Exponential-type Inequalities Involving Ratios of the Modified Bessel Function of the First Kind and their Applications

Abstract

The modified Bessel function of the first kind, $I_{\nu }(x)$, arises in numerous areas of study, such as physics, signal processing, probability, statistics, etc. As such, there has been much interest in recent years in deducing properties of functionals involving $I_{\nu }(x)$, in particular, of the ratio $I_{\nu +1}(x)/I_{\nu }(x)$, when $\nu ,x\ge 0$. In this paper we establish sharp upper and lower bounds on $H(\nu ,x)={\sum }_{k=1}^{\infty }{I}_{\nu +k}(x)/I_{\nu }(x)$ for $\nu ,x\ge 0$ that appears as the complementary cumulative hazard function for a Skellam$(\lambda ,\lambda )$ probability distribution in the statistical analysis of networks. Our technique relies on bounding existing estimates of $I_{\nu +1}(x)/I_{\nu }(x)$ from above and below by quantities with nicer algebraic properties, namely exponentials, to better evaluate the sum, while optimizing their rates in the regime when $\nu +1\le x$ in order to maintain their precision. We demonstrate the relevance of our results through applications, providing an improvement for the well-known asymptotic $\exp (-x)I_{\nu }(x)\sim 1/\sqrt{2\pi x}$ as $x\to \infty$, upper and lower bounding $\mathbb{P}\left[W=\nu \right]$ for $W\sim Skellam({\lambda }_1,{\lambda }_2)$, and deriving a novel concentration inequality on the $Skellam(\lambda ,\lambda )$ probability distribution from above and below.

18 pages, 3 figures

Cited by in corpus (3)

• Asymptotics for the number of spanning trees in circulant graphs and degenerating d-dimensional discrete tori
• On the Propagation of Low-Rate Measurement Error to Subgraph Counts in Large Networks
• A Spherical Hidden Markov Model for Semantics-Rich Human Mobility Modeling