laws of a stochastic process are related to those of a process embedded in it. The results herein unify and extend many existing limit laws for Markov, semi-Markov, queueing, regenerative, semi-stationary, and subordinated processes.

Functional limit theorems for stochastic processes based on embedded processes

1975 ◽  
Vol 7 (1) ◽  
pp. 123-139 ◽  
Author(s):  
Richard F. Serfozo
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Limit Theorems ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Limit Laws ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
Subordinated Processes ◽  
Analogous Functional

The techniques used by Doeblin and Chung to obtain ordinary limit laws (central limit laws, weak and strong laws of large numbers, and laws of the iterated logarithm) for Markov chains, are extended to obtain analogous functional limit laws for stochastic processes which have embedded processes satisfying these laws. More generally, it is shown how functional limit laws of a stochastic process are related to those of a process embedded in it. The results herein unify and extend many existing limit laws for Markov, semi-Markov, queueing, regenerative, semi-stationary, and subordinated processes.

Embedded renewal processes in the GI/G/s queue

1972 ◽  
Vol 9 (3) ◽  
pp. 650-658 ◽  
Author(s):  
Ward Whitt
Keyword(s):  
Limit Theorems ◽  
Sufficient Conditions ◽  
Interarrival Time ◽  
Renewal Processes ◽  
Positive Probability ◽  
Functional Limit ◽  
Light Traffic ◽  
Functional Limit Theorems ◽  
Cumulative Processes ◽  
Rule Out

The stable GI/G/s queue (ρ < 1) is sometimes studied using the “fact” that epochs just prior to an arrival when all servers are idle constitute an embedded persistent renewal process. This is true for the GI/G/1 queue, but a simple GI/G/2 example is given here with all interarrival time and service time moments finite and ρ < 1 in which, not only does the system fail to be empty ever with some positive probability, but it is never empty. Sufficient conditions are then given to rule out such examples. Implications of embedded persistent renewal processes in the GI/G/1 and GI/G/s queues are discussed. For example, functional limit theorems for time-average or cumulative processes associated with a large class of GI/G/s queues in light traffic are implied.

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.

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