On Generalized Binomial and Negative Binomial Distributions for Dependent Bernoulli Variables

We study the distributions of the random variables S_n and V_r related to a sequence of dependent Bernoulli variables, where S_n denotes the number of successes in n trials and V_r the number of trials necessary to obtain r successes. The purpose of this article is twofold: (1) Generalizing some results on the “nature” of the binomial and negative binomial distributions we show that S_n and V_r can follow any prescribed discrete distribution. The corresponding joint distributions of the Bernoulli variables are characterized as the solutions of systems of linear equations. (2) We consider a specific type of dependence of the Bernoulli variables, where the probability of a success depends only on the number of previous successes. We develop some theory based on new closed-form representations for the probability mass functions of S_n and V_r which enable direct computations of the probabilities.     

On moments of the supremum of normed weighted averages

Let {X, X_n; n ≥ 1} be a sequence of real-valued iid random variables, 0 < r < 2 and p > 0. Let D = { A = (a_nk; 1 ≤ k ≤ n, n ≥ 1); a_nk, ? R and sup_{n, k} |a_n,k| < ∞}. Set S_n( A ) = ∑ⁿ_k=1a_{n, k}X_k for A ? D and n ≥ 1. This paper is devoted to determining conditions whereby E{sup_{n ≥ 1}, |S_n( A )|/n^1/r}^p < ∞ or E{sup_{n ≥ 2} |S_n( A )|/2n log n)^1/2}^p < ∞ for every A ? D. This generalizes some earlier results, including those of Burkholder (1962), Choi and Sung (1987), Davis (1971), Gut (1979), Klass (1974), Siegmund (1969) and Teicher (1971).     

Strassen type limit points for moving averages of a wiener process

Let {W(s); 8 ≥ 0} be a standard Wiener process, and let β_N = (2a_N (log (N/a_N) + log log N)^-1/2, 0 < α_N ≤ N < ∞, where α_N↑ and α_N/N is a non-increasing function of N, and define γ_N(t) = β_N[W(n_N + ta_N) ? W(n_N)), 0 ≥ t ≥ 1, with n_N = N – a_N. Let K = {x ? C[0,1]: x is absolutely continuous, x(0) = 0 and }. We prove that, with probability one, the sequence of functions {γ_N(t), t ? [0,1]} is relatively compact in C[0,1] with respect to the sup norm ||·||, and its set of limit points is K. With a_N = N, our result reduces to Strassen's well-known theorem. Our method of proof follows Strassen's original, direct approach. The latter, however, contains an oversight which, in turn, renders his proof invalid. Strassen's theorem is true, of course, and his proof can also be rectified. We do this in a somewhat more general context than that of his original theorem. Applications to partial sums of independent identically distributed random variables are also considered.     

Asymptotic Properties of Random Weighted Empirical Distribution Function

This article studies the asymptotic properties of the random weighted empirical distribution function of independent random variables. Suppose X₁, X₂, ???, X_n is a sequence of independent random variables, and this sequence is not required to be identically distributed. Denote the empirical distribution function of the sequence by F_n(x). Based on the random weighting method and F_n(x), the random weighted empirical distribution function H_n(x) is constructed and the asymptotic properties of H_n are discussed. Under weak conditions, the Glivenko–Cantelli theorem and the central limit theorem for the random weighted empirical distribution function are obtained. The obtained results have also been applied to study the distribution functions of random errors of multiple sensors.     

Large deviation local limit theorems for ratio statistics

Let {T_n, n ≥ 1} be an arbitrary sequence of nonlattice random variables and let {S_n, n ≥ 1} be another sequence of positive random variables. Assume that the sequences are independent. In this paper we obtain asymptotic expression for the density function of the ratio statistic R_n = T_n/S_n based on simple conditions on the moment generating functions of T_n and S_n. When S_n = re, our main result reduces to that of Chaganty and Sethura-man[Ann. Probab. 13(1985):97-114]. We also obtain analogous results when T_n and S_n are both lattice random variables. We call our theorems large deviation local limit theorems for R_n, since the conditions of our theorems imply that R_n → c in probability for some constant c. We present some examples to illustrate our theorems.     

