Les fonctions sphériques d'un couple de Gelfand symétrique et les chaînes de Markov

1982 ◽  
Vol 14 (2) ◽  
pp. 272-294 ◽  
Author(s):  
Gérard Letac
Keyword(s):  
Markov Chains ◽  
Random Walks ◽  
Spherical Symmetry ◽  
Spherical Functions ◽  
Plancherel Measure ◽  
Sufficient Condition ◽  
Gelfand Pairs ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

After an elementary description of Gelfand pairs, spherical functions and Plancherel measure, some explicit computations on the related Markov chains are performed. Random walks on polyhedra belong to this class of Markov chains; two more examples of chains on graphs are worked out, and the necessary and sufficient condition of transcience of random walks on p-adic numbers with spherical symmetry is given as an application of the techniques of the paper.

Les fonctions sphériques d'un couple de Gelfand symétrique et les chaînes de Markov

1982 ◽  
Vol 14 (02) ◽  
pp. 272-294 ◽  
Author(s):  
Gérard Letac
Keyword(s):  
Markov Chains ◽  
Random Walks ◽  
Spherical Symmetry ◽  
Spherical Functions ◽  
Plancherel Measure ◽  
Sufficient Condition ◽  
Gelfand Pairs ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

After an elementary description of Gelfand pairs, spherical functions and Plancherel measure, some explicit computations on the related Markov chains are performed. Random walks on polyhedra belong to this class of Markov chains; two more examples of chains on graphs are worked out, and the necessary and sufficient condition of transcience of random walks on p-adic numbers with spherical symmetry is given as an application of the techniques of the paper.

On the moments of ladder epochs for driftless random walks

1994 ◽  
Vol 31 (A) ◽  
pp. 201-205
Author(s):  
Yuan S. Chow
Keyword(s):  
Random Walks ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Let X, X1, X2, … be i.i.d. Sn =Σ1 n Xj , E|X| > 0, E(X) = 0 and τ = inf {n ≥ 1: Sn ≥ 0}. By Wald's equation, E(τ) =∞. If E(X 2) <∞, then by a theorem of Burkholder and Gundy (1970), E(τ 1/2) =∞. In this paper, we prove that if E((X– ) 2 ) <∞, then E(τ 1/2) =∞. When X is integer-valued and X ≥ −1 a.s., a necessary and sufficient condition for E(τ 1–1/p ) <∞, p > 1, is Σn–1–1p E|Sn| <∞.

scholarly journals SCHUR MULTIPLIERS AND SPHERICAL FUNCTIONS ON HOMOGENEOUS TREES

2010 ◽  
Vol 21 (10) ◽  
pp. 1337-1382 ◽  
Author(s):  
U. HAAGERUP ◽  
T. STEENSTRUP ◽  
R. SZWARC
Keyword(s):  
Fourier Multiplier ◽  
Spherical Functions ◽  
Sufficient Condition ◽  
Homogeneous Tree ◽  
Necessary And Sufficient Condition ◽  
Closed Expression ◽  
Radial Functions ◽  
Schur Multipliers ◽  
Homogeneous Trees ◽  
Necessary And Sufficient

Let X be a homogeneous tree of degree q + 1 (2 ≤ q ≤ ∞) and let ψ : X × X → ℂ be a function for which ψ(x, y) only depends on the distance between x, y ∈ X. Our main result gives a necessary and sufficient condition for such a function to be a Schur multiplier on X × X. Moreover, we find a closed expression for the Schur norm ||ψ||S of ψ. As applications, we obtaina closed expression for the completely bounded Fourier multiplier norm ||⋅||M0A(G) of the radial functions on the free (non-abelian) group 𝔽N on N generators (2 ≤ N ≤ ∞) and of the spherical functions on the q-adic group PGL2(ℚq) for every prime number q.

