An Inequality and Almost Sure Convergence

Jakimovski Methods and Almost-Sure Convergence

The classical summability methods of Borel (B) and Euler (E(λ), λ > 0) play an important role in many areas of mathematics. For instance, in summability theory they are perhaps the most important methods other than the Cesàro (Cα) and Abel (A) methods, and two chapters of the classic book of Hardy (1949) are devoted to them. In probability, the distinction between methods of Cesàro-Abel and Eul...

Almost sure convergence of the Bartlett estimator

We study the almost sure convergence of the Bartlett estimator for the asymptotic variance of the sample mean of a stationary weekly dependent process. We also study the a. s. behavior of this estimator in the case of long-range dependent observations. In the weakly dependent case, we establish conditions under which the estimator is strongly consistent. We also show that, after appropriate nor...

Almost sure convergence, convergence in probability and asymptotic normality

In the previous chapter we considered estimator of several different parameters. The hope is that as the sample size increases the estimator should get ‘closer’ to the parameter of interest. When we say closer we mean to converge. In the classical sense the sequence {xk} converges to x (xk → x), if |xk − x| → 0 as k → ∞ (or for every ε > 0, there exists an n where for all k > n, |xk − x| < ε). ...

Saturated Particle Filter: Almost sure convergence and improved resampling

Nonlinear stochastic dynamical systems are widely used to model physical processes. In many practical applications, the state variables are defined on a compact set of the state space, i.e., they are bounded or saturated. To estimate the states of systems with saturated variables, the Saturated Particle Filter (SPF) has recently been developed. This filter exploits the structure of the saturate...

Local Times and Almost Sure Convergence of Semi-Martingales