ON THE NUMBER OF RECORDS NEAR THE MAXIMUM

Recent work has considered properties of the number of observations X_j, independently drawn from a discrete law, which equal the sample maximum X_(n) The natural analogue for continuous laws is the number K_n(a) of observations in the interval (X_(n)–a, X_(n)], where a > 0. This paper derives general expressions for the law, first moment, and probability generating function of K_n(a), mentioning examples where evaluations can be given. It seeks limit laws for n→ and finds a central limit result when a is fixed and the population law has a finite right extremity. Whenever the population law is attracted to an extremal law, a limit theorem can be found by letting a depend on n in an appropriate manner; thus the limit law is geometric when the extremal law is the Gumbel type. With these results, the paper obtains limit laws for ‘top end’ spacings X_(n) - X_(n-j) with j fixed.     

Tolerance Limits Based on the Multivariate MaxMin Chart

In a previous paper, we have showed how to obtain sequences of number proved random. With this aim, we used sequences of noises y_n such that the conditional probabilities have Lipschitz coefficients not too large. We transformed them using Fibonacci congruences. Then, we obtained sequences x_n which admit the IID model for correct model. This method consisted to value the work of Marsaglia in order to build his CD-ROM. But we did not use Rap Music (as Marsaglia), but texts files. This method also uses an extractor and at the same time the notion of correct models. In this paper, we apply this method to numbers provided by machines or chips. Unfortunately, it is less sure than they have Lipschtiz coefficient not too large. But we can solve this problem: it suffices to use the Central Limit Theorem. We do it modulo 1. In this case, we use a new limit theorem, the XOR Limit theorem : asymptotic distribution of sum of random vectors modulo 1 are asymptotically independent. Then Lipschtiz coefficient of associated sequences are not too large and we can obtain IID sequences by using Fibonacci congruences.     

Central Limit Theorem for ISE of Kernel Density Estimators in Censored Dependent Model

In some long-term studies, a series of dependent and possibly censored failure times may be observed. Suppose that the failure times have a common continuous distribution function F. A popular stochastic measure of the distance between the density function f of the failure times and its kernel estimate f _n is the integrated square error(ISE). In this article, we derive a central limit theorem for the integrated square error of the kernel density estimators under a censored dependent model.     

Poisson approximations in selected metrics by coupling and semigroup methods with applications

Let S_n = X₁ + … + X_n, where X₁,…, X_n are independent Bernoulli random variables. In this paper, we evaluate probability metrics of the Wasserstein type between the distribution of S_n and a Poisson distribution. Our results show that, if E(S_n) = O(1) and if the individual probabilities of success of the X_i's tend uniformly to zero, then the general rate of convergence of the above mentioned metrics to zero is O(∑ⁿ_i = ₁P²_i). We also show that this rate is sharp and discuss applications of these results.     

On a random series in a hilbert space

Let (S_n) be partial sums of a non-degenerate sequence of Identically and independently distributed random variables taking values in a separable Hilbert space. Then for 0 ≤ β ≤ 3/2, the series converges almost nowhere. For β > 3/2 this may not be true.     

Central limit theorems for functionals of large sample covariance matrix and mean vector in matrix‐variate location mixture of normal distributions

In this paper, we consider the asymptotic distributions of functionals of the sample covariance matrix and the sample mean vector obtained under the assumption that the matrix of observations has a matrix‐variate location mixture of normal distributions. The central limit theorem is derived for the product of the sample covariance matrix and the sample mean vector. Moreover, we consider the product of the inverse sample covariance matrix and the mean vector for which the central limit theorem is established as well. All results are obtained under the large‐dimensional asymptotic regime, where the dimension p and the sample size n approach infinity such that p/n→c ∈ [0, + ∞) when the sample covariance matrix does not need to be invertible and p/n→c ∈ [0,1) otherwise.     

A multivariate extension of poisson's theorem

For n ≥ 1, let X_nl,…, X_nn be independent integer-valued random variables, and define S_n = X_nl+···+X_nn. In a recent paper, we obtained a simple proof for the convergence of the distribution of S_n to a Poisson distribution under very general conditions. In this paper, we extend that result to the multidimensional case.     

