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 ν (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 ν (x), in particular, of the ratio I ν+1 (x)/I ν (x), when ν, x ≥ 0. In this paper we establish sharp upper and lower bounds on H(ν, x) = ∞ k=1 I ν+k (x)/I ν (x) for ν, x ≥ 0 that appears as the complementary cumulative hazard function for a Skellam(λ, λ) probability distribution in the statistical analysis of networks. Our technique relies on bounding existing estimates of I ν+1 (x)/I ν (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 ν + 1 ≤ 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 ν (x) ∼ 1/ √ 2πx as x → ∞, upper and lower bounding P [W = ν] for W ∼ Skellam(λ 1 , λ 2 ), and deriving a novel concentration inequality on the Skellam(λ, λ) probability distribution from above and below.

Asymptotics for the Number of Spanning Trees in Circulant Graphs and Degenerating d-Dimensional Discrete Tori

“…This means that we need a sharp integrable upper bound of e −t I ν (t). To the best of our knowledge, the best upper bound of e −t I ν (t) is given in [3] and is not sharp enough.…”

“…They again improved this in [2] by finding an efficient way of solving the recurrence relation of a n . and C 1, 3 13 .…”

Modified Pólya-Gamma data augmentation for Bayesian analysis of directional data

“…Therefore, the series in (A.5) satisfies the monotonicity condition in its terms when t is larger than log (4) 3π 2 = 0.04682033.…”