ch2 - 1 Chapter 2 Some Basic Large Sample Theory 1. Modes...

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1 Chapter 2 Some Basic Large Sample Theory 1. Modes of Convergence Convergence in distribution, d Convergence in probability, p Convergence almost surely, a.s. Convergence in r th mean, r 2. Classical Limit Theorems Weak and strong laws of large numbers Classical (Lindeberg) CLT Liapounov CLT Lindeberg-Feller CLT Cram´ er-Wold device; Mann-Wald theorem; Slutsky’s theorem Delta-method 3. Replacing d by a.s. 4. Empirical Measures and Empirical Processes The empirical distribution function; the uniform empirical process The Brownian bridge process Finite-dimensional convergence A hint of Donsker’s Mann-Wald type theorem General empirical measures and processes 5. The Partial Sum Process and Brownian motion Brownian motion Existence of Brownian motion and Brownian bridge as continuous processes 6. Sample Quantiles
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Chapter 2 Some Basic Large Sample Theory 1M o des of Convergence Consider a probability space (Ω , A ,P ). For our frst three defnitions we suppose that X , X n , n 1 are all random variables defned on this one probability space. Defnition 1.1 We say that X n converges a.s. to X , denoted by X n a.s. X ,i ± X n ( ω ) X ( ω ) ±or all ω A where P ( A c )=0 , (1) or, equivalently, i±, ±or every ²> 0 P (sup m n | X m X | ) 0a s n →∞ . (2) Defnition 1.2 We say that X n converges in probability to X and write X n p X i± ±or every 0 P ( | X n X | ) s n . (3) Defnition 1.3 Let 0 <r< .W esay that X n converges in r th mean to X , denoted by X n r X ± E | X n X | r s n ±or ±unctions X n ,X L r ( P ) . (4) Defnition 1.4 We say that X n converges in distribution to X , denoted by X n d X ,or F n F , or L ( X n ) L ( X ) with L re±erring to the the “law” or “distribution”, i± the distribution ±unctions F n and F X n and X satis±y F n ( x ) F ( x )a s n ±or each continuity point x F. (5) Note that F n 1 [1 /n, ) d 1 [0 , ) F even though F n (0) = 0 does not converge to 1 = F (0). The statement d will carry with it the implication that F corresponds to a (proper) probability measure P . Defnition 1.5 A sequence o± random variables { X n } is uniformly integrable lim λ →∞ limsup n →∞ E © | X n | 1 [ | X n |≥ λ ] ª =0 . (6) 3
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4 CHAPTER 2. SOME BASIC LARGE SAMPLE THEORY Theorem 1.1 (Convergence implications). A. If X n a.s. X , then X n p X . B. If X n p X , then X n 0 a.s. X for some subsequence { n 0 } . C. If X n r X , then X n p X . D. If X n p X and | X n | r is uniformly integrable, then X n r X . If X n p X and limsup n E | X n | r E | X | r , then X n r X . E. If X n r X then X n r 0 X for all 0 <r 0 r . F. If X n p X , then X n d X . G. X n p X if and only if every subsequence { n 0 } contains a further subsequence { n 0 } for which X n 0 a.s.
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ch2 - 1 Chapter 2 Some Basic Large Sample Theory 1. Modes...

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