Large deviations for a Markov chain in a random landscape

2002 ◽  
Vol 34 (2) ◽  
pp. 375-393 ◽  
Author(s):  
Nadine Guillotin-Plantard
Keyword(s):  
Markov Chain ◽  
Large Deviations ◽  
State Space ◽  
Random Vectors ◽  
Independent And Identically Distributed

Let (Sk)k≥0 be a Markov chain with state space E and (ξx)x∊E be a family of ℝp-valued random vectors assumed independent of the Markov chain. The ξx could be assumed independent and identically distributed or could be Gaussian with reasonable correlations. We study the large deviations of the sum

Large deviations for a Markov chain in a random landscape

2002 ◽  
Vol 34 (02) ◽  
pp. 375-393
Author(s):  
Nadine Guillotin-Plantard
Keyword(s):  
Markov Chain ◽  
Large Deviations ◽  
State Space ◽  
Random Vectors ◽  
Independent And Identically Distributed

Let (Sk)k≥0be a Markov chain with state spaceEand (ξx)x∊Ebe a family of ℝp-valued random vectors assumed independent of the Markov chain. The ξxcould be assumed independent and identically distributed or could be Gaussian with reasonable correlations. We study the large deviations of the sum

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.

scholarly journals On multivariate records from random vectors with independent components

2018 ◽  
Vol 55 (1) ◽  
pp. 43-53 ◽  
Author(s):  
M. Falk ◽  
A. Khorrami Chokami ◽  
S. A. Padoan
Keyword(s):  
Distribution Function ◽  
Markov Chain ◽  
State Space ◽  
Random Vector ◽  
Waiting Times ◽  
Continuous Distribution ◽  
The State ◽  
Random Vectors ◽  
Independent Components ◽  
Carry Over

Abstract Let X1, X2, . . . be independent copies of a random vector X with values in ℝd and a continuous distribution function. The random vector Xn is a complete record, if each of its components is a record. As we require X to have independent components, crucial results for univariate records clearly carry over. But there are substantial differences as well. While there are infinitely many records in the d = 1 case, they occur only finitely many times in the series if d ≥ 2. Consequently, there is a terminal complete record with probability 1. We compute the distribution of the random total number of complete records and investigate the distribution of the terminal record. For complete records, the sequence of waiting times forms a Markov chain, but unlike the univariate case now the state at ∞ is an absorbing element of the state space.

Monotonicity of Positive Dependence with Time for Stationary Reversible Markov Chains

1995 ◽  
Vol 9 (2) ◽  
pp. 227-237 ◽  
Author(s):  
Taizhong Hu ◽  
Harry Joe
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Real Line ◽  
Continuous Time ◽  
Marginal Distribution ◽  
Random Vectors ◽  
Positive Dependence ◽  
Reversible Markov Chain ◽  
Reversible Markov Chains

Let (X1, X2) and (Y1, Y2) be bivariate random vectors with a common marginal distribution (X1, X2) is said to be more positively dependent than (Y1, Y2) if E[h(X1)h(X2)] ≥ E[h(Y1)h(Y2)] for all functions h for which the expectations exist. The purpose of this paper is to study the monotonicity of positive dependence with time for a stationary reversible Markov chain [X1]; that is, (Xs, Xl+s) is less positively dependent as t increases. Both discrete and continuous time and both a denumerable set and a subset of the real line for the state space are considered. Some examples are given to show that the assertions established for reversible Markov chains are not true for nonreversible chains.

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].

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. I. Survival for a long time

2005 ◽  
Vol 37 (4) ◽  
pp. 1015-1034 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
State Space ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (Xt)t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E→ℝ\{0}, and let (φt)t≥0 be an additive functional defined by φt=∫0tv(Xs)d s. We consider the case in which the process (φt)t≥0 is oscillating and that in which (φt)t≥0 has a negative drift. In each of these cases, we condition the process (Xt,φt)t≥0 on the event that (φt)t≥0 is nonnegative until time T and prove weak convergence of the conditioned process as T→∞.

scholarly journals Large Deviations for Exchangeable Random Vectors

1992 ◽  
Vol 20 (3) ◽  
pp. 1147-1166 ◽  
Author(s):  
I. H. Dinwoodie ◽  
S. L. Zabell
Keyword(s):  
Large Deviations ◽  
Random Vectors
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