Some duality results for a class of multivariate semi-markov processes

1982 ◽  
Vol 19 (01) ◽  
pp. 90-98
Author(s):  
J. Janssen ◽  
J. M. Reinhard
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Processes ◽  
Queueing Theory ◽  
Random Variables ◽  
Classical Case ◽  
Risk Theory ◽  
Finite Markov Chain ◽  
Duality Results ◽  
Semi Markov Processes

The duality results well known for classical random walk and generalized by Janssen (1976) for (J-X) processes (or sequences of random variables defined on a finite Markov chain) are extended to a class of multivariate semi-Markov processes. Just as in the classical case, these duality results lead to connections between some models of risk theory and queueing theory.

Some duality results for a class of multivariate semi-markov processes

1982 ◽  
Vol 19 (1) ◽  
pp. 90-98 ◽  
Author(s):  
J. Janssen ◽  
J. M. Reinhard
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Processes ◽  
Queueing Theory ◽  
Random Variables ◽  
Classical Case ◽  
Risk Theory ◽  
Finite Markov Chain ◽  
Duality Results ◽  
Semi Markov Processes

The duality results well known for classical random walk and generalized by Janssen (1976) for (J-X) processes (or sequences of random variables defined on a finite Markov chain) are extended to a class of multivariate semi-Markov processes. Just as in the classical case, these duality results lead to connections between some models of risk theory and queueing theory.

First passage to a general threshold for a process corresponding to sampling at Poisson times

1971 ◽  
Vol 8 (03) ◽  
pp. 573-588 ◽  
Author(s):  
Barry Belkin
Keyword(s):  
Markov Processes ◽  
Classical Case ◽  
First Passage ◽  
Explicit Result ◽  
Threshold Functions ◽  
Linear Threshold Functions ◽  
Linear Threshold ◽  
Special Cases ◽  
Weiner Process ◽  
Semi Markov Processes

The problem of computing the distribution of the time of first passage to a constant threshold for special classes of stochastic processes has been the subject of considerable study. For example, Baxter and Donsker (1957) have considered the problem for processes with stationary, independent increments, Darling and Siegert (1953) for continuous Markov processes, Mehr and McFadden (1965) for Gauss-Markov processes, and Stone (1969) for semi-Markov processes. The results, however, generally express the first passage distribution in terms of transforms which can be inverted only in a relatively few special cases, such as in the classical case of the Weiner process and for certain stable and compound Poisson processes. For linear threshold functions and processes with non-negative interchangeable increments the first passage problem has been studied by Takács (1957) (an explicit result was obtained by Pyke (1959) in the special case of a Poisson process). Again in the case of a linear threshold, an explicit form for the first passage distribution was found by Slepian (1961) for the Weiner process. For the Ornstein-Uhlenbeck process and certain U-shaped thresholds the problem has recently been studied by Daniels (1969).

scholarly journals Asymptotic Probabilities of an Exceedance Over Renewal Thresholds with an Application to Risk Theory

2005 ◽  
Vol 42 (01) ◽  
pp. 153-162 ◽  
Author(s):  
Christian Y. Robert
Keyword(s):  
Random Walk ◽  
Stopping Time ◽  
Random Variables ◽  
Risk Theory ◽  
Heavy Tailed Distribution ◽  
Negative Drift ◽  
Heavy Tailed ◽  
Independent And Identically Distributed

Let (Y n , N n ) n≥1 be independent and identically distributed bivariate random variables such that the N n are positive with finite mean ν and the Y n have a common heavy-tailed distribution F. We consider the process (Z n ) n≥1 defined by Z n = Y n - Σ n-1, where It is shown that the probability that the maximum M = max n≥1 Z n exceeds x is approximately as x → ∞, where F' := 1 - F. Then we study the integrated tail of the maximum of a random walk with long-tailed increments and negative drift over the interval [0, σ], defined by some stopping time σ, in the case in which the randomly stopped sum is negative. Finally, an application to risk theory is considered.

