131B+Handout+2 - ⇒ g Z n-→ g Z in probability Z n-→ Z...

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STA 131B Handout 2 Winter 2016 Let { Z n } be a sequence of random variables. Definition : Modes of Convergence 1. Z n -→ Z in probability ⇐⇒ Z n - Z -→ 0 in probability. ⇐⇒ For any > 0 , P ( | Z n - Z | > ) -→ 0, as n -→ ∞ . 2. Z n -→ Z in distribution ⇐⇒ Z n - Z -→ 0 in distribution. ⇐⇒ F Z n ( t ) -→ F Z ( t ) , t that is a point of continuity of F Z ( · ). 3. Z n -→ Z in quadratic mean ⇐⇒ E { ( Z n - Z ) 2 } -→ 0 . Properties : 1. Z n -→ Z in quadratic mean = Z n -→ Z in probability. 2. Z n -→ Z in probability = Z n -→ Z in distribution. * So convergence in quadratic mean and convergence in distribution are, respectively, the strongest and weakest mode of convergence among the three modes of convergence. 3. Z n -→ c (a constant) in probability ⇐⇒ Z n -→ c in distribution. 4. Z n -→ Z in probability, and g is a continuous function =
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Unformatted text preview: ⇒ g ( Z n )-→ g ( Z ) in probability. Z n-→ Z in distribution, and g is a continuous function = ⇒ g ( Z n ) = ⇒ g ( Z ) in distribution. 5. X n-→ X in probability, Y n-→ Y in probability = ⇒ X n ± Y n-→ X ± Y in probability; X n Y n-→ XY in probability; and X n /Y n-→ X/Y , if P ( Y = 0) = 0 . 6. Slutsky’s Theorem : X n-→ X in distribution, Y n-→ c (a constant) = ⇒ X n ± Y n-→ X ± c in distribution; X n Y n-→ cX in distribution; and X n /Y n-→ X/c , if c 6 = 0 . * This is a very important theorem to find the asymptotic distribution of several estimators....
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