80 research outputs found
On the asymptotic normality of the Legendre-Stirling numbers of the second kind
For the Legendre-Stirling numbers of the second kind asymptotic formulae are
derived in terms of a local central limit theorem. Thereby, supplements of the
recently published asymptotic analysis of the Chebyshev-Stirling numbers are
established. Moreover, we provide results on the asymptotic normality and
unimodality for modified Legendre-Stirling numbers
Mod-phi convergence I: Normality zones and precise deviations
In this paper, we use the framework of mod- convergence to prove
precise large or moderate deviations for quite general sequences of real valued
random variables , which can be lattice or
non-lattice distributed. We establish precise estimates of the fluctuations
, instead of the usual estimates for the rate of
exponential decay . Our approach provides us with a
systematic way to characterise the normality zone, that is the zone in which
the Gaussian approximation for the tails is still valid. Besides, the residue
function measures the extent to which this approximation fails to hold at the
edge of the normality zone.
The first sections of the article are devoted to a proof of these abstract
results and comparisons with existing results. We then propose new examples
covered by this theory and coming from various areas of mathematics: classical
probability theory, number theory (statistics of additive arithmetic
functions), combinatorics (statistics of random permutations), random matrix
theory (characteristic polynomials of random matrices in compact Lie groups),
graph theory (number of subgraphs in a random Erd\H{o}s-R\'enyi graph), and
non-commutative probability theory (asymptotics of random character values of
symmetric groups). In particular, we complete our theory of precise deviations
by a concrete method of cumulants and dependency graphs, which applies to many
examples of sums of "weakly dependent" random variables. The large number as
well as the variety of examples hint at a universality class for second order
fluctuations.Comment: 103 pages. New (final) version: multiple small improvements ; a new
section on mod-Gaussian convergence coming from the factorization of the
generating function ; the multi-dimensional results have been moved to a
forthcoming paper ; and the introduction has been reworke
Thirty-two Goldbach Variations
We give thirty-two diverse proofs of a small mathematical gem--the
fundamental Euler sum identity zeta(2,1)=zeta(3) =8zeta(\bar 2,1). We also
discuss various generalizations for multiple harmonic (Euler) sums and some of
their many connections, thereby illustrating both the wide variety of
techniques fruitfully used to study such sums and the attraction of their
study.Comment: v1: 34 pages AMSLaTeX. v2: 41 pages AMSLaTeX. New introductory
material added and material on inequalities, Hilbert matrix and Witten zeta
functions. Errors in the second section on Complex Line Integrals are
corrected. To appear in International Journal of Number Theory. Title change
Classical and Almost Sure Local Limit Theorems
We present and discuss the many results obtained concerning a famous limit
theorem, the local limit theorem, which has many interfaces, with Number Theory
notably, and for which, in spite of considerable efforts, the question
concerning conditions of validity of the local limit theorem, has up to now no
satisfactory solution. These results mostly concern sufficient conditions for
the validity of the local limit theorem and its interesting variant forms:
strong local limit theorem, strong local limit theorem with convergence in
variation. Quite importantly are necessary conditions, and the results obtained
are sparse, essentially: Rozanov's necessary condition, Gamkrelidze's necessary
condition, and Mukhin's necessary and sufficient condition. Extremely useful
and instructive are the counter-examples due to Azlarov and Gamkrelidze, as
well as necessary and sufficient conditions obtained for a class of random
variables, such as Mitalauskas' characterization of the local limit theorem in
the strong form for random variables having stable limit distributions. The
method of characteristic functions and the Bernoulli part extraction method,
are presented and compared. A second part of the survey is devoted to the more
recent study of the almost sure local limit theorem, instilled by Denker and
Koch. The almost sure local limit theorems established already cover the i.i.d.
case, the stable case, Markov chains, the model of the Dickman function. Our
aim in writing this monograph was notably to bring to knowledge many
interesting results obtained by the Lithuanian and Russian Schools of
Probability during the sixties and after, and which are essentially written in
Russian, and moreover often published in Journals of difficult access
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