Using a Chen-Stein identity to obtain low variance simulation estimators

This paper is concerned with developing low variance simulation estimators of probabilities related to the sum of Bernoulli random variables. It shows how to utilize an identity used in the Chen-Stein approach to bounding Poisson approximations to obtain low variance estimators. Applications and numerical examples in such areas as pattern occurrences, generalized coupon collecting, system reliability, and multivariate normals are presented. We also consider the problem of estimating the probability that a positive linear combination of Bernoulli random variables is greater than some specified value, and present a simulation estimator that is always less than the Markov inequality bound on that probability.

Several Different Types of Convergence for ND Random Variables under Sublinear Expectations

The goal of this paper is to build average convergence and almost sure convergence for ND (negatively dependent) sequences of random variables under sublinear expectation space. By using the basic definition of sublinear expectation space, Markov inequality, and C r inequality, we extend average convergence and almost sure convergence theorems for ND sequences of random variables under sublinear expectation space, and we provide a way to learn this subject.

Least Squares Estimation for Discretely Observed Stochastic Lotka–Volterra Model Driven by Small α -Stable Noises

Stochastic Lotka–Volterra model driven by small α -stable noises is used to describe population dynamics perturbed by random environment. However, parameters in the model are always unknown. The contrast function is given to obtain least squares estimators. The consistency and the rate of convergence of the least squares estimators are proved, and the asymptotic distribution of the estimators are derived by Markov inequality, Cauchy–Schwarz inequality, and Gronwall’s inequality. Some numerical examples are provided to verify the effectiveness of the estimators.

Markov inequality on the graph of holomorphic function

The purpose of this paper is to show that the Markov inequality does not hold on the graph of holomorphic function.