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Any of these conditions can be waived by permission of the Corresponding Author. ecp@iam.uni-bonn.de (Anton Bovier (Chief Editor)) ejpecp@chafai.net (Djalil Chafaï) Thu, 09 Jan 2014 00:05:12 -0800 OJS http://blogs.law.harvard.edu/tech/rss 60 Recurrence for the frog model with drift on $\mathbb{Z}^d$ http://ecp.ejpecp.org/article/view/3740 In this paper we present a recurrence criterion for the frog model on $\mathbb{Z}^d$ with an i.i.d. initial configuration of sleeping frogs and such that the underlying random walk has a drift to the right. Christian Döbler, Lorenz Pfeifroth http://ecp.ejpecp.org/article/view/3740 Fri, 14 Nov 2014 22:26:29 -0800 Wald for non-stopping times: the rewards of impatient prophets http://ecp.ejpecp.org/article/view/3609 Let $X_1,X_2,\ldots$ be independent identically distributed nonnegative random variables. Wald's identity states that the random sum $S_T:=X_1+\cdots+X_T$ has expectation $\mathbb{E} T \cdot \mathbb{E} X_1$ provided $T$ is a stopping time. We prove here that for any $1&lt;\alpha\leq 2$, if $T$ is an arbitrary nonnegative random variable, then $S_T$ has finite expectation provided that $X_1$ has finite $\alpha$-moment and $T$ has finite $1/(\alpha-1)$-moment. We also prove a variant in which $T$ is assumed to have a finite exponential moment. These moment conditions are sharp in the sense that for any i.i.d. sequence $X_i$ violating them, there is a $T$ satisfying the given condition for which $S_T$ (and, in fact, $X_T$) has infinite expectation.An interpretation of this is given in terms of a prophet being more rewarded than a gambler when a certain impatience restriction is imposed. Alexander E Holroyd, Yuval Peres, Jeffrey E Steif http://ecp.ejpecp.org/article/view/3609 Wed, 12 Nov 2014 21:21:38 -0800 Some limit results for Markov chains indexed by trees http://ecp.ejpecp.org/article/view/3601 We consider a sequence of Markov chains $(\mathcal X^n)_{n=1,2,...}$ with $\mathcal X^n = (X^n_\sigma)_{\sigma\in\mathcal T}$, indexed by the full binary tree $\mathcal T = \mathcal T_0 \cup \mathcal T_1 \cup ...$, where $\mathcal T_k$ is the $k$th generation of $\mathcal T$. In addition, let $(\Sigma_k)_{k=0,1,2,...}$ be a random walk on $\mathcal T$ with $\Sigma_k \in \mathcal T_k$ and $\widetilde{\mathcal R}^n = (\widetilde R_t^n)_{t\geq 0}$ with $\widetilde R_t^n := X_{\Sigma_{[tn]}}$, arising by observing the Markov chain $\mathcal X^n$ along the random walk. We present a law of large numbers concerning the empirical measure process $\widetilde{\mathcal Z}^n = (\widetilde Z_t^n)_{t\geq 0}$ where $\widetilde{Z}_t^n = \sum_{\sigma\in\mathcal T_{[tn]}} \delta_{X_\sigma^n}$ as $n\to\infty$. Precisely, we show that if $\widetilde{\mathcal R}^n \Rightarrow{n\to\infty} \mathcal R$ for some Feller process $\mathcal R = (R_t)_{t\geq 0}$ with deterministic initial condition, then $\widetilde{\mathcal Z}^n \Rightarrow{n\to\infty} \mathcal Z$ with $Z_t = \delta_{\mathcal L(R_t)}$. Peter Czuppon, Peter Pfaffelhuber http://ecp.ejpecp.org/article/view/3601 Tue, 11 Nov 2014 09:12:35 -0800 Monotone interaction of walk and graph: recurrence versus transience http://ecp.ejpecp.org/article/view/3607 We consider recurrence versus transience for models of random walks on growing in time, connected subsets $\mathbb{G}_t$ of some fixed locally finite, connected graph, in which monotone interaction enforces such growth as a result of visits by the walk (or probes it sent), to the neighborhood of the boundary of $\mathbb{G}_t$. Amir Dembo, Ruojun Huang, Vladas Sidoravicius http://ecp.ejpecp.org/article/view/3607 Thu, 06 Nov 2014 23:27:32 -0800 A note on the times of first passage for `nearly right-continuous' random walks http://ecp.ejpecp.org/article/view/3735 A natural extension of a right-continuous integer-valued random walk is one which can jump to the right by one or two units. First passage times above a given fixed level then admit - on each of the two events, which correspond to overshoot zero and one, separately - a tractable probability generating function. Some applications are considered. Matija Vidmar http://ecp.ejpecp.org/article/view/3735 Sat, 01 Nov 2014 04:01:02 -0700