Schlomilch and Bell Series for Bessel's Functions, with Probabilistic Applications

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.

Cited by in corpus (20)

• Sharp moment estimates for polynomial martingales
• Central Limit Theorem and exponential tail estimations in mixed (anisotropic) Lebesgue spaces
• Moment and tail estimates for martingales and martingale transform, with application to the martingale limit theorem in Banach spaces
• Non-Improved Uniform Tail Estimates for Normed Sums of Independent Random Variables with Heavy Tails, with applications
• Central limit theorem and exponential tail estimations in hybrid Lebesgue-continuous spaces
• Moment and Tail Inequalities for Polynomial Martingales
• Multidimensional Probabilistic Rearrangement Invariant Spaces: A New Approach
• Equivalence between tails, Grand Lebesgue Spaces and Orlicz norms for random variables without Cramer's condition
• Central Limit Theorem in Holder spaces in the terms of majorizing measures
• Sharp moment and exponential tail estimates for U-statistics
• Entropy and Grand Lebesgue Spaces approach for Prokhorov-Skorokhod continuity of random processes, with tail estimates
• Uniform Limit Theorem and tail estimates for parametric u-statistics
• Non-asymptotic estimation for Bell function, with probabilistic applications
• Exact exponential tail estimations in the Law of Iterated Logarithm for Bochner's mixed Lebesgue spaces
• Moment and tail estimates and Banach space valued Non-Central Limit Theorem (NCLT) for sums of multi-indexed random variables, processes and fields
• Random processes and Central Limit Theorem in Besov spaces
• Moment and tail estimation for U-statistics with positive kernels
• Prokhorov-Skorokhod continuity of random fields. A natural approach
• Lower boundaries for parametric estimations in different norms
• Bilateral tail estimate for distribution of self normalizes sums of independent centered random variables under natural norming