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0809Convergence2H - O.H Probability and Markov Chains MATH...

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O.H. Probability and Markov Chains – MATH 2561/2571 E09 3 Convergence of random variables In probability theory one uses many different modes of convergence of random vari- ables, many of which are crucial for applications. In this section we shall consider some of the most important of them: convergence in L p , convergence in probability and almost sure convergence. 3.1 Weak laws of large numbers Definition 3.1. Let p > 0 be fixed. We say that a sequence X j , j 1 , of random variables converges to a random variable X in L p (write X n L p X ) as n → ∞ , if E ˛ ˛ X n X ˛ ˛ p 0 as n → ∞ . Example 3.2. Let ` X n ´ n 1 be a sequence of random variables such that for some real numbers ( a n ) n 1 , we have P ` X n = a n ´ = r n , P ` X n = 0 ´ = 1 r n . (3.1) Then X n L p 0 iff E ˛ ˛ X n ˛ ˛ p ( a n ) p r n 0 as n → ∞ . The following result is often referred to as the L 2 weak law of large numbers ( L 2 - WLLN ) Theorem 3.3. Let X j , j 1 , be a sequence of uncorrelated random variables with E X j = μ and Var ( X j ) C < . Denote S n = X 1 + · · · + X n . Then 1 n S n L 2 μ as n → ∞ . Proof. Immediate from E 1 n S n μ 2 = E ( S n ) 2 n 2 = Var ( S n ) n 2 Cn n 2 0 as n → ∞ . Definition 3.4. We say that a sequence X j , j 1 , of random variables converges to a random variable X in probability (write X n P X ) as n → ∞ , if for every fixed ε > 0 P ` | X n X | ≥ ε ´ 0 as n → ∞ . Example 3.5. Let the sequence ` X n ´ n 1 be as in (3.1). Then for every ε > 0 P ` | X n | ≥ ε ´ P ` X n negationslash = 0) = r n , so that X n P 0 if r n 0 as n → ∞ . In fact, the usual Weak Law of Large Numbers ( WLLN ) is nothing else than a convergence in probability result: Theorem 3.6. Under the conditions of Theorem 3.3, 1 n S n P μ as n → ∞ . Exercise 3.7. Derive Theorem 3.6 from the Chebyshev inequality. We prove Theorem 3.6 using the following simple fact: 26
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O.H. Probability and Markov Chains – MATH 2561/2571 E09 Lemma 3.8. Let X j , j 1 , be a sequence of random variables. If X n L p X for some fixed p > 0 , then X n P X as n → ∞ . Proof. Applying the generalized Markov inequality with g ( x ) = x p and Z n = | X n X | ≥ 0, we get: for every fixed ε > 0 P ` Z n ε ´ P ` | X n X | p ε p ´ E | X n X | p ε p 0 as n → ∞ . Proof of Theorem 3.6. The result follows immediately from Theorem 3.3 and Lemma 3.8. square As the following example shows, a high dimensional cube is almost a sphere. Example 3.9. Let X j , j 1 be iid with X j U ( 1 , 1) . Then the variables Y j = ( X j ) 2 satisfy E Y j = 1 3 , Var ( Y j ) E [( Y 2 j )] = E [( X j ) 4 ] 1 . Fix ε > 0 and consider the set A n,ε def = n z R n : (1 ε ) p n/ 3 < | z | < (1 + ε ) p n/ 3 o , where | z | denotes the usual Euclidean length in R n , | z | 2 = P n j =1 ( z j ) 2 . By WLLN , 1 n n X j =1 Y j 1 n n X j =1 ( X j ) 2 P 1 3 ; in other words, for every fixed ε > 0 , a point X = ( X 1 , . . . , X n ) chosen uniformly at random in ( 1 , 1) n satisfies P “˛ ˛ ˛ 1 n n X j =1 ( X j ) 2 1 3 ˛ ˛ ˛ ε P ` X negationslash∈ A n,ε ´ 0 as n → ∞ , ie., with probability close to one, X is located near the boundary of the
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