Small deviation probabilities of weighted sums under minimal moment assumptions

We examine small deviation probabilities of weighted sums of i.i.d.r.v. with a power decay at zero under moment assumptions close to necessary.     

Generation of Bernoulli processes

We consider five different algorithms for generating Bernoulli processes and discuss their efficiency. Simulation results are presented as well.     

An Efficient Algorithm for Exact Distribution of Discrete Scan Statistics

Waiting time random variables and related scan statistics have a wide variety of interesting and useful applications. In this paper, exact distribution of discrete scan statistics for the cases of homogeneous two-state Markov dependent trials as well as i.i.d. Bernoulli trials are discussed by utilizing probability generating functions. A simple algorithm has been developed to calculate the distributions. Numerical results show that the algorithm is very efficient and is capable of handling large problems. AMS 2000 Subject Classification 60J22, 60E05, 60J10     

关于非负独立同分布随机变量停止和的精确不等式的注记(英语)

Hitczenko[2]证明了不等式 E(sum from i=1 to γ(ζ_i))~γ≤2(γ-1)E(sum from i=1 to γ~2(ζ_i))~γ,1≤γ〈∞,其中(ζ_i)为非负独立随机变量,γ为停时,γ′为停时γ的一个复制品,且与(ζ_i)独立,2(γ-1)是最佳常数,我们证明了,对于非负独立同分布的(ζ_i),2(γ-1)也是最佳常数,从而解决了Hitczenko[2]提出的问题。     

Almost Sure Local Limit Theorems with Rate

We establish almost sure versions, with rate, of the local limit theorem for lattice distributed random variables. We also prove a new delicate correlation inequality for sums of i.i.d. lattice distributed random variables.     

Asymptotics for self-normalized random products of sums of i.i.d. random variables

Let be a sequence of independent and identically distributed positive random variables, which is in the domain of attraction of the normal law, and tn be a positive, integer random variable. Denote , , where denotes the sample mean. Then we show that the self-normalized random product of the partial sums, , is still asymptotically lognormal under a suitable condition about tn.     

Stability of Maxima of Random Variables with Multidimensional Indices

The paper discusses the stability of suitably-defined maxima of a set of i.i.d. random variables with multidimensional indices.It is shown that theorems of Gnedenko (1943) and Tomkins (1986) concerning relative stability and complete relative stability of maxima remain valid in the new setting.Moreover, a criterion for almost sure relative stability for maxima with multidimensional indices is presented, extending a result of Barndorff-Nielsen (1963).AMS 2000 Subject Classification. Primary—60F15, 60G60, 62G30, Secondary—10A25, 60G99     

Laws of the single logarithm for delayed sums of random fields II

In an earlier paper we extended Lai's (1974) law of the single logarithm for delayed sums to a class of delayed sums of random fields. In this paper we allow for more general windows.     

Lim sup behavior of sums of geometrically weighted i.i.d. random variables

Geometrically weighted i.i.d. random variables {Y_n} which are bounded above are shown to exhibit iterated logarithm type behavior. Specifically, if b > 1 and if the lower tail of the distribution of Y₁ approaches 0 fast enough, then lim sup_n→∞(b?1) Σⁿ_j=1b¹Y_j?bⁿ⁺¹=L, almost certainly, where L is the essential supremum of Y₁.     

Moderate deviations and Erdös-Rényi-Shepp laws for weighted sums

Let X₀,X₁,… be i.i.d. random variables with E(X₀)=0, E(X²₀)=1 and E(exp{tX₀})<∞ for any |t|<t₀. We prove that the weighted sums V(n)=∑_j=0^∞a_j(n)X_j, n?1 obey a moderately large deviation principle if the weights satisfy certain regularity conditions. Then we prove a new version of the Erdös-Rényi-Shepp laws for the weighted sums.     

Saddlepoint approximation for moments of random variables

In this paper, we introduce a saddlepoint approximation method for higher-order moments like E(S − a)₊ ^m , a>0, where the random variable S in these expectations could be a single random variable as well as the average or sum of some i.i.d random variables, and a > 0 is a constant. Numerical results are given to show the accuracy of this approximation method.     

Degenerate Bernoulli polynomials, generalized factorial sums, and their applications

We prove a general symmetric identity involving the degenerate Bernoulli polynomials and sums of generalized falling factorials, which unifies several known identities for Bernoulli and degenerate Bernoulli numbers and polynomials. We use this identity to describe some combinatorial relations between these polynomials and generalized factorial sums. As further applications we derive several identities, recurrences, and congruences involving the Bernoulli numbers, degenerate Bernoulli numbers, generalized factorial sums, Stirling numbers of the first kind, Bernoulli numbers of higher order, and Bernoulli numbers of the second kind.     

Complete moment convergence for i.i.d. random variables

In the paper, the complete moment convergence is obtained for i.i.d. random variables such that all moments exist, but the moment generating function does not exist. The main results extend the related known works due to Gut and Stadtmüller.     

Precise rates in the law of logarithm for the moment convergence of i.i.d. random variables

Let be a sequence of i.i.d. random variables with EX=0 and EX²=σ²<∞. Set , Mn=maxk?n|Sk|, n?1. Let r>1, then we obtain

     

An Extension of the Kolmogorov–Feller Weak Law of Large Numbers with an Application to the St. Petersburg Game

The Kolmogorov–Feller weak law of large numbers for i.i.d. random variables without finite mean is extended to a larger class of distributions, requiring regularly varying normalizing sequences. As an application we show that the weak law of large numbers for the St. Petersburg game is an immediate consequence of our result.     

Precise asymptotics for a series of T. L. Lai

Let be i.i.d. random variables with , and set . We prove that, for

under the assumption that and Necessary and sufficient conditions for the convergence of the sum above were established by Lai (1974).

     

On irregular prime power divisors of the Bernoulli numbers

Let ( ) denote the usual th Bernoulli number. Let be a positive even integer where or . It is well known that the numerator of the reduced quotient is a product of powers of irregular primes. Let be an irregular pair with . We show that for every the congruence has a unique solution where and . The sequence defines a -adic integer which is a zero of a certain -adic zeta function originally defined by T. Kubota and H. W. Leopoldt. We show some properties of these functions and give some applications. Subsequently we give several computations of the (truncated) -adic expansion of for irregular pairs with below 1000.

     

Estimates for the Rapid Decay of Concentration Functions of n-Fold Convolutions

We estimate the concentration functions of n-fold convolutions of one-dimensional probability measures. The Kolmogorov–Rogozin inequality implies that for nondegenerate distributions these functions decrease at least as O(n ^–1/2). On the other hand, Esseen⁽³⁾ has shown that this rate is o(n ^–1/2) iff the distribution has an infinite second moment. This statement was sharpened by Morozova.⁽⁹⁾ Theorem 1 of this paper provides an improvement of Morozova's result. Moreover, we present more general estimates which imply the rates o(n ^–1/2).