2013/11/06 by Prakash Balachandran, Balachandran, Prakash, Weston D. Viles +3 · 2 citations
Biochemistry, Genetics and Molecular Biology · Mathematics · #Diffusion and Search Dynamics #FOS: Mathematics #Graph theory and applications #Probability (math.PR) #Random Matrices and Applications #Statistical Methods and Inference
paper · pdf · doi:10.48550/arxiv.1311.1450
openalex publication_date 2013/11/06 · openalex created_date 2022/09/02 · openalex updated_date 2026/07/28
The modified Bessel function of the first kind, I\ν(x), arises in\nnumerous areas of study, such as physics, signal processing, probability,\nstatistics, etc. As such, there has been much interest in recent years in\ndeducing properties of functionals involving I\ν(x), in particular, of\nthe ratio I\ν+1(x)/I\ν(x), when \ν,x\≥ 0. In this paper we\nestablish sharp upper and lower bounds on H(\ν,x)=\∑k=1\∞\nI\ν+k(x)/I_\ν(x) for \ν,x\≥ 0 that appears as the complementary\ncumulative hazard function for a Skellam(\λ,\λ) probability\ndistribution in the statistical analysis of networks. Our technique relies on\nbounding existing estimates of I\ν+1(x)/I\ν(x) from above and\nbelow by quantities with nicer algebraic properties, namely exponentials, to\nbetter evaluate the sum, while optimizing their rates in the regime when\n\ν+1\≤ x in order to maintain their precision. We demonstrate the\nrelevance of our results through applications, providing an improvement for the\nwell-known asymptotic \exp(-x)I\ν(x)\∼ 1/\√(2\π x) as\nx\→ \∞, upper and lower bounding \ℙ\[W=\ν\]\nfor W\∼ Skellam(\λ1,\λ2), and deriving a novel concentration\ninequality on the Skellam(\λ,\λ) probability distribution from\nabove and below.\n