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 $\nu,x\geq 0$. 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 $\nu,x\geq 0$ 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 $\nu+1\leq x$ 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 $x\rightarrow \infty$, 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.