Sheet11 - 1 n ↓ 1 where P 1 2 = 0. (v) Show that there...

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Statistics 330/600 Due: Thursday 14 April 2005: Sheet 11 *(11.1) Suppose { X n : n N 0 } is a sequence of N 0 -valued random variables that converges almost surely to a random variable X . Suppose the sequence is adapted to a Fltration { F n : n N 0 } and that there exist numbers { p ij : i N 0 , j N 0 } such that P { X n = j | F n 1 }= X i N 0 { X n 1 = i } p ij . Suppose p 00 = 1 and p ii < 1 for all i 1. Show that P { X = i }= 0if i / ∈{ 0 , ∞} . Hint: Show that { X = i }=∪ k n k { X n = i } and P { X k = X k + 1 = ... = X k + m = i }= p m ii 0. *(11.2) Suppose { ( X n , F n ) : n N 0 } is a martingale with sup n P X 2 n < . As usual, write X n as X 0 + i n ξ i . (i) ±or each n and m with n < m , show that P | X m X n | 2 = m i = n + 1 σ 2 i , where σ 2 i : = P ξ 2 i . (ii) Show that there exists an X in L 2 (Ä, F , P ) for which P | X n X | 2 0as n →∞ . (iii) DeFne 1 n : = sup min ( i , j ) n | X i X j | . Show that P 1 2 n C i = n + 1 σ 2 i for some constant C . (iv) Show that
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Unformatted text preview: 1 n ↓ 1 where P 1 2 = 0. (v) Show that there exists an F ∞-measurable random variable Z for which X n → Z almost surely. (vi) Show that Z = X ∞ almost surely. (vii) ±or each n , show that X n = P ( X ∞ | F n ) almost surely. Hint: Read UGMTP §6.6. *(11.3) (branching process, µ > 1, σ 2 < ∞ ) UGMTP Problem 6.13. (11.4) (second moment SLLN via martingales) UGMTP Problem 6.15. (Kronecker’s lemma is given in UGMTP Problem 4.22. There is no need to prove it.) (11.5) (densities via martingales) UGMTP Problem 6.10....
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This note was uploaded on 11/21/2009 for the course STAT 330 taught by Professor Davidpollard during the Spring '09 term at Yale.

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