Seminar on Stochastic Processes, 1991 pp 7-14 | Cite as

# A Correlation Inequality for Tree-Indexed Markov Chains

## Abstract

Let *T* be a tree, i.e. an infinite, locally finite graph without loops or cycles. One vertex of *T*, designated 0, is called the root; we assume all other vertices have degree at least 2. Attach a random variable *S* _{σ} to each vertex *σ* of *T* as follows. Take *S* _{0} = 0 and for *σ* ≠ 0 choose *S* _{σ} randomly, with equal probabilities, from \( \left\{ {{S_{{\tilde{\sigma }}}} - 1,{S_{{\tilde{\sigma }}}} + 1} \right\} \) where σ is the “predecessor” of *σ* in *T* (see §2 for precise definitions). We call the process \( \left\{ {{S_{\sigma }}:\sigma \in T} \right\} \) a *T*-walk on Z; note that taking *T* = {0,1,2,…} with consecutive integers connected, we recover the (ordinary) simple random walk on Z. Allowing richer trees *T* we may observe quite different asymptotic behaviour. One can study this behaviour either by considering the levels \( \left\{ {\sigma :\left| \sigma \right| = n} \right\} \) = *T* _{ n } of *T* (*|σ|* is the distance from 0 to *σ*) or by observing the rays (infinite non-self-intersecting paths) in *T*. The first approach, which we adopt here, was initiated by Joffe and Moncayo [JM] who gave conditions for asymptotic normality of the empirical measures determined by \( \left\{ {{S_{\sigma }}:\left| \sigma \right| = n} \right\} \). The second approach is used in [E], [LP] and [BP1]. To elucidate both approaches, we quote a theorem which relates three notions of “speed” for a *T*-walk, to dimensional properties of the boundary *∂T* of *T* (*∂T* is the collection of rays in *T*, emanating from 0). Equip *∂T* with the metric *ρ* given by *ρ*(*ξ, η*) = *e* ^{ −n } if *ξ, η* ∈ *∂T* intersect in a path of length precisely *n* from 0.

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