Natalia Shenkman | Searcharxiv

arXiv2020

A Constructive Proof of a Concentration Bound for Real-Valued Random Variables

Almost 10 years ago, Impagliazzo and Kabanets (2010) gave a new combinatorial proof of Chernoff's bound for sums of bounded independent random variables. Unlike previous methods, their proof is constructive. This means that it provides an efficient randomized algorithm for the following task: given a set of Boolean random variables whose sum is not concentrated around its expectation, find a subset of statistically dependent variables. However, the algorithm of Impagliazzo and Kabanets (2010) is given only for the Boolean case. On the other hand, the general proof technique works also for real-valued random variables, even though for this case, Impagliazzo and Kabanets (2010) obtain a concentration bound that is slightly suboptimal. Herein, we revisit both these issues and show that it is relatively easy to extend the Impagliazzo-Kabanets algorithm to real-valued random variables and to improve the corresponding concentration bound by a constant factor.

cs.DM↗

arXiv2020

Multivariate geometric distributions, (logarithmically) monotone sequences, and infinitely divisible laws (with erratum by Natalia Shenkman)

Two stochastic representations of multivariate geometric distributions are analyzed, both are obtained by lifting the lack-of-memory (LM) property of the univariate geometric law to the multivariate case. On the one hand, the narrow-sense multivariate geometric law can be considered a discrete equivalent of the well-studied Marshall-Olkin exponential law. On the other hand, the more general wide-sense geometric law is shown to be characterized by the LM property and can differ significantly from its continuous counterpart, e.g., by allowing for negative pairwise correlations. For both families of distributions, their $d$-dimensional exchangeable subclass is characterized analytically via $d$-log-monotone, resp.\ $d$-monotone, sequences of parameters. Using this reparameterization, the subfamilies of distributions with conditionally i.i.d.\ components in the sense of de Finetti's theorem are determined. For these, a third stochastic construction based on a non-decreasing random walk is presented. The narrow-sense family is embedded in this construction when the increments of the involved random walk are infinitely divisible. The exchangeable narrow-sense law is furthermore shown to exhibit the multivariate right tail increasing (MRTI) dependence.

math.PR↗