On Lerch’s transcendent and the Gaussian random walk

Random walk kernels and learning curves for Gaussian process regression on random graphs

We consider learning on graphs, guided by kernels that encode similarity between vertices. Our focus is on random walk kernels, the analogues of squared exponential kernels in Euclidean spaces. We show that on large, locally treelike, graphs these have some counter-intuitive properties, specifically in the limit of large kernel lengthscales. We consider using these kernels as covariance matrice...

Random walk on random walks

In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density ρ ∈ (0,∞). At each step the random walk performs a nearest-neighbour jump, moving to the right with probability p◦ when it is on a vacant site and probability p• when it is on an oc...

On the Random Walk and Brownian Motion

Introduction. Consider on the one hand a separable Brownian motion (Wiener process) X(t), 0^t< «>, with A(0) = 0, and on the other a classical random walk S(n) = E"-i -^<> 1 a« < °°, where Xi, X2, • • • is a sequence of Bernoulli trials with probability 1/2 for A,= +1 and for A,= — 1. It is well known that for the sequence of processes Rk(t), l^k<<», defined by the formula 74(0 = 2~kS([22"t]), ...

A Random Walk with Exponential Travel Times

Consider the random walk among N places with N(N - 1)/2 transports. We attach an exponential random variable Xij to each transport between places Pi and Pj and take these random variables mutually independent. If transports are possible or impossible independently with probability p and 1-p, respectively, then we give a lower bound for the distribution function of the smallest path at point log...

Autoregressive Gaussian Random Vectors of First Order