[Paper Review] Schlomilch and Bell Series for Bessel's Functions, with Probabilistic Applications

E. Ostrovsky, L. Sirota|ArXiv.org|Apr 1, 2008

Mathematical Inequalities and Applications11 references22 citations

TL;DR

This paper introduces Schlömilch and Bell series for modified Bessel functions, establishing their asymptotic and non-asymptotic properties and linking them to moment inequalities for sums of independent random variables. It derives exact constants in Rosenthal-type moment inequalities, improves asymptotic bounds, and provides numerical algorithms for computation, with applications to Poisson and symmetrically distributed random variables in Banach and Hilbert spaces.

ABSTRACT

We have introduced and investigated so-called Shlomilchs and Bells series for modified Bessel's functions, namely, their asymptotic and non-asymptotic properties, connection with Stirling's and Bell's numbers etc. We have obtained exact constants in the moment inequalities for sums of centered independent random variables, improved their asymptotical properties, found lower and upper bounds, calculated a more exact approximation, elaborated the numerical algorithm for their calculation, studied the class of smoothing, etc.

Motivation & Objective

• To develop and analyze Schlömilch and Bell series for modified Bessel functions, particularly their asymptotic and non-asymptotic behavior.
• To establish exact constants in moment inequalities for sums of independent, centered random variables, improving upon existing asymptotic bounds.
• To connect these series with Stirling numbers, Bell numbers, and Poisson distributions to enable precise moment estimation.
• To provide a numerical algorithm for computing these constants and to study the class of smoothing functions in this context.
• To generalize results to Hilbert space-valued random variables and derive sharp bounds for p-norms of sums of symmetric random vectors.

Proposed method

• Define Schlömilch functions $ F_3(p;\theta,\beta) = \sum_{k=-\infty}^{\infty} |k|^p \theta^k I_k(\beta) $ and $ G_3(p;\theta,\beta) = \sum_{k=-\infty}^{\infty} k^p \theta^k I_k(\beta) $, where $ I_k(\beta) $ is the modified Bessel function of the first kind.
• Establish probabilistic interpretations: $ \mathbb{E}|\tau|^p = e^{-(\lambda+\mu)} F_3(p; \sqrt{\lambda/\mu}, 2\sqrt{\lambda\mu}) $, where $ \tau = \xi - \eta $, $ \xi, \eta \sim \text{Poisson}(\lambda, \mu) $.
• Introduce generalized Bell functions $ B_4(p;a,\lambda,\gamma) = \sum_{n=0}^{\infty} \frac{|n-a|^p \lambda^n}{e^\lambda \Gamma(n+\gamma+1)} $, with $ B_3, B_2, B_1 $ as special cases.
• Use saddle-point methods and generating function techniques to derive asymptotic expansions for $ F_3, G_3, B_1(p) $, and $ D_1(p) $ as $ p \to \infty $.
• Relate the moments to Stirling numbers of the second kind via identities such as $ \mathbb{E}\xi^{(r)} = \lambda^r $ and $ x^n = \sum_r s(n,r) x_{(r)} $.
• Apply these expansions to derive sharp bounds for $ \mathbb{E}\left|\sum \eta(i)\right|^p $ in Hilbert spaces, using $ Z(p) = S(p) $, where $ S(p) $ is the sharp constant in Rosenthal-type inequalities.

Experimental results

Research questions

• RQ1 What are the asymptotic and non-asymptotic properties of Schlömilch and Bell series for modified Bessel functions?
• RQ2 How can these series be used to derive exact constants in moment inequalities for sums of independent, centered random variables?
• RQ3 What is the connection between these series and Stirling numbers, Bell numbers, and Poisson distributions?
• RQ4 How do the derived bounds improve upon existing asymptotic estimates in Rosenthal-type inequalities?
• RQ5 What are the sharp constants for the p-norm of sums of symmetric, Hilbert space-valued random variables, and how are they related to Bessel functions?

Key findings

• The exact constant in the Rosenthal moment inequality for sums of independent, centered random variables is given by $ G(p) = \sum_{n=0}^{\infty} \frac{n^p}{e \cdot n!} $, which satisfies $ \exp(X(p)) \cdot (1 + C_{20} \log p / p) \leq G(p) \leq \exp(X(p)) \cdot (1 + C_{19} \log p / p) $, with $ X(p) = \frac{p}{e} \log \left( \frac{p}{e} \right) $.
• For $ p \to \infty $, $ G(p) \sim \frac{p}{e \log p} \left[ 1 + \frac{\log \log p}{\log p} + \frac{1 + \log t}{\log p} + \frac{\log^2 \log p}{\log^2 p} + o\left( \frac{\log \log p}{\log^2 p} \right) \right] $, where $ t $ is a parameter related to the variance.
• The sharp constant $ Z(p) = S(p) $ for the p-norm of sums of symmetric, Hilbert space-valued random variables satisfies $ \sup_{p \geq 4} Z(p)/g(p) = C_9 \approx 1.53572 $, and $ \sup_{p \geq 15} Z(p)/h(p) = C_{11} \approx 1.03734 $.
• For $ t \in (0, 1/2] $, the quantity $ R(p,t) = \mathbb{E}|\nu_1 - \nu_2|^p $, where $ \nu_1, \nu_2 \sim \text{Poisson}(t) $, satisfies $ R(p,t) = 2e^{-2t} \sum_{n=1}^{\infty} n^p I_n(2t) = e^{-2t} F_2(p; 2) $, and $ \lim_{p \to \infty} R(p,t)/g(p) = 1 $.
• The function $ Q^{p}(p,A,D) \cdot (D^2/A)^{1/(p-2)} = \mathbb{E}|\nu_1 - \nu_2|^p = e^{-2t} F_2(p;2) $, with $ t = 0.5(A/D^p)^{1/(p-2)} $, and $ R(p,t) $ is asymptotically $ \left[ t M(p/t) \right]^{1 - tM(p/t)/p} \left(1 + o(\log p / p) \right) $.
• Numerical tables show that $ u(t) = \sup_{p \geq 4} R(p,t)/g(p) $ decreases from 1.53572 at $ t=0.5 $ to 1.2163 at $ t=0.2 $, indicating improved concentration as $ t $ decreases.