Two counterexamples, addressing questions raised in Adamczak (2019) and Poly Zheng (2019), are provided. Both counterexamples related to chaoses. Let F n = Y + Z , where the random variables belong different chaoses of uniformly bounded degree. It may be that ? a . s 0 L 2 ? E { sup | } < ? for some > yet fails converge a.s.

In this paper, we define the almost uniform convergence and the almost everywhere convergence for cone-valued functions with respect to an operator valued measure. We prove the Egoroff theorem for Pvalued functions and operator valued measure &theta; : R &rarr; L(P, Q), where R is a &sigma;-ring of subsets of X&ne; &empty;, (P, V) is a quasi-full locally convex cone and (Q, W) is a locally ...

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). ...