A Characterization of the Gamma Distribution from a Random Difference Equation.

1985 ◽  
Author(s):  
Eric S. Tollar
Keyword(s):  
Difference Equation ◽  
Gamma Distribution ◽  
Random Difference Equation ◽  
Random Difference

A characterization of the gamma distribution from a random difference equation

1988 ◽  
Vol 25 (1) ◽  
pp. 142-149 ◽  
Author(s):  
Eric S. Tollar
Keyword(s):  
Difference Equation ◽  
Gamma Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Characteristic Functions ◽  
Random Difference Equation ◽  
Random Difference

A characterization of the gamma distribution is considered which arises from a random difference equation. A proof without characteristic functions is given that if V and Y are independent random variables, then the independence of V · Y and (1 – V) · Y results in a characterization of the gamma distribution (after excluding the trivial cases).

A characterization of the gamma distribution from a random difference equation

1988 ◽  
Vol 25 (01) ◽  
pp. 142-149 ◽  
Author(s):  
Eric S. Tollar
Keyword(s):  
Difference Equation ◽  
Gamma Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Characteristic Functions ◽  
Random Difference Equation ◽  
Random Difference

A characterization of the gamma distribution is considered which arises from a random difference equation. A proof without characteristic functions is given that if V and Y are independent random variables, then the independence of V · Y and (1 – V) · Y results in a characterization of the gamma distribution (after excluding the trivial cases).

A perpetuity and the M/M/∞ ranked server system

1997 ◽  
Vol 34 (02) ◽  
pp. 508-513 ◽  
Author(s):  
J. Preater
Keyword(s):  
Difference Equation ◽  
Generating Function ◽  
Classical Result ◽  
Equilibrium Size ◽  
Server Systems ◽  
Random Difference Equation ◽  
Random Difference ◽  
Server System

We relate the equilibrium size of an M/M/8 type queue having an intermittent arrival stream to a perpetuity, the solution of a random difference equation. One consequence is a classical result for ranked server systems, previously obtained by generating function methods.

Extreme Value Theory for a Class of Markov Chains with Values in ℝd

1997 ◽  
Vol 29 (1) ◽  
pp. 138-164 ◽  
Author(s):  
Roland Perfekt
Keyword(s):  
Difference Equation ◽  
Markov Chains ◽  
Extreme Value Theory ◽  
Marginal Distribution ◽  
Transition Probabilities ◽  
Value Theory ◽  
Extreme Value ◽  
Extremal Index ◽  
Random Difference Equation ◽  
Random Difference

We consider extreme value theory for a class of stationary Markov chains with values in ℝd. The asymptotic distribution of Mn, the vector of componentwise maxima, is determined under mild dependence restrictions and suitable assumptions on the marginal distribution and the transition probabilities of the chain. This is achieved through computation of a multivariate extremal index of the sequence, extending results of Smith [26] and Perfekt [21] to a multivariate setting. As a by-product, we obtain results on extremes of higher-order, real-valued Markov chains. The results are applied to a frequently studied random difference equation.

A perpetuity and the M/M/∞ ranked server system

1997 ◽  
Vol 34 (2) ◽  
pp. 508-513 ◽  
Author(s):  
J. Preater
Keyword(s):  
Difference Equation ◽  
Generating Function ◽  
Classical Result ◽  
Equilibrium Size ◽  
Server Systems ◽  
Random Difference Equation ◽  
Random Difference ◽  
Server System

We relate the equilibrium size of an M/M/8 type queue having an intermittent arrival stream to a perpetuity, the solution of a random difference equation. One consequence is a classical result for ranked server systems, previously obtained by generating function methods.

scholarly journals Null recurrence and transience of random difference equations in the contractive case

2017 ◽  
Vol 54 (4) ◽  
pp. 1089-1110 ◽  
Author(s):  
Gerold Alsmeyer ◽  
Dariusz Buraczewski ◽  
Alexander Iksanov
Keyword(s):  
Markov Chain ◽  
Difference Equation ◽  
Difference Equations ◽  
Random Vectors ◽  
Positive Recurrence ◽  
Null Recurrence ◽  
Random Difference Equation ◽  
Random Difference ◽  
Random Difference Equations ◽  
Independent And Identically Distributed

Abstract Given a sequence (Mk, Qk)k ≥ 1 of independent and identically distributed random vectors with nonnegative components, we consider the recursive Markov chain (Xn)n ≥ 0, defined by the random difference equation Xn = MnXn - 1 + Qn for n ≥ 1, where X0 is independent of (Mk, Qk)k ≥ 1. Criteria for the null recurrence/transience are provided in the situation where (Xn)n ≥ 0 is contractive in the sense that M1 ⋯ Mn → 0 almost surely, yet occasional large values of the Qn overcompensate the contractive behavior so that positive recurrence fails to hold. We also investigate the attractor set of (Xn)n ≥ 0 under the sole assumption that this chain is locally contractive and recurrent.

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