1. Weak convergence of stochastic processes

Abstract

The purpose of this course was to present results on weak convergence and invariance principle with statistical applications. As various techniques used to obtain different statistical applications, I have made an effort to introduce students to embedding technique of Skorokhod in chapter 1 and 5. Most of the material is from the book of Durrett [3]. In chapter 2, we relate this convergence to weak convergence on C [0,1] following the book of Billingsley [1]. In addition, we present the work in [1] on weak convergence on D[0,1] and D[0,∞) originated in the work of Skorokhod. In particular, we present the interesting theorem of Aldous for determining compactness in D[0,∞) as given in [1]. This is then exploited in chapter 4 to obtain central limit theorems for continuous semi-martingale due to Lipster and Shiryayev using ideas from the book of Jacod and Shiryayev [5]. As an application of this work we present the work of R. Gill [4], Kaplan-Meier estimate of life distributin with censored data using techniques in [2]. Finally in the last chapter we present the work on empirical processes using recent book of Van der Vaart and Wellner [6]. I thank Mr. J. Kim for taking careful notes and typing them. Finally, I dedicated these notes to the memory of A. V. Skorokhod from whom I learned a lot. 2 Weak Convergence In Metric Spaces 2.1 Cylindrical Measures Let {Xt, t ∈ T} be family of random variable on a probability space (Ω,F , P ). Assume that Xt takes values in (Xt,At). For any finite set S ⊂ T , XS = ∏ t∈S Xt,AS = ⊗ t∈S At, QS = P ◦ (Xt, t ∈ S)−1 where QS is the induced measure. Check if ΠS′S : XS′ → XS for S ⊂ S′, then QS = QS′ ◦Π−1 S′S (1) Suppose we are given a family {QS , S ⊂ T finite dimensional} where QS on (XS ,AS). Assume they satisfy (1). Then, there exists probability measure on (XT ,AT ) such that Q ◦Π−1 S = QS where XT = ∏ t∈T Xt,AT = σ (⋃ S⊂T CS ) , CS = Π−1 S (AS). Remark. For S ⊂ T and C ∈ CS , define Q0(C) = QS(A) C = Π−1 S (A) CS = Π−1 S (AS) We can define Q0 on ⋃ S⊂T CS . Then, for C ∈ CS and CS′ , Q0(C) = QS(A) = QS′(A), and hence Q0 is well-defined. Note that CS1 ∪ CS2 ∪ · · · ∪ CSk ⊂ CS1∪···∪Sk Q0 is finitely additive on ⋃ S⊂T CS . We have to show the countable additivity. Definition 2.1 A collection of subsets K ⊂ X is called a compact class if for every sequence {Ck} ⊂ K, for all n <∞, n ⋂