Pointwise Estimates for Transition Probabilities of Random Walks on Infinite Graphs

2003 ◽  
pp. 119-134 ◽  
Author(s):  
Thierry Coulhon ◽  
Alexander Grigor’yan
Keyword(s):  
Random Walks ◽  
Transition Probabilities ◽  
Infinite Graphs ◽  
Pointwise Estimates

scholarly journals Random Walks with Invariant Loop Probabilities: Stereographic Random Walks

Entropy ◽  
2021 ◽  
Vol 23 (6) ◽  
pp. 729
Author(s):  
Miquel Montero
Keyword(s):  
Random Walks ◽  
Stereographic Projection ◽  
Transition Probabilities ◽  
Simple Random Walk ◽  
General Introduction ◽  
Elliptic Case ◽  
One Step ◽  
Underlying Geometry ◽  
Step Transition ◽  
Wide Family

Random walks with invariant loop probabilities comprise a wide family of Markov processes with site-dependent, one-step transition probabilities. The whole family, which includes the simple random walk, emerges from geometric considerations related to the stereographic projection of an underlying geometry into a line. After a general introduction, we focus our attention on the elliptic case: random walks on a circle with built-in reflexing boundaries.

Optimal control of random walks, birth and death processes, and queues

1981 ◽  
Vol 13 (01) ◽  
pp. 61-83 ◽  
Author(s):  
Richard Serfozo
Keyword(s):  
Optimal Control ◽  
Random Walks ◽  
Queue Length ◽  
Transition Probabilities ◽  
Transition Rates ◽  
Average Reward ◽  
Birth And Death Processes ◽  
Optimal Policies ◽  
One Step ◽  
Increasing Functions

This is a study of simple random walks, birth and death processes, and M/M/s queues that have transition probabilities and rates that are sequentially controlled at jump times of the processes. Each control action yields a one-step reward depending on the chosen probabilities or transition rates and the state of the process. The aim is to find control policies that maximize the total discounted or average reward. Conditions are given for these processes to have certain natural monotone optimal policies. Under such a policy for the M/M/s queue, for example, the service and arrival rates are non-decreasing and non-increasing functions, respectively, of the queue length. Properties of these policies and a linear program for computing them are also discussed.

scholarly journals Hypergroups derived from random walks on some infinite graphs

2019 ◽  
Vol 189 (2) ◽  
pp. 321-353
Author(s):  
Tomohiro Ikkai ◽  
Yusuke Sawada
Keyword(s):  
Random Walks ◽  
Infinite Graphs

Random walks on a dodecahedron

1980 ◽  
Vol 17 (02) ◽  
pp. 373-384 ◽  
Author(s):  
G. Letac ◽  
L. Takács
Keyword(s):  
Markov Chain ◽  
Random Walks ◽  
Transition Probabilities ◽  
Regular Dodecahedron

We consider the general Markov chain on the vertices of a regular dodecahedron D such that P[Xn +1 = j | Xn = i] depends only on the distance between i and j. We consider also a Markov chain on the oriented edges (i, j) of D for which the only non-zero transition probabilities are and fix a vertex A. This paper computes explicitly P[Xn = A | X 0 = A] and P[In = A | I 0 = A]. The methods used are applicable to other solids.

Central limit theorems for a class of polynomial hypergroups

1990 ◽  
Vol 22 (01) ◽  
pp. 68-87 ◽  
Author(s):  
Michael Voit
Keyword(s):  
Random Walks ◽  
Central Limit ◽  
Limit Distribution ◽  
Limit Theorems ◽  
Order Of Convergence ◽  
Transition Probabilities ◽  
Recursion Formula ◽  
Rayleigh Distribution ◽  
Central Limit Theorems ◽  
Polynomial Hypergroups

Central limit theorems are proved for random walks on the non-negative integers where the transition probabilities are homogeneous with respect to a sequence of orthogonal polynomials. Assuming some restrictions concerning the three-term recursion formula of these polynomials, one gets a Rayleigh distribution as limit distribution where bounds of the order of convergence can be computed explicitly. These central limit theorems are applied to generalized birth and death random walks and random walks on polynomial hypergroups. Finally some examples of polynomial hypergroups are discussed in view of the limit theorems above.

Random walk in a random environment with correlated sites

2001 ◽  
Vol 38 (4) ◽  
pp. 1018-1032 ◽  
Author(s):  
T. Komorowski ◽  
G. Krupa
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Field ◽  
Random Environment ◽  
Law Of Large Numbers ◽  
Transition Probabilities ◽  
Direct Application ◽  
Random Environments ◽  
Large Numbers ◽  
Stationary Random Field

We prove the law of large numbers for random walks in random environments on the d-dimensional integer lattice Zd. The environment is described in terms of a stationary random field of transition probabilities on the lattice, possessing a certain drift property, modeled on the Kalikov condition. In contrast to the previously considered models, we admit possible correlation of transition probabilities at different sites, assuming however that they become independent at finite distances. The possible dependence of sites makes impossible a direct application of the renewal times technique of Sznitman and Zerner.

Random walks and periodic continued fractions

1985 ◽  
Vol 17 (1) ◽  
pp. 67-84 ◽  
Author(s):  
Wolfgang Woess
Keyword(s):  
Random Walks ◽  
Harmonic Functions ◽  
Limit Theorems ◽  
Continued Fraction ◽  
Transition Probabilities ◽  
Local Limit ◽  
Periodic Sequence ◽  
Nearest Neighbour ◽  
Continued Fraction Expansions ◽  
Local Limit Theorems

Nearest-neighbour random walks on the non-negative integers with transition probabilities p0,1 = 1, pk,k–1 = gk, pk,k+1 = 1– gk (0 < gk < 1, k = 1, 2, …) are studied by use of generating functions and continued fraction expansions. In particular, when (gk) is a periodic sequence, local limit theorems are proved and the harmonic functions are determined. These results are applied to simple random walks on certain trees.

scholarly journals A further study of an approximation for last-exit and first-passage probabilities of a random walk

1994 ◽  
Vol 7 (3) ◽  
pp. 411-422
Author(s):  
D. J. Daley ◽  
L. D. Servi
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Transition Probabilities ◽  
Continuous Lattice ◽  
First Passage ◽  
Continuity Properties ◽  
Exit Probabilities

Identities between first-passage or last-exit probabilities and unrestricted transition probabilities that hold for left- or right-continuous lattice-valued random walks form the basis of an intuitively based approximation that is demonstrated by computation to hold for certain random walks without either the left- or right-continuity properties. The argument centers on the use of ladder variables; the identities are known to hold asymptotically from work of Iglehart leading to Brownian meanders.

Sign in / Sign up

Export Citation Format

Share Document