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.

scholarly journals Functional limit theorems for the euler characteristic process in the critical regime

2021 ◽  
Vol 53 (1) ◽  
pp. 57-80
Author(s):  
Andrew M. Thomas ◽  
Takashi Owada
Keyword(s):  
Euler Characteristic ◽  
Limit Theorems ◽  
Law Of Large Numbers ◽  
Critical Regime ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Strong Law ◽  
Large Numbers ◽  
Nontrivial Topology ◽  
Characteristic Process

AbstractThis study presents functional limit theorems for the Euler characteristic of Vietoris–Rips complexes. The points are drawn from a nonhomogeneous Poisson process on $\mathbb{R}^d$ , and the connectivity radius governing the formation of simplices is taken as a function of the time parameter t, which allows us to treat the Euler characteristic as a stochastic process. The setting in which this takes place is that of the critical regime, in which the simplicial complexes are highly connected and have nontrivial topology. We establish two ‘functional-level’ limit theorems, a strong law of large numbers and a central limit theorem, for the appropriately normalized Euler characteristic process.

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.

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