Weak convergence of convex stochastic processes

✍ Scribed by Miguel A. Arcones

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 530 KB
• Volume: 37
• Category: Article
• ISSN: 0167-7152
• DOI: 10.1016/s0167-7152(97)00115-6

✦ Synopsis

We discuss the weak convergence of convex stochastic processes. Let {Z.(t): t e T }, n >/ 1, be a sequence of stochastic processes, where T is an open convex set of R e, such that Z, : T ~ R is a convex function (for each ~o and each n), We show that {Z,(t):ts To} converges weakly to {Z(t):t ~ T}, for each compact set T o of T, if and only if, the finite dimensional distributions of {Z.(t): t s T} converge to those of {Z(t):t ~ T }. This is applied to triangular arrays of empirical processes. In particular, we consider random series and central limit theorems with normal and stable limits. The uniform compact law of the iterated logarithm is also discussed.