In this note we show how the properties of association and negative association can be combined with Stein’s method for compound Poisson approximation. Applications include k–runs in iid Bernoulli trials, an urn model with urns of limited capacity and extremes of random variables.

Abstract: For each n ≥ 1, let {Xn,j , j ≥ 1} be i.i.d. Bernoulli random variables with P{Xn,1 = 1} = pn = 1 − qn = 1 − P{Xn,1 = 0}, 0 < pn < 1. Define Wn,mn = ∑mn j=1 (Rn,j −an), where Rn,j = inf{k ≥ 0 : Xn,j+k = 0} is the number of success runs starting at j, mn is a sequences of positive integers, and an = pn/qn. We show that, under the condition mnpn → ∞, the central limit theorem Wn,mn σmn ...

The static pull-in instability of beam-type micro-electromechanical systems is theoretically investigated. Two engineering cases including cantilever and double cantilever micro-beam are considered. Considering the mid-plane stretching as the source of the nonlinearity in the beam behavior, a nonlinear size-dependent Euler-Bernoulli beam model is used based on a modified couple stress theory, c...

In this paper, the stability of a functionally graded magneto-electro-elastic (FG-MEE) micro-beam under actuation of electrostatic pressure is studied. For this purpose Euler-Bernoulli beam theory and constitutive relations for magneto-electro-elastic (MEE) materials have been used. We have supposed that material properties vary exponentially along the thickness direction of the micro-beam. Gov...

Using the integral expression of Mathieu's series and some integral and analytic inequalities involving periodic functions and the generating function of Bernoulli numbers, we present several new inequalities and estimates for Mathieu's series and generalize Math-ieu's series. Two open problems are proposed.

Scan statistics have been widely applied to test for unusual cluster of events in many scientific areas. It has been of practical interest on how to select the window size of a scan statistic. An adaptive procedure for multiple window scan statistics is proposed and the distributions are studied for independent identically distributed Bernoulli trials and uniform observations on (0, 1) in one-d...

Using Szenes formula for multiple Bernoulli series, we explain how to compute Witten series associated to classical Lie algebras. Particular instances of these series compute volumes of moduli spaces of flat bundles over surfaces, and also multiple zeta values.

We discuss the asymptotic expansions of certain products of Bernoulli numbers and factorials, e.g., n ∏

In [7] we further proved that H1(p−1) ≡ 0 (mod p ) if and only if p divides the numerator of Bp−3, which never happens for primes less than 12 million [2]. There is another important equivalent statement of Wolstenholme’s Theorem by using combinatorics. D.F. Bailey [1] generalizes it to the following form. Theorem 1.1. ([1, Theorem 4]) Let n and r be non-negative integers and p ≥ 5 be a prime. ...