On the Problem of Establishing the Existence of Stationary Distributions for Continuous-Time Markov Chains

1993 ◽  
Vol 7 (4) ◽  
pp. 529-543 ◽  
Author(s):  
P. K. Pollett ◽  
P. G. Taylor
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Transition Rates ◽  
Necessary And Sufficient Condition ◽  
Stationary Distributions ◽  
Continuous Time Markov Chains ◽  
Necessary And Sufficient ◽  
Networks Of Queues

We consider the problem of establishing the existence of stationary distributions for continuous-time Markov chains directly from the transition rates Q. Given an invariant probability distribution m for Q, we show that a necessary and sufficient condition for m to be a stationary distribution for the minimal process is that Q be regular. We provide sufficient conditions for the regularity of Q that are simple to verify in practice, thus allowing one to easily identify stationary distributions for a variety of models. To illustrate our results, we shall consider three classes of multidimensional Markov chains, namely, networks of queues with batch movements, semireversible queues, and partially balanced Markov processes.

On the moments of ladder epochs for driftless random walks

1994 ◽  
Vol 31 (A) ◽  
pp. 201-205 ◽  
Author(s):  
Yuan S. Chow
Keyword(s):  
Random Walks ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

Let X, X1, X2, … be i.i.d. Sn=Σ1nXj, E|X| > 0, E(X) = 0 and τ = inf {n ≥ 1: Sn ≥ 0}. By Wald's equation, E(τ) =∞. If E(X2) <∞, then by a theorem of Burkholder and Gundy (1970), E(τ1/2) =∞. In this paper, we prove that if E((X–)2) <∞, then E(τ1/2) =∞. When X is integer-valued and X ≥ −1 a.s., a necessary and sufficient condition for E(τ1–1/p) <∞, p > 1, is Σn–1–1p E|Sn| <∞.

On Group Inverses of Matrices with Order and Rank Perturbations

1994 ◽  
Vol 44 (3-4) ◽  
pp. 209-222 ◽  
Author(s):  
P.S.S.N.V.P. Rao
Keyword(s):  
Markov Chains ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Group Inverse ◽  
Necessary And Sufficient Condition ◽  
Matrix Formula ◽  
Group Inverses ◽  
Finite Markov Chains ◽  
Necessary And Sufficient

Group inverse of a square matrix A exists if and only if rank of A is equal to rank of A2. Group inverses have many applications, prominent among them is in the analysis of finite Markov chains discussed by Meyer (1982). In this note necessary and sufficient conditions for the existence of group inverses of bordered matrix, [Formula: see text] are obtained and expressions for the group inverses in terms of group inverse of A are given, whenever they exist. Also necessary and sufficient condition for the existence of group inverse of A in terms of group inverse of B and C are given. An application to perturbation in Markov chains is illustrated.

On the arc-sine laws for Lévy processes

1994 ◽  
Vol 31 (01) ◽  
pp. 76-89 ◽  
Author(s):  
R. K. Getoor ◽  
M. J. Sharpe
Keyword(s):  
Random Walks ◽  
Real Line ◽  
Lévy Process ◽  
Elementary Proof ◽  
Levy Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Necessary And Sufficient ◽  
Spitzer's Theorem

Let X be a Lévy process on the real line, and let Fc denote the generalized arcsine law on [0, 1] with parameter c. Then t −1 ⨍0 t P 0(X s &gt; 0) ds → c as t → ∞ is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs &gt;0} ds to converge in P 0 law to Fc. Moreover, P 0(Xt &gt; 0) = c for all t &gt; 0 is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs &gt;0} ds under P 0 to have law Fc for all t &gt; 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.

On the arc-sine laws for Lévy processes

1994 ◽  
Vol 31 (1) ◽  
pp. 76-89 ◽  
Author(s):  
R. K. Getoor ◽  
M. J. Sharpe
Keyword(s):  
Random Walks ◽  
Real Line ◽  
Lévy Process ◽  
Elementary Proof ◽  
Levy Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Necessary And Sufficient ◽  
Spitzer's Theorem

Let X be a Lévy process on the real line, and let Fc denote the generalized arcsine law on [0, 1] with parameter c. Then t−1 ⨍0tP0(Xs > 0) ds → c as t → ∞ is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds to converge in P0 law to Fc. Moreover, P0(Xt > 0) = c for all t > 0 is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds under P0 to have law Fc for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.

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