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• W2523901629 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 ⋂" @default.
• W2523901629 created "2016-10-07" @default.
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• W2523901629 date "2016-09-26" @default.
• W2523901629 modified "2023-09-26" @default.
• W2523901629 title "1. Weak convergence of stochastic processes" @default.
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• W2523901629 doi "https://doi.org/10.1515/9783110476316-001" @default.
• W2523901629 hasPublicationYear "2016" @default.
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