Two-boundary problems for a random walk with negative geometric jumps are considered, and the corresponding results for a usual semicontinuous random walk are generalized for them. The following results are obtained: the probability distribution of ruin is found and expressed in terms of the lower and upper boundaries; formulas are given for the joint distribution of the infimum, supremum, and ...

In this paper, we study a graph parameter that was recently introduced, the burning number, focusing on a few probabilistic aspects of the problem. The original burning number is revisited and analyzed for binomial random graphs G(n, p), random geometric graphs, and the Cartesian product of paths. Moreover, new variants of the burning number are introduced in which a burning sequence of vertice...

We study a random graph Gn that combines certain aspects of geometric random graphs and preferential attachment graphs. This model yields a graph with power-law degree distribution where the expansion property depends on a tunable parameter of the model. The vertices of Gn are n sequentially generated points x1, x2, . . . , xn chosen uniformly at random from the unit sphere in R. After generati...

Motivated by the problem of efficiently collecting data from wireless sensor networks via a mobile sink, we present an accelerated random walk on Random Geometric Graphs. Random walks in wireless sensor networks can serve as fully local, lightweight strategies for sink motion that significantly reduce energy dissipation but introduce higher latency in the data collection process. In most cases ...

In this work we give precise asymptotic expressions on the probability of the existence of fixed-size components at the threshold of connectivity for random geometric graphs.

Projective transformations are a fundamental part of imaging with a camera. While the geometric properties of the transformation have been well studied, statistical estimation under the transformation has not been sufficiently addressed. In particular, the properties of random variables transformed by a projective transformation have not been adequately explored. In this paper, we study the eff...

A Markov chain is geometrically ergodic if it converges to its invariant distribution at a geometric rate in total variation norm. We study geometric ergodicity of deterministic and random scan versions of the two-variable Gibbs sampler. We give a sufficient condition which simultaneously guarantees both versions are geometrically ergodic. We also develop a method for simultaneously establishin...

‎The main purpose of this note is to establish some bounds in Poisson approximation for row-wise arrays of independent geometric distributed random variables using the operator method‎. ‎Some results related to random sums of independent geometric distributed random variables are also investigated.

We study d–records in sequences generated by independent geometric random variables and derive explicit and asymptotic formulæ for expectation and variance. Informally speaking, a d–record occurs, when one computes the d–largest values, and the variable maintaining it changes its value while the sequence is scanned from left to right. This is done for the “strict model,” but a “weak model” is a...