Michael Keane

An elementary proof of the hitting time theorem

In this note, we give an elementary proof of the random walk hitting time theorem, which states that, for a left-continuous random walk on Z starting at a nonnegative integer k, the conditional probability that the walk hits the origin for the first time at time n, given that it does hit zero at time n, is equal to k/n. Here, a walk is called left-continuous when its steps are bounded from below by −1. We start by introducing some notation. Let Pk denote the law of a random walk starting at k ≥ 0, let {Yi } ∞=1 be the independent and identically distributed (i.i.d.) steps of the random walk, let Sn = k + Y1 +···+ Yn be the position of the random walk starting at k after n steps, and let T0 = inf{n : Sn = 0}

Ergodicity of the adic transformation on the Euler graph

The Euler graph has vertices labelled (n,k) for n=0,1,2,... and k=0,1,...,n, with k+1 edges from (n,k) to (n+1,k) and n-k+1 edges from (n,k) to (n+1,k+1). The number of paths from (0,0) to (n,k) is the Eulerian number A(n,k), the number of permutations of 1,2,...,n+1 with exactly n-k falls and k rises. We prove that the adic (Bratteli-Vershik) transformation on the space of infinite paths in this graph is ergodic with respect to the symmetric measure.

Finitary orbit equivalence of odometers

A finitary orbit equivalence mapping between the binary and ternary odometers is constructed.

A characterization of the Morse minimal set up to topological conjugacy

We establish necessary and sufficient conditions for a dynamical system to be topologically conjugate to the Morse minimal set, the shift orbit closure of the Morse sequence, and conditions for topological conjugacy to the closely related Teoplitz minimal set.

Easy and nearly simultaneous proofs of the Ergodic Theorem and Maximal Ergodic Theorem

We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem.

Routing on Delay Tolerant Sensor Networks

Delay (or disruption) tolerant sensor networks may be modeled as Markovian evolving graphs [1]. We present experimental evidence showing that considering multiple (possibly not shortest) paths instead of one fixed (greedy) path can decrease the expected time to deliver a packet on such a network by as much as 65 per cent depending on the probability that an edge exists in a given time interval. We provide theoretical justification for this result by studying a special case of the Markovian evolving grid graph. We analyze a natural algorithm for routing on such networks and show that it is possible to improve the expected time of delivery by up to a factor of two depending upon the probability of an edge being up during a time step and the relative positions of the source and destination. Furthermore we show that this is optimal, i.e., no other algorithm can achieve a better expected running time. As an aside, our results give high probability bounds for Knuths toilet paper problem [11].

Reinforced Random Walk

One of the distinguishing properties of the present scientific method is reproducibility. In one of its guises, probability theory is based on statistical reproduction, near certainty being obtained of truth of statements by averaging over long term to remove randomness occurring in individual experiments. When one assumes, as is often the case, that events farther and farther in the past have less and less influence on the present, the probabilistic paradigm is currently well understood and is successful in many scientific and technological applications. Recently, however, we have come to realize that precisely in these applications important stocahstic processes occur whose present outcomes are significantly influenced by events in the remote past. This behaviour is not at all well understood and some of the simplest questions remain today irritatingly beyond reach. A salient example occurs in the theory of random walks, where there is a dichotomy between recurrent and transient behaviour. After explaining this classical dichotomy, we present a very simple example with infinite memory which is neither known to be transient nor recurrent. Then, using a reinforcement mechanism due to Polya, we explain the nature of a particular infinite memory process in terms of spontaneous emergence of opinions. Finally we would like to discuss briefly some of our recent results towards understanding the recurrence-transience dichotomy for reinforced random walks.