The number of minima in a discrete sample

The number of times is considered that the minimum occurs in a sample from a discrete distribution. The special case of the geometric distribution is considered in some detail, and applied to the computation of the expected maximum of a sample from the Cantor distribution

A curious implication of Spitzers identity

Spitzer's identity can be read as follows: Let W_n denote the waiting time of the n-th customer in a G|G|1-queue, and let N be geometrically distributed on (0, 1, ...) and independent of $(W_n)^\infty_1$. Then $W_{N+1}$ is infinitely divisible

Poisson processes and a Bessel function integral

The probability of winning a simple game of competing Poisson processes turns out to be equal to the well-known Bessel function integral J(x, y) (cf. Y. L. Luke, Integrals of Bessel Functions, McGraw-Hill, New York, 1962). Several properties of J, some of which seem to be new, follow quite easily from this probabilistic interpretation. The results are applied to the random telegraph process as considered by Kac [Rocky Mountain J. Math., 4 (1974), pp. 497-509

Mod-phi convergence I: Normality zones and precise deviations

In this paper, we use the framework of mod-$\phi$ convergence to prove precise large or moderate deviations for quite general sequences of real valued random variables $(X_{n})_{n \in \mathbb{N}}$, which can be lattice or non-lattice distributed. We establish precise estimates of the fluctuations $P[X_{n} \in t_{n}B]$, instead of the usual estimates for the rate of exponential decay $\log( P[X_{n}\in t_{n}B])$. 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

On the degree of approximation of functions in C2π1 with operators of the Jackson type

AbstractLet C2π1 be the class of real functions of a real variable that are 2π-periodic and have a continuous derivative. The positive linear operators of the Jackson type are denoted by Ln,p(n ∈ N), where p is a fixed positive integer. The object of this paper is to determine the exact degree of approximation when approximating functions f ϵ C2π1 with the operators Ln,p. The value of maxx¦Ln,p(f x) − f(x)¦ is estimated in terms of ω1(f; δ), the modulus of continuity of f′, with δ = πn. Exact constants of approximation are obtained for the operators Ln,p (n ∈ N, p ≥ 2) and for the Fejér operators Ln,1 (n ∈ N). Furthermore, the limiting behaviour of these constants is investigated as n → ∞, and p → ∞, separately or simultaneously