scholarly journals The random difference equation $X\sb n=A\sb nX\sb {n-1}+B\sb n$ in the critical case

1997 ◽  
Vol 25 (1) ◽  
pp. 478-493 ◽  
Author(s):  
Martine Babillot ◽  
Philippe Bougerol ◽  
Laure Elie
Keyword(s):  
Difference Equation ◽  
Critical Case ◽  
Random Difference Equation ◽  
Random Difference

On uniqueness of invariant measures for random walks on

2021 ◽  
pp. 1-32
Author(s):  
SARA BROFFERIO ◽  
DARIUSZ BURACZEWSKI ◽  
TOMASZ SZAREK
Keyword(s):  
Random Walk ◽  
Invariant Measure ◽  
Difference Equation ◽  
Random Walks ◽  
Critical Case ◽  
Invariant Measures ◽  
Fundamental Question ◽  
Symmetric Random Walk ◽  
Random Difference Equation ◽  
Random Difference

Abstract We consider random walks on the group of orientation-preserving homeomorphisms of the real line ${\mathbb R}$ . In particular, the fundamental question of uniqueness of an invariant measure of the generated process is raised. This problem was studied by Choquet and Deny [Sur l’équation de convolution $\mu = \mu * \sigma $ . C. R. Acad. Sci. Paris250 (1960), 799–801] in the context of random walks generated by translations of the line. Nowadays the answer is quite well understood in general settings of strongly contractive systems. Here we focus on a broader class of systems satisfying the conditions of recurrence, contraction and unbounded action. We prove that under these conditions the random process possesses a unique invariant Radon measure on ${\mathbb R}$ . Our work can be viewed as following on from Babillot et al [The random difference equation $X_n=A_n X_{n-1}+B_n$ in the critical case. Ann. Probab.25(1) (1997), 478–493] and Deroin et al [Symmetric random walk on $\mathrm {HOMEO}^{+}(\mathbb {R})$ . Ann. Probab.41(3B) (2013), 2066–2089].

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.

Large deviations of branching process in a random environment

2021 ◽  
Vol 31 (4) ◽  
pp. 281-291
Author(s):  
Aleksandr V. Shklyaev
Keyword(s):  
Difference Equation ◽  
Large Deviations ◽  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Random Variables ◽  
Asymptotic Formulas ◽  
Random Difference Equation ◽  
Random Difference ◽  
Independent Identically Distributed

Abstract In this first part of the paper we find the asymptotic formulas for the probabilities of large deviations of the sequence defined by the random difference equation Y n+1=A n Y n + B n , where A 1, A 2, … are independent identically distributed random variables and B n may depend on { ( A k , B k ) , 0 ⩽ k < n } $ \{(A_k,B_k),0\leqslant k \lt n\} $ for any n≥1. In the second part of the paper this results are applied to the large deviations of branching processes in a random environment.

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