scholarly journals Symmetric Random Walks on Regular Tetrahedra, Octahedra, and Hexahedra

2017 ◽  
Vol 69 (1) ◽  
pp. 110-128
Author(s):  
Jyotirmoy Sarkar ◽  
Saran Ishika Maiti
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Equal Probability ◽  
Symmetric Random Walk ◽  
Regular Polyhedra ◽  
Standard Deviations

We study a symmetric random walk on the vertices of three regular polyhedra. Starting from the origin, at each step the random walk moves, independently of all previous moves, to one of the vertices adjacent to the current vertex with equal probability. We find the distributions, or at least the means and the standard deviations, of the number of steps needed (a) to return to origin, (b) to visit all vertices, and (c) to return to origin after visiting all vertices. We also find the distributions of (i) the number of vertices visited before return to origin, (ii) the last vertex visited, and (iii) the number of vertices visited during return to origin after visiting all vertices.

scholarly journals Optimal Coadapted Coupling for a Random Walk on the Hyper-Complete Graph

2013 ◽  
Vol 50 (4) ◽  
pp. 1117-1130
Author(s):  
Stephen Connor
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Complete Graph ◽  
Continuous Time ◽  
Symmetric Random Walk ◽  
Coupling Inequality ◽  
Maximal Coupling

The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks on Z2d was considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such coadapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal coadapted coupling for the continuous-time symmetric random walk on Knd, where Kn is the complete graph with n vertices. Moreover, we show that although this coupling is not maximal for any n (i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling as n → ∞.

scholarly journals Optimal Coadapted Coupling for a Random Walk on the Hyper-Complete Graph

2013 ◽  
Vol 50 (04) ◽  
pp. 1117-1130
Author(s):  
Stephen Connor
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Complete Graph ◽  
Continuous Time ◽  
Symmetric Random Walk ◽  
Coupling Inequality ◽  
Maximal Coupling

The problem of constructing an optimal coadapted coupling for a pair of symmetric random walks onZ2dwas considered by Connor and Jacka (2008), and the existence of a coupling which is stochastically fastest in the class of all such coadapted couplings was demonstrated. In this paper we show how to generalise this construction to an optimal coadapted coupling for the continuous-time symmetric random walk onKnd, whereKnis the complete graph withnvertices. Moreover, we show that although this coupling is not maximal for anyn(i.e. it does not achieve equality in the coupling inequality), it does tend to a maximal coupling asn→ ∞.

scholarly journals Optimal Co-Adapted Coupling for the Symmetric Random Walk on the Hypercube

2008 ◽  
Vol 45 (03) ◽  
pp. 703-713 ◽  
Author(s):  
Stephen Connor ◽  
Saul Jacka
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Continuous Time ◽  
Symmetric Random Walk ◽  
Continuous Time Random Walks

Let X and Y be two simple symmetric continuous-time random walks on the vertices of the n-dimensional hypercube, Z 2 n . We consider the class of co-adapted couplings of these processes, and describe an intuitive coupling which is shown to be the fastest in this class.

Spitzer's condition for asymptotically symmetric random walk

1980 ◽  
Vol 17 (03) ◽  
pp. 856-859
Author(s):  
R. A. Doney
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Length Distribution ◽  
Domain Of Attraction ◽  
Step Length ◽  
Sufficient Condition ◽  
Symmetric Form ◽  
Stable Law ◽  
Symmetric Random Walk ◽  
Image Position

If the step-length distribution function F for a random walk {Sn, n ≧ 0} is either continuous and symmetric or belongs to the domain of attraction of a symmetric stable law, then it is clear that the symmetric form of ‘Spitzer's condition' holds, i.e. The question considered in this note is whether or not (⋆) can hold for other random walks. The answer is in the affirmative, for we show that (⋆) holds for a large class of random walks for which F is neither symmetric nor belongs to any domain of attraction; all such random walks are asymptotically symmetric, in the sense that lim x→∞ {F(–x)| 1 – F(x)} = 1, but we show by an example that this is not a sufficient condition for (⋆) to hold.

Random walk and the existence of evolutionary rates

Paleobiology ◽  
1987 ◽  
Vol 13 (4) ◽  
pp. 446-464 ◽  
Author(s):  
Fred L. Bookstein
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Empirical Data ◽  
Fossil Record ◽  
Null Model ◽  
Evolutionary Rates ◽  
Square Root ◽  
Symmetric Random Walk ◽  
Elapsed Time ◽  
Mathematical Literature

Before one can study evolutionary rates one must reject the null model of symmetric random walk. for which the requisite quantity does not exist. As random walks reliably simulate all the features we find so compelling in the fossil record—jumps, trends, and irregular cycles—rejection of this irritating hypothesis is much more difficult than one might hope. This paper reviews principal theorems from the mathematical literature of random walk and shows how they may be applied to empirical data by scaling net changes according to the square root of elapsed time. The notorious pair of “opposite” findings, equilibrium and anagenesis, may be construed as deviations from random walk in opposite directions. Malmgren's data on Globorotalia tumida, previously interpreted as an example of punctuated anagenesis, are consistent with a random walk showing neither punctuation nor anagenesis, but instead varying in speed over four subsequences.

scholarly journals Persistent random walks may have arbitrarily large tails

1989 ◽  
Vol 21 (1) ◽  
pp. 229-230 ◽  
Author(s):  
D. R. Grey
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Size Distribution ◽  
Symmetric Random Walk ◽  
Jump Size ◽  
Persistent Random Walks ◽  
Probabilistic Proof

We give a probabilistic proof of a result of Shepp, that a symmetric random walk may have jump size distribution with arbitrarily large tails and yet still be persistent.

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].

scholarly journals Optimal Co-Adapted Coupling for the Symmetric Random Walk on the Hypercube

2008 ◽  
Vol 45 (3) ◽  
pp. 703-713 ◽  
Author(s):  
Stephen Connor ◽  
Saul Jacka
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Continuous Time ◽  
Symmetric Random Walk ◽  
Continuous Time Random Walks

Let X and Y be two simple symmetric continuous-time random walks on the vertices of the n-dimensional hypercube, Z2n. We consider the class of co-adapted couplings of these processes, and describe an intuitive coupling which is shown to be the fastest in this class.

scholarly journals Persistent random walks may have arbitrarily large tails

1989 ◽  
Vol 21 (01) ◽  
pp. 229-230
Author(s):  
D. R. Grey
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Size Distribution ◽  
Symmetric Random Walk ◽  
Jump Size ◽  
Persistent Random Walks ◽  
Probabilistic Proof

We give a probabilistic proof of a result of Shepp, that a symmetric random walk may have jump size distribution with arbitrarily large tails and yet still be persistent.

Spitzer's condition for asymptotically symmetric random walk

1980 ◽  
Vol 17 (3) ◽  
pp. 856-859 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Length Distribution ◽  
Domain Of Attraction ◽  
Step Length ◽  
Sufficient Condition ◽  
Symmetric Form ◽  
Stable Law ◽  
Symmetric Random Walk ◽  
Image Position

If the step-length distribution function F for a random walk {Sn, n ≧ 0} is either continuous and symmetric or belongs to the domain of attraction of a symmetric stable law, then it is clear that the symmetric form of ‘Spitzer's condition' holds, i.e. The question considered in this note is whether or not (⋆) can hold for other random walks. The answer is in the affirmative, for we show that (⋆) holds for a large class of random walks for which F is neither symmetric nor belongs to any domain of attraction; all such random walks are asymptotically symmetric, in the sense that limx→∞ {F(–x)| 1 – F(x)} = 1, but we show by an example that this is not a sufficient condition for (⋆) to hold.

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