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Journal of Probability and Statistics
Volume 2009, Article ID 241539, 4 pages
http://dx.doi.org/10.1155/2009/241539
Research Article

On the Moments of Hitting Times for Random Walks on Trees

Departamento de Cómputo Científico y Estadística, Universidad Simón Bolívar, Apartado 89000, Caracas, Venezuela

Received 24 June 2009; Accepted 31 August 2009

Academic Editor: Nikolaos E. Limnios

Copyright © 2009 José Luis Palacios. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Using classical arguments we derive a formula for the moments of hitting times for an ergodic Markov chain. We apply this formula to the case of simple random walk on trees and show, with an elementary electric argument, that all the moments are natural numbers.

1. Introduction

On a connected undirected graph such that the edge between vertices and is given a resistance (or equivalently, a conductance ), we can define the general random walk (GRW) on as the Markov chain , that from its current vertex jumps to the neighboring vertex with probability , where , and means that is a neighbor of . There may be a conductance from a vertex to itself, giving rise to a transition probability from to itself, though the most studied case of these random walks on graphs, the simple random walk (SRW), excludes the loops and considers all 's to be equal to 1.

The hitting time of the vertex is the number of jumps that the walk takes until it lands on , and its th moment when the walk starts at is denoted by . Chen and Zhang [1] found a closed-form formula for the expected hitting times of SRW on trees that yielded as a corollary the fact that these expected times are all natural numbers. Furthermore, Chen [2] studied the moment generating function of hitting times for SRW on trees and showed that the second moments are also natural numbers.

The purpose of this note is to give a recursive formula for the moments of the hitting times of any ergodic finite Markov chain, using as tools classical material found in Kemeny and Snell [3] and Chung [4]. When this formula is applied to the case of SRW on trees, an elementary argument from the electrical approach found in Doyle and Snell [5] shows that all these moments are natural numbers.

To ease the notation, let us assume that and that we are interested in finding for and . We consider the vectors and , where the superscript means “transpose.” The classical approach to finding in an ergodic Markov chain with state space , due to Kemeny and Snell [3], consists of considering the matrix which results from deleting the -th row and column of the transition probability matrix and finding the “fundamental matrix” which yields

on account of the fact that the entry of the fundamental matrix is the expected number of visits to state , , by the walk started at before hitting the state . We can write this fact as

An additional fact that we will need regarding the matrix can be found in Kemeny and Snell [3, page 49]:

Of the electrical approach we will quote the fact that

where is the sum of all conductances emanating from and is the voltage at when a battery is placed between and such that the current entering at is 1 and the voltage at is 0. The details can be found in Doyle and Snell [5, Section , page 49].

2. The Formula and Its Corollaries

In the spirit of Corollary in Chung [4, page 64] we find a recurrence for the moments of the hitting times of an ergodic Markov chain. Our formula differs from that of Chung in that his is presented neither as a recurrence nor in vector form, and more importantly, in that it involves neither taboo probabilities nor mean recurrence times, and this latter fact is crucial for our purposes, because for SRW on graphs the mean recurrence time of a vertex is given by , where is the number of neighbors of , and for the case of a tree this expression becomes which may or may not be a natural number, and we need that all coefficients in the vectors and matrices of our formula to be natural numbers. This recurrence should perhaps be better known and we present it here in the following

Theorem 2.1. For any ergodic Markov chain with state space one has for .

Proof. For any and any , conditioning gives us In vector form the above set of equations yields For , (2.3) becomes (1.1). For , from (2.3) we obtain Solving for and using (1.1) in (2.4) we get
Finally using (1.3) in (2.5) finishes the proof.

Now we can give sufficient conditions for all moments to be natural numbers.

Corollary 2.2. Under the hypotheses of the theorem, if is a vector of natural numbers and all the entries of the fundamental matrix are natural numbers, then for all , is a vector of natural numbers.

Proof. Use induction and the recursion (2.1), and verify that all the summands in the right-hand side of (2.1), under the hypotheses, turn out to be natural numbers.

Corollary 2.3. In the case of SRW on trees, is a vector of natural numbers for all .

Proof. It is known [1] that is a vector of natural numbers. By the previous corollary if suffices to prove that all entries of the fundamental matrix are natural numbers. But (1.2) and (1.4) tell us that , and for SRW is just the number of neighbors of , so all that is left to check is that the voltage is a natural number, when a battery is placed between and such that the current entering is 1 and the voltage . Since the graph is a tree, it is immediate from Ohm's law that for all vertices in the unique path from to , , the distance from to . Also, if we denote by the set of edges in the path , it is plain to see that all vertices in the connected component of that contains share the same voltage . So the voltages in all vertices are natural numbers and we are done.

The condition in Corollary 2.2 that all the entries of are natural numbers is not necessary. For example, in the case of SRW in the complete graph we have that

so that no entry of the fundamental matrix is a natural number. However, (1.1) gives us that . Moreover, by symmetry, for some constant , and it is plain to see that . Therefore, induction and formula (2.1) imply that all moments in this case are natural numbers, which can be given explicitely: , , and so forth.

References

  1. H. Chen and F. Zhang, “The expected hitting times for graphs with cutpoints,” Statistics & Probability Letters, vol. 66, no. 1, pp. 9–18, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. H. Chen, “The generating functions of hitting times for random walk on trees,” Statistics & Probability Letters, vol. 77, no. 15, pp. 1574–1579, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. J. G. Kemeny and J. L. Snell, Finite Markov Chains, Van Nostrand, Princeton, NJ, USA, 1960. View at MathSciNet
  4. K. L. Chung, Markov Chains with Stationary Transition Probabilities, Springer, New York, NY, USA, 1967. View at MathSciNet
  5. P. G. Doyle and J. L. Snell, Random Walks and Electric Networks, vol. 22, Mathematical Association of America, Washington, DC, USA, 1984. View at MathSciNet