Limiting behavior of randomly weighted averages of symmetric heavy-tailed random variables

In this paper we consider a sequence of independent continuous symmetric random variables X₁, X₂, …, with heavy-tailed distributions. Then we focus on limiting behavior of randomly weighted averages S_n = R⁽ⁿ⁾₁X₁ + ??? + R⁽ⁿ⁾_nX_n, where the random weights R⁽ⁿ⁾₁, …, R_n⁽ⁿ⁾ which are independent of X₁, X₂, …, X_n, are the cuts of (0, 1) by the n ? 1 order statistics from a uniform distribution. Indeed we prove that c_nS_n converges in distribution to a symmetric α-stable random variable with c_n = n^{1 ? 1/α}/Γ^1/α(α + 1).     

Asymptotics for multisample statistics

Consider a family of square-integrable R^d-valued statistics S_k = S_k(X_1,k1; X_2,k2;…; X_m,km), where the independent samples X_i,kj respectively have k_i i.i.d. components valued in some separable metric space X_i. We prove a strong law of large numbers, a central limit theorem and a law of the iterated logarithm for the sequence {S_k}, including both the situations where the sample sizes tend to infinity while m is fixed and those where the sample sizes remain small while m tends to infinity. We also obtain two almost sure convergence results in both these contexts, under the additional assumption that S_k is symmetric in the coordinates of each sample X_i,kj. Some extensions to row-exchangeable and conditionally independent observations are provided. Applications to an estimator of the dimension of a data set and to the Henze-Schilling test statistic for equality of two densities are also presented.     

An extension of almost sure central limit theorem for self-normalized products of sums for mixing sequences

ABSTRACT

Let X, X₁, X₂, … be a sequence of strictly stationary φ-mixing random variables with EX = μ > 0. In this paper, we show that a self-normalized version of almost sure central limit theorem (ASCLT) holds under the assumptions that the mixing coefficients satisfy ∑^∞_{n = 1}φ^1/2(2ⁿ) < ∞ and the weight sequence {d_k} satisfies a mild growth condition similar to Kolmogorov’s condition for the LIL. This shows that logarithmic averages, used traditionally in ASCLT for products of sums, can be replaced by other averages, leading to considerably sharper results.     

Estimators for the Drift of Subfractional Brownian Motion

In this paper, we consider, using technique based on Girsanov theorem, the problem of efficient estimation for the drift of subfractional Brownian motion S^H ? (S^H_t)_{t ∈ [0, T]}. We also construct a class of biased estimators of James-Stein type which dominate, under the usual quadratic risk, the natural maximum likelihood estimator.     

Randomly weighted sums and their maxima with heavy-tailed increments and dependence structure

Consider the randomly weighted sums S_m(θ) = ∑^m_{i = 1}θ_iX_i, 1 ? m ? n, and their maxima M_n(θ) = max?_{1 ? m ? n}S_m(θ), where X_i, 1 ? i ? n, are real-valued and dependent according to a wide type of dependence structure, and θ_i, 1 ? i ? n, are non negative and arbitrarily dependent, but independent of X_i, 1 ? i ? n. Under some mild conditions on the right tails of the weights θ_i, 1 ? i ? n, we establish some asymptotic equivalence formulas for the tail probabilities of S_n(θ) and M_n(θ) in the case where X_i, 1 ? i ? n, are dominatedly varying, long-tailed and subexponential distributions, respectively.     

The Rate of Convergence of some Asymptotic Expansions for Distribution Approximations via an Esseen Type Estimate

Some asymptotic expansions not necessarily related to the central limit theorem are studied. We first observe that the smoothing inequality of Esseen implies the proximity, in the Kolmogorov distance sense, of the distributions of the random variables of two random sequences satisfying a sort of general asymptotic relation. We then present several instances of this observation. A first example, partially motivated by the the statistical theory of high precision measurements, is given by a uniform asymptotic approximation to (g(X + μ _n )) _n∈?, where g is some smooth function, X is a random variable and (μ _n ) _n∈? is a sequence going to infinity; a multivariate version is also stated and proved. We finally present a second class of examples given by a randomization of the interesting parameter in some classical asymptotic formulas; namely, a generic Laplace's type integral, randomized by the sequence (μ _n X) _n∈?, X being a Gamma distributed random variable.