## Recursos de colección

#### Project Euclid (Hosted at Cornell University Library) (191.996 recursos)

Electronic Communications in Probability

1. #### Convergence of complex martingales in the branching random walk: the boundary

Biggins [Uniform convergence of martingales in the branching random walk. Ann. Probab., 20(1):137–151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex parameters $\lambda$ from an open set $\Lambda \subseteq \mathbb{C} ^d$. We investigate the martingales corresponding to parameters from the boundary $\partial \Lambda$ of $\Lambda$. The boundary can be decomposed into several parts. We demonstrate by means of an example that there may be a part of the boundary, on which the martingales do not exist. Where the martingales exist, they may diverge, vanish in the limit or converge...

2. #### Some connections between permutation cycles and Touchard polynomials and between permutations that fix a set and covers of multisets

Pinsky, Ross G.
We present a new proof of a fundamental result concerning cycles of random permutations which gives some intuition for the connection between Touchard polynomials and the Poisson distribution. We also introduce a rather novel permutation statistic and study its distribution. This quantity, indexed by $m$, is the number of sets of size $m$ fixed by the permutation. This leads to a new and simpler derivation of the exponential generating function for the number of covers of certain multisets.

3. #### Self-averaging sequences which fail to converge

Cator, Eric; Don, Henk
We consider self-averaging sequences in which each term is a weighted average over previous terms. For several sequences of this kind it is known that they do not converge to a limit. These sequences share the property that $n$th term is mainly based on terms around a fixed fraction of $n$. We give a probabilistic interpretation to such sequences and give weak conditions under which it is natural to expect non-convergence. Our methods are illustrated by application to the group Russian roulette problem.

4. #### Erratum: Optimal linear drift for the speed of convergence of an hypoelliptic diffusion

Guillin, Arnaud; Monmarché, Pierre
Erratum for Optimal linear drift for the speed of convergence of an hypoelliptic diffusion, A. Guillin, and P. Monmarché, Electron. Commun. Probab. 21 (2016), paper no. 74, 14 pp. doi:10.1214/16-ECP25.

5. #### First passage percolation on a hyperbolic graph admits bi-infinite geodesics

Benjamini, Itai; Tessera, Romain
Given an infinite connected graph, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges. An open question attributed to Furstenberg ([14]) is whether there exists a bi-infinite geodesic in first passage percolation on the euclidean lattice of dimension at least 2. Although the answer is generally conjectured to be negative, we give a positive answer for graphs satisfying some negative curvature assumption. Assuming only strict positivity and finite expectation of the random lengths, we prove that if a graph $X$ has bounded degree and contains a Morse geodesic (e.g. is non-elementary Gromov hyperbolic), then almost...

6. #### Indicable groups and $p_c<1$

A conjecture of Benjamini & Schramm from 1996 states that any finitely generated group that is not a finite extension of $\mathbb{Z}$ has a non-trivial percolation phase. Our main results prove this conjecture for certain groups, and in particular prove that any group with a non-trivial homomorphism into the additive group of real numbers satisfies the conjecture. We use this to reduce the conjecture to the case of hereditary just-infinite groups. ¶ The novelty here is mainly in the methods used, combining the methods of EIT and evolving sets, and using the algebraic properties of the group to apply these methods.

7. #### The set of connective constants of Cayley graphs contains a Cantor space

Martineau, Sébastien
The connective constant of a transitive graph is the exponential growth rate of its number of self-avoiding walks. We prove that the set of connective constants of the so-called Cayley graphs contains a Cantor set. In particular, this set has the cardinality of the continuum.

8. #### Product space for two processes with independent increments under nonlinear expectations

Gao, Qiang; Hu, Mingshang; Ji, Xiaojun; Liu, Guomin
In this paper, we consider the product space for two processes with independent increments under nonlinear expectations. By introducing a discretization method, we construct a nonlinear expectation under which the given two processes can be seen as a new process with independent increments.

9. #### Necessary and sufficient conditions for the $r$-excessive local martingales to be martingales

Urusov, Mikhail; Zervos, Mihail
We consider the decreasing and the increasing $r$-excessive functions $\varphi _r$ and $\psi _r$ that are associated with a one-dimensional conservative regular continuous strong Markov process $X$ with values in an interval with endpoints $\alpha < \beta$. We prove that the $r$-excessive local martingale $\bigl ( e^{-r (t \wedge T_\alpha )} \varphi _r (X_{t \wedge T_\alpha }) \bigr )$ $\bigl ($resp., $\bigl ( e^{-r (t \wedge T_\beta )} \psi _r (X_{t \wedge T_\beta }) \bigr ) \bigr )$ is a strict local martingale if the boundary point $\alpha$ (resp., $\beta$) is inaccessible and entrance, and a martingale...

10. #### Stable limit theorem for $U$-statistic processes indexed by a random walk

Franke, Brice; Pène, Françoise; Wendler, Martin

20. #### Palm measures and rigidity phenomena in point processes

Ghosh, Subhroshekhar
We study the mutual regularity properties of Palm measures of point processes, and establish that a key determining factor for these properties is the rigidity-tolerance behaviour of the point process in question (for those processes that exhibit such behaviour). Thereby, we extend the results of [23], [2], [20] to new ensembles, particularly those that are devoid of any determinantal structure. These include the zeroes of the standard planar Gaussian analytic function and several others.

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