Martingale Convergence Theorems
In Chapter 17 we studied convergence theorems, but they were all of the type that one form of convergence, plus perhaps an extra condition, implies another type of convergence. What is unusual about martingale convergence theorems is that no type of conve
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In Chapter 17 we studied convergence theorems, but they were all of the type that one form of convergence, plus perhaps an extra condition, implies another type of convergence. What is unusual about martingale convergence theorems is that no type of convergence is assumed – only a certain structure – yet convergence is concluded. This makes martingale convergence theorems special in analysis; the only similar situation arises in ergodic theory. Theorem 27.1 (Martingale Convergence Theorem). Let (Xn )n≥1 be a submartingale such that supn E{Xn+ } < ∞. Then limn→∞ Xn = X exists a.s. (and is finite a.s.). Moreover, X is in L1 . [Warning: we do not assert here that Xn converges to X in L1 ; this is not true in general.] Proof. Let Un be the number of upcrossings of [a, b] before time n, as defined in (26.2). Then Un is non-decreasing hence U (a, b) = limn→∞ Un exists. By the Monotone Convergence Theorem E{U (a, b)} = lim E{Un } n→∞
1 sup E{(Xn − a)+ } b−a n
c 1 sup E{Xn+ } + |a| ≤ 0. Let Yn = Xn + c, then Yn is a nonnegative martingale and hence a nonnegative supermartingale, and we need only to apply the first part of this corollary. If (Xn )n≥1 is a martingale bounded above, then (−Xn )n≥1 is a martingale bounded below and again we are done. Theorem 27.1 gives the a.s. convergence to a r.v. X, which is in L1 . But it does not give L1 convergence of Xn to X. To obtain that we need a slightly stronger hypothesis, and we need to introduce the concept of uniform integrability. Definition 27.1. A subset H of L1 is said to be a uniformly integrable collection of random variables if lim sup E{1{|X|≥c} |X|} = 0.
c→∞ X∈H
Next we present two sufficient conditions to ensure uniform integrability. Theorem 27.2. Let H be a class of random variables a) If supX∈H E{|X|p } < ∞ for some p > 1, then H is uniformly integrable. b) If there exists a r.v. Y such that |X| ≤ Y a.s. for all X ∈ H and E{Y } < ∞, then H is uniformly integrable.
27 Martingale Convergence Theorems
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Proof. (a) Let k be a constant such that supX∈H E{|X|p } < k < ∞. If x ≥ c > 0, then x1−p ≤ c1−p , and multiplying by xp yields x ≤ c1−p xp . Therefore we have 0 1 1 0 E |X|1{|X|>c} ≤ c1−p E |X|p 1{|X|>c} ≤ hence limc→∞ supX∈H E{|X|1{|X|>c} } ≤ limc→∞ (b) Since |X| ≤ Y a.s. for all X ∈ H, we have
k cp −1
k , cp−1
= 0.
|X|1{|X|>c} ≤ Y 1{Y >c} . But limc→∞ Y 1{Y >c} = 0 a.s.; thus by Lebesgue’s dominated convergence theorem we have lim sup E{|X|1{|X|>c} } ≤ lim E{Y 1{Y >c} }
c→∞ X∈H
c→∞
= E{ lim Y 1{Y >c} } = 0. c→∞
For more results on uniform integrability we recommend [15, pp. 16–21]. We next give a strengthening of Theorem 27.1 for the martingale case. Theorem 27.3 (Martingale Convergence Theorem). a) Let (Mn )n≥1 be a martingale and suppose (Mn )n≥1 is a uniformly integrable collection of random variables. Then lim Mn = M∞ exists a.s.,
n→∞
M∞ is in L1 , and Mn converges to M∞ in L1 . Moreover Mn = E{M∞ | Fn }. b) Conversely let Y ∈ L1 and consider the martingale Mn = E{Y |Fn }. Then (Mn )n≥1 is a uniformly integrable collection of r.v.’s. In oth
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