Renewal theory in a random environment

1994 ◽  
Vol 116 (1) ◽  
pp. 179-190
Author(s):  
Laurence A. Baxter ◽  
Linxiong Li
Keyword(s):  
Stochastic Process ◽  
Random Environment ◽  
Limit Theorems ◽  
Random Variables ◽  
Renewal Theory ◽  
Renewal Processes ◽  
Conditional Distributions ◽  
The Future ◽  
Standard Limit

AbstractA random environment is modelled by an arbitrary stochastic process, the future of which is described by a σ-algebra. Renewal processes and alternating renewal processes are defined in this environment by considering the conditional distributions of random variables generated by the processes with respect to the σ-algebra. Generalizations of several of the standard limit theorems of renewal theory are derived.

A central limit theorem by remainder term for renewal processes

1992 ◽  
Vol 24 (2) ◽  
pp. 267-287 ◽  
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Renewal Processes

Three different definitions of the renewal processes are considered. For each of them, a central limit theorem with a remainder term is proved. The random variables that form the renewal processes are independent but not necessarily identically distributed and do not have to be positive. The results obtained in this paper improve and extend the central limit theorems obtained by Ahmad (1981) and Niculescu and Omey (1985).

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

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 →∞.

On some characteristics of quality for systems operating in a random environment

2018 ◽  
Vol 233 (2) ◽  
pp. 257-267
Author(s):  
Ji Hwan Cha ◽  
Maxim Finkelstein
Keyword(s):  
Stochastic Process ◽  
Failure Rate ◽  
Random Environment ◽  
Process Performance ◽  
Output Function ◽  
Quality Function ◽  
Shock Process ◽  
Monotonicity Properties ◽  
The Future

We consider systems that are operating in a random environment modeled by an external shock process. Performance of a system is characterized by a quality (output) function that is decreasing (due to degradation) in the absence of shocks. The important feature of our model is that shocks affect the failure rate of a system directly, and at the same time, each shock contributes to the additional decrease in the quality function forming the corresponding stochastic process. Expectations (unconditional and conditional on survival) and variability of this process are analyzed. Some monotonicity properties of the conditional quality function are discussed and expressions for the future values of this function are derived.

A central limit theorem by remainder term for renewal processes

1992 ◽  
Vol 24 (02) ◽  
pp. 267-287
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Renewal Processes

Three different definitions of the renewal processes are considered. For each of them, a central limit theorem with a remainder term is proved. The random variables that form the renewal processes are independent but not necessarily identically distributed and do not have to be positive. The results obtained in this paper improve and extend the central limit theorems obtained by Ahmad (1981) and Niculescu and Omey (1985).

scholarly journals On renewal theory, counter problems, and quasi-Poisson processes

1957 ◽  
Vol 53 (1) ◽  
pp. 175-193 ◽  
Author(s):  
Walter L. Smith
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Renewal Theory ◽  
Special Property ◽  
Poisson Processes ◽  
Partial Sums ◽  
Renewal Processes ◽  
Negative Exponential Distribution ◽  
Negative Exponential ◽  
Image Position

The power and appropriateness of renewal theory as a tool for the solution of general problems concerning counters has been amply demonstrated by Feller (7), who considered a variety of counter problems and reduced them to special renewal processes. The use of what may be called renewal-type arguments had certainly been made by authors other than Feller (e.g. in § 3 of Domb (3)), but it was only in (7) that the simplicity of the renewal approach to counter problems was recognized and systematically applied. More recently, Hammersley (8) was concerned with the generalization of a counter problem previously studied by Domb (2). This problem may be introduced, mathematically, as follows. Let {xi}, {yi} be two independent sequences of independent non-negative random variables which are non-zero with probability one (i.e. two independent renewal processes). The {xi}, are distributed in a negative-exponential distribution with mean λ-1, and we write Eλ for their distribution function and say ≡ {xi} is a Poisson process to imply this special property of ; the {yi} have a distribution function ‡ B(x) with mean b1 ≤ ∞. Form the partial sums and define ni to be the greatest integer k such that Xk ≥ t, taking X0 0 and nt = 0 if x1 > t. Then define the stochastic processHammersley'sx counter problem concerns the stochastic process

Representations and limit theorems for extreme value distributions

1978 ◽  
Vol 15 (3) ◽  
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

Let Xn1 ≦ Xn2 ≦ ··· ≦ Xnn denote the order statistics from a sample of n independent, identically distributed random variables, and suppose that the variables Xnn, Xn, n–1, ···, when suitably normalized, have a non-trivial limiting joint distribution ξ1, ξ2, ···, as n → ∞. 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 for ξ n as n →∞.

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