Asymptotic location and recurrence properties of maxima of a sequence of random variables defined on a Markov chain

1971 ◽  
Vol 18 (3) ◽  
pp. 197-217 ◽  
Author(s):  
Sidney I. Resnick
Keyword(s):  
Markov Chain ◽  
Random Variables

Infinite dams with inputs forming a Markov chain

1968 ◽  
Vol 5 (1) ◽  
pp. 72-83 ◽  
Author(s):  
M. S. Ali Khan ◽  
J. Gani
Keyword(s):  
Markov Chain ◽  
Storage Systems ◽  
Random Variables ◽  
Time Intervals ◽  
Storage Theory ◽  
Annual Time ◽  
Theory Of Storage ◽  
Independent And Identically Distributed

Moran's [1] early investigations into the theory of storage systems began in 1954 with a paper on finite dams; the inputs flowing into these during consecutive annual time-intervals were assumed to form a sequence of independent and identically distributed random variables. Until 1963, storage theory concentrated essentially on an examination of dams, both finite and infinite, fed by inputs (discrete or continuous) which were additive. For reviews of the literature in this field up to 1963, the reader is referred to Gani [2] and Prabhu [3].

Criteria for the non-ergodicity of stochastic processes: application to the exponential back-off protocol

1987 ◽  
Vol 24 (02) ◽  
pp. 347-354 ◽  
Author(s):  
Guy Fayolle ◽  
Rudolph Iasnogorodski
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Stochastic Processes ◽  
State Space ◽  
Discrete Time ◽  
Random Variables ◽  
Countable State Space ◽  
Countable State ◽  
New Criteria

In this paper, we present some simple new criteria for the non-ergodicity of a stochastic process (Yn ), n ≧ 0 in discrete time, when either the upward or downward jumps are majorized by i.i.d. random variables. This situation is encountered in many practical situations, where the (Yn ) are functionals of some Markov chain with countable state space. An application to the exponential back-off protocol is described.

scholarly journals Approximations: replacing random variables with their means

2014 ◽  
Vol 51 (A) ◽  
pp. 57-62
Author(s):  
Joe Gani
Keyword(s):  
Stochastic Process ◽  
Markov Chain ◽  
Continuous Time ◽  
Random Variables ◽  
Original Form ◽  
Continuous Time Markov Chain ◽  
Standard Methods ◽  
Sis Epidemics

One of the standard methods for approximating a bivariate continuous-time Markov chain {X(t), Y(t): t ≥ 0}, which proves too difficult to solve in its original form, is to replace one of its variables by its mean, This leads to a simplified stochastic process for the remaining variable which can usually be solved, although the technique is not always optimal. In this note we consider two cases where the method is successful for carrier infections and mutating bacteria, and one case where it is somewhat less so for the SIS epidemics.

Sums and weighted sums of a gamma Markov sequence

1988 ◽  
Vol 25 (01) ◽  
pp. 204-209 ◽  
Author(s):  
Ravindra M. Phatarfod
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Random Variables ◽  
Laplace Transforms ◽  
Weighted Sums ◽  
Sums Of Random Variables ◽  
Markov Sequence

We derive the Laplace transforms of sums and weighted sums of random variables forming a Markov chain whose stationary distribution is gamma. Both seasonal and non-seasonal cases are considered. The results are applied to two problems in stochastic reservoir theory.

Infinite dams with inputs forming a Markov chain

1968 ◽  
Vol 5 (01) ◽  
pp. 72-83 ◽  
Author(s):  
M. S. Ali Khan ◽  
J. Gani
Keyword(s):  
Markov Chain ◽  
Storage Systems ◽  
Random Variables ◽  
Time Intervals ◽  
Storage Theory ◽  
Annual Time ◽  
Theory Of Storage ◽  
Independent And Identically Distributed

Moran's [1] early investigations into the theory of storage systems began in 1954 with a paper on finite dams; the inputs flowing into these during consecutive annual time-intervals were assumed to form a sequence of independent and identically distributed random variables. Until 1963, storage theory concentrated essentially on an examination of dams, both finite and infinite, fed by inputs (discrete or continuous) which were additive. For reviews of the literature in this field up to 1963, the reader is referred to Gani [2] and Prabhu [3].

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.

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