scholarly journals Random walks on free products, quotients and amalgams

1986 ◽  
Vol 102 ◽  
pp. 163-180 ◽  
Author(s):  
Donald I. Cartwright ◽  
P. M. Soardi
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Asymptotic Behaviour ◽  
Random Walks ◽  
Green's Function ◽  
Discrete Group ◽  
Transition Probabilities ◽  
Free Products ◽  
Step Transition ◽  
Independent And Identically Distributed

Suppose that G is a discrete group and p is a probability measure on G. Consider the associated random walk {Xn} on G. That is, let Xn = Y1Y2 … Yn, where the Yj’s are independent and identically distributed G-valued variables with density p. An important problem in the study of this random walk is the evaluation of the resolvent (or Green’s function) R(z, x) of p. For example, the resolvent provides, in principle, the values of the n step transition probabilities of the process, and in several cases knowledge of R(z, x) permits a description of the asymptotic behaviour of these probabilities.

scholarly journals Stochastic LU factorizations, Darboux transformations and urn models

2018 ◽  
Vol 55 (3) ◽  
pp. 862-886 ◽  
Author(s):  
F. Alberto Grünbaum ◽  
Manuel D. de la Iglesia
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Free Parameter ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Urn Models ◽  
Triangular Matrices ◽  
New Family ◽  
Step Transition

Abstract We consider upper‒lower (UL) (and lower‒upper (LU)) factorizations of the one-step transition probability matrix of a random walk with the state space of nonnegative integers, with the condition that both upper and lower triangular matrices in the factorization are also stochastic matrices. We provide conditions on the free parameter of the UL factorization in terms of certain continued fractions such that this stochastic factorization is possible. By inverting the order of the factors (also known as a Darboux transformation) we obtain a new family of random walks where it is possible to state the spectral measures in terms of a Geronimus transformation. We repeat this for the LU factorization but without a free parameter. Finally, we apply our results in two examples; the random walk with constant transition probabilities, and the random walk generated by the Jacobi orthogonal polynomials. In both situations we obtain urn models associated with all the random walks in question.

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.

scholarly journals An Abramov formula for stationary spaces of discrete groups

2013 ◽  
Vol 34 (3) ◽  
pp. 837-853 ◽  
Author(s):  
YAIR HARTMAN ◽  
YURI LIMA ◽  
OMER TAMUZ
Keyword(s):  
Random Walk ◽  
Probability Measure ◽  
Finite Index ◽  
Discrete Group ◽  
Return Time ◽  
Discrete Groups ◽  
Poisson Boundary ◽  
Index Subgroup ◽  
Expected Return ◽  
Finite Index Subgroup

AbstractLet $(G, \mu )$ be a discrete group equipped with a generating probability measure, and let $\Gamma $ be a finite index subgroup of $G$. A $\mu $-random walk on $G$, starting from the identity, returns to $\Gamma $ with probability one. Let $\theta $ be the hitting measure, or the distribution of the position in which the random walk first hits $\Gamma $. We prove that the Furstenberg entropy of a $(G, \mu )$-stationary space, with respect to the action of $(\Gamma , \theta )$, is equal to the Furstenberg entropy with respect to the action of $(G, \mu )$, times the index of $\Gamma $ in $G$. The index is shown to be equal to the expected return time to $\Gamma $. As a corollary, when applied to the Furstenberg–Poisson boundary of $(G, \mu )$, we prove that the random walk entropy of $(\Gamma , \theta )$ is equal to the random walk entropy of $(G, \mu )$, times the index of $\Gamma $ in $G$.

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.

The correlated random walk

1981 ◽  
Vol 18 (2) ◽  
pp. 403-414 ◽  
Author(s):  
Eric Renshaw ◽  
Robin Henderson
Keyword(s):  
Random Walk ◽  
Transition Probabilities ◽  
Correlated Random Walk ◽  
Limiting Distributions ◽  
One Dimensional ◽  
Step Transition

A one-dimensional random walk is studied in which, at each stage, the probabilities of continuing in the same direction or of changing direction are p and 1 – p, respectively. Exact expressions are derived for the n-step transition probabilities, and various limiting distributions are investigated.