Absorption Probabilities for Sums of Random Variables Defined on a Finite Markov Chain

1962 ◽  
Vol 58 (2) ◽  
pp. 286-298 ◽  
Author(s):  
H. D. Miller
Keyword(s):  
Markov Chain ◽  
Asymptotic Expression ◽  
Random Variables ◽  
Partial Sums ◽  
Finite Markov Chain ◽  
Initial State ◽  
Main Result Theorem ◽  
Sums Of Random Variables ◽  
Result Theorem ◽  
Image Position

SummaryThis paper is essentially a continuation of the previous one (5) and the notation established therein will be freely repeated. The sequence {ξr} of random variables is defined on a positively regular finite Markov chain {kr} as in (5) and the partial sums and are considered. Let ζn be the first positive ζr and let πjk(y), the ‘ruin’ function or absorption probability, be defined by The main result (Theorem 1) is an asymptotic expression for πjk(y) for large y in the case when , the expectation of ξ1 being computed under the unique stationary distribution for k0, the initial state of the chain, and unconditional on k1.

scholarly journals On the Study of Transience and Recurrence of the Markov Chain Defined by Directed Weighted Circuits Associated with a Random Walk in Fixed Environment

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Chrysoula Ganatsiou
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Representation Theory ◽  
Markov Processes ◽  
Directed Cycles

By using the cycle representation theory of Markov processes, we investigate proper criterions regarding transience and recurrence of the corresponding Markov chain represented uniquely by directed cycles (especially by directed circuits) and weights of a random walk with jumps in a fixed environment.

Stability of maxima of random variables defined on a Markov chain

1972 ◽  
Vol 4 (2) ◽  
pp. 285-295 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Regularity Condition ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Finite Product ◽  
Normalizing Constants ◽  
Necessary And Sufficient

Consider maxima Mn of a sequence of random variables defined on a finite Markov chain. Necessary and sufficient conditions for the existence of normalizing constants Bn such that are given. The problem can be reduced to studying maxima of i.i.d. random variables drawn from a finite product of distributions πi=1mHi(x). The effect of each factor Hi(x) on the behavior of maxima from πi=1mHi is analyzed. Under a mild regularity condition, Bn can be chosen to be the maximum of the m quantiles of order (1 - n-1) of the H's.

Truncation approximation of the limit probabilities for denumerable semi-Markov processes

1975 ◽  
Vol 12 (1) ◽  
pp. 161-163 ◽  
Author(s):  
Richard L. Tweedie
Keyword(s):  
Markov Chain ◽  
Markov Processes ◽  
Transition Matrix ◽  
Embedded Markov Chain ◽  
Semi Markov Processes ◽  
Approximate Limit

It is shown that methods used by the author to approximate limit probabilities for Markov processes from their Q-matrices extend to semi-Markov processes. The limit probabilities for semi-Markov processes can be approximated using only truncations of the embedded Markov chain transition matrix and the vector of mean holding times.

Stability of maxima of random variables defined on a Markov chain

1972 ◽  
Vol 4 (02) ◽  
pp. 285-295 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Regularity Condition ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Finite Product ◽  
Normalizing Constants ◽  
Necessary And Sufficient

Consider maxima M n of a sequence of random variables defined on a finite Markov chain. Necessary and sufficient conditions for the existence of normalizing constants B n such that are given. The problem can be reduced to studying maxima of i.i.d. random variables drawn from a finite product of distributions π i=1 m H i (x). The effect of each factor H i (x) on the behavior of maxima from π i=1 m H i is analyzed. Under a mild regularity condition, B n can be chosen to be the maximum of the m quantiles of order (1 - n -1) of the H's.

Almost sure stability of maxima

1973 ◽  
Vol 10 (2) ◽  
pp. 387-401 ◽  
Author(s):  
Sidney I. Resnick ◽  
R. J. Tomkins
Keyword(s):  
Markov Chain ◽  
Stability Problem ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Necessary And Sufficient Conditions ◽  
Finite Markov Chain ◽  
Almost Sure Stability ◽  
Normalizing Constants ◽  
The Stability ◽  
Necessary And Sufficient

For random variables {Xn, n ≧ 1} unbounded above set Mn = max {X1, X2, …, Xn}. When do normalizing constants bn exist such that Mn/bn→ 1 a.s.; i.e., when is {Mn} a.s. stable? If {Xn} is i.i.d. then {Mn} is a.s. stable iff for all and in this case bn ∼ F–1 (1 – 1/n) Necessary and sufficient conditions for lim supn→∞, Mn/bn = l > 1 a.s. are given and this is shown to be insufficient in general for lim infn→∞Mn/bn = 1 a.s. except when l = 1. When the Xn are r.v.'s defined on a finite Markov chain, one shows by means of an analogue of the Borel Zero-One Law and properties of semi-Markov matrices that the stability problem for this case can be reduced to the i.i.d. case.

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