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MS&E 321 Stochastic Systems Prof. Peter W. Glynn Appendix A: A.1 Spring 12-13 June 1, 2013 Page 1 of 3 Convergence Concepts for Random Variables Convergence Definitions Recall that a real-valued rv X is a function X : Ω → R. Given that a rv is a function, there are many different mechanisms for determining whether a sequence of rvs/functions (Xn : n ≥ 0) converges to a limit rv/function X∞ . Almost Sure Convergence: We say that (Xn : n ≥ 1) converges almost surely to X∞ if P (A) = 1, where A = {ω : Xn (ω) → X∞ (ω) as n → ∞} and write Xn → X∞ a.s. as n → ∞ when this convergence holds. This type of convergence is equivalently called: convergence with probability one (written Xn → X∞ w.p. 1 as n → ∞); convergence almost everywhere (written Xn → X∞ a.e. as n → ∞); convergence almost certainly (written Xn → X∞ a.c. as n → ∞). Remark A.1.1 Note that the event A depends on the infinite-dimensional joint distribution of (Xn : 1 ≤ n ≤ ∞). As a result, rigorous discussion of probability assignments to events like A needed to await the development of measure-theoretic probability in the early 20th century. Convergence in p’th Mean: We say that (Xn : n ≥ 1) converges in p’th mean (for p > 0) to X∞ if E|Xn |p < ∞ for n ≥ 1 and kXn − X∞ kp → 0 as n → ∞, where kY kp = E 1/p |Y |p Lp for E|Y |p < ∞. When this convergence holds, we write Xn −→ X∞ as n → ∞. Convergence in Probability We say that (Xn : n ≥ 1) converges in probability to X∞ if, for each > 0, P (|Xn − X∞ | > ) → 0 p as n → ∞, in which case we write Xn → − X∞ as n → ∞. Remark A.1.2 Convergence in probability and convergence in p’th mean is a statement about the joint distribution of the two rv’s Xn and X∞ for n large. Remark A.1.3 Convergence in p’th mean implies convergence in probability, since Markov’s inequality implies that E|Xn − X∞ |p P (|Xn − X∞ | > ) ≤ . p 1 § APPENDIX A: CONVERGENCE CONCEPTS FOR RANDOM VARIABLES Exercise A.1.1 Prove that almost sure convergence implies convergence in probability. Weak Convergence We say that (Xn : n ≥ 1) converges weakly to X∞ if X∞ is a finite-valued rv for which Ef (Xn ) → Ef (X∞ ) as n → ∞ for each bounded and continuous function f : R → R, in which case we write Xn ⇒ X∞ as n → ∞. Weak convergence is equivalently called “convergence in distribution”. Remark A.1.4 Weak convergence is a statement about the distribution of Xn when n is large. Remark A.1.5 Weak convergence can be equivalently formulated as: Xn ⇒ X∞ as n → ∞ if and only if X∞ is a finite-valued rv for which P (Xn ≤ x) → P (X∞ ≤ x) as n → ∞ whenever x is a continuity point of P (X∞ ≤ ·). p Exercise A.1.2 Prove that Xn → − X∞ as n → ∞ implies that Xn ⇒ X∞ as n → ∞. Remark A.1.6 Convergence in probability does not imply almost sure convergence. Note that if Y = (Yn : n ≥ 0) is a nearest neighbor symmetric random walk on Z, then the recurrence of Y implies that Xn , I(Yn = 0) = 1 infinitely often a.s. so Xn 9 0 a.s. as n → ∞. But p P (Xn = 0) = P (Yn 6= 0) → 1 as n → ∞, so Xn → − 0 as n → ∞. The following diagram makes clear the relationship between these convergence concepts: Almost Sure Convergence Convergence in p’th mean @ @ @ R @ Convergence in Probability ? Weak Convergence A.2 Basic Facts on Almost Sure Convergence Fact 1: Suppose that Xn → X∞ a.s. as n → ∞ and Yn → Y∞ a.s. as n → ∞, where X∞ and Y∞ are finite-valued. If f : R2 → R is continuous, then h(Xn , Yn ) → h(X∞ , Y∞ ) a.s. as n → ∞. 2 § APPENDIX A: CONVERGENCE CONCEPTS FOR RANDOM VARIABLES Fact 2: Suppose that Xn → X∞ a.s. as n → ∞ and (Tn : n ≥ 1) is a sequence of Z+ -valued rv’s for which Tn → ∞ a.s. as n → ∞. Then, XTn → X∞ a.s. as n → ∞. A.3 Basic Facts on Convergence in Probability p Fact 1: Suppose that Xn → − X∞ , where X∞ is finite-valued. If h : R → R is continuous, then p h(Xn ) → − h(X∞ ) as n → ∞. p p Remark A.3.1 If Xn → − X∞ and Yn → − Y∞ as n → ∞ and ((Xn , Yn ) : 1 ≤ n ≤ ∞) are all defined p on a common sample space (i.e. are jointly distributed), then (Xn , Yn ) → − (X∞ , Y∞ ) as n → ∞. Fact 2: n → ∞. p p Suppose that Xn → X∞ a.s. as n → ∞ and Tn → − ∞ as n → ∞. Then, XTn → − X∞ as p p p Remark A.3.2 If Xn → − X∞ and Tn → − ∞ as n → ∞, it is not always the case that XTn → − X∞ as n → ∞. A.4 Basic Facts on Weak Convergence Fact 1: Suppose that Xn ⇒ X∞ as n → ∞. If h : R → R is continuous, then h(Xn ) ⇒ h(X∞ ) as n → ∞. In fact, if P (X∞ ∈ Dh ) = 0 (where Dh = {x ∈ R : h(·) is discontinuous at x}), then h(Xn ) ⇒ h(X∞ ) as n → ∞. Remark A.4.1 The above result is called the continuous mapping principle. p Fact 2: Suppose that Xn ⇒ X∞ as n → ∞ and Yn → − c as n → ∞, where c is deterministic. If 2 h : R → R is such that P ((X∞ , c) ∈ Dh ) = 0, then h(Xn , Yn ) ⇒ h(X∞ , c) as n → ∞. p Remark A.4.2 The above fact implies that if Xn ⇒ X∞ and Yn → − c, then Xn + Yn ⇒ X∞ + c and Xn Yn ⇒ cX∞ as n → ∞. p Remark A.4.3 If Xn ⇒ X∞ and Yn → − Y∞ as n → ∞, it is not always the case that h(Xn , Yn ) ⇒ 2 h(X∞ , Y∞ ) (even if h : R → R is continuous everywhere). p Remark A.4.4 If Xn ⇒ X∞ as n → ∞ and Tn → − ∞ as n → ∞, it is not always the case that XTn ⇒ X∞ as n → ∞. 3