  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.