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.

scholarly journals Empirical processes for recurrent and transient random walks in random scenery

2020 ◽  
Vol 24 ◽  
pp. 127-137
Author(s):  
Nadine Guillotin-Plantard ◽  
Françoise Pène ◽  
Martin Wendler
Keyword(s):  
Random Walk ◽  
Asymptotic Behaviour ◽  
Random Walks ◽  
Stable Distribution ◽  
Random Variables ◽  
Empirical Processes ◽  
Independent Random Variables ◽  
Random Scenery ◽  
Recurrent Random Walk ◽  
Transient Random Walk

In this paper, we are interested in the asymptotic behaviour of the sequence of processes (Wn(s,t))s,t∈[0,1] with \begin{equation*} W_n(s,t):=\sum_{k=1}^{\lfloor nt\rfloor}\big(\mathds{1}_{\{\xi_{S_k}\leq s\}}-s\big) \end{equation*} where (ξx, x ∈ ℤd) is a sequence of independent random variables uniformly distributed on [0, 1] and (Sn)n ∈ ℕ is a random walk evolving in ℤd, independent of the ξ’s. In M. Wendler [Stoch. Process. Appl. 126 (2016) 2787–2799], the case where (Sn)n ∈ ℕ is a recurrent random walk in ℤ such that (n−1/αSn)n≥1 converges in distribution to a stable distribution of index α, with α ∈ (1, 2], has been investigated. Here, we consider the cases where (Sn)n ∈ ℕ is either: (a) a transient random walk in ℤd, (b) a recurrent random walk in ℤd such that (n−1/dSn)n≥1 converges in distribution to a stable distribution of index d ∈{1, 2}.

scholarly journals Random Walks in Hypergraph

2021 ◽  
Vol 15 ◽  
pp. 13-20
Author(s):  
Abdelghani Bellaachia ◽  
Mohammed Al-Dhelaan
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Transition Probabilities ◽  
Graph Representation ◽  
Real World Data ◽  
Data Set ◽  
Random Walks On Graphs ◽  
Text Ranking ◽  
Relationship Structure ◽  
The Relationship

Random walks on graphs have been extensively used for a variety of graph-based problems such as ranking vertices, predicting links, recommendations, and clustering. However, many complex problems mandate a high-order graph representation to accurately capture the relationship structure inherent in them. Hypergraphs are particularly useful for such models due to the density of information stored in their structure. In this paper, we propose a novel extension to defining random walks on hypergraphs. Our proposed approach combines the weights of destination vertices and hyperedges in a probabilistic manner to accurately capture transition probabilities. We study and analyze our generalized form of random walks suitable for the structure of hypergraphs. We show the effectiveness of our model by conducting a text ranking experiment on a real world data set with a 9% to 33% improvement in precision and a range of 7% to 50% improvement in Bpref over other random walk approaches.

scholarly journals Rate of Escape of Random Walks on Regular Languages and Free Products by Amalgamation of Finite Groups

2008 ◽  
Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽  
Author(s):  
Lorenz A. Gilch
Keyword(s):  
Random Walks ◽  
Finite Groups ◽  
Transition Probabilities ◽  
Finite Alphabet ◽  
Free Products ◽  
Rate Of Escape ◽  
International Audience ◽  
Special Case ◽  
Current Word ◽  
Natural Way

International audience We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities depend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural length functions for these random walks using different techniques.

A correlated random walk model for two-dimensional diffusion

1984 ◽  
Vol 21 (2) ◽  
pp. 233-246 ◽  
Author(s):  
Robin Henderson ◽  
Eric Renshaw ◽  
David Ford
Keyword(s):  
Random Walk ◽  
Transition Probabilities ◽  
Exact Expression ◽  
Dispersion Matrix ◽  
Two Dimensional ◽  
Correlated Random Walk ◽  
Dimensional Diffusion ◽  
Step Transition ◽  
Lattice Random Walk ◽  
Previous Step

A two-dimensional lattice random walk is studied in which, at each stage, the direction of the next step is correlated to that of the previous step. An exact expression is obtained from the characteristic function for the dispersion matrix of the n-step transition probabilities, and the limiting diffusion equation is derived.

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