introduction-probability.pdf

For the above types of convergence the random

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For the above types of convergence the random variables have to be defined on the same probability space. There is a variant without this assumption. Definition 3.8.2 [Convergence in distribution] Let (Ω n , F n , P n ) and , F , P ) be probability spaces and let f n : Ω n R and f : Ω R be random variables. Then the sequence ( f n ) n =1 converges in distribution to f ( f n d f ) if and only if E ψ ( f n ) E ψ ( f ) as n → ∞ for all bounded and continuous functions ψ : R R . We have the following relations between the above types of convergence. Proposition 3.8.3 Let , F , P ) be a probability space and f, f 1 , f 2 , ... : Ω R be random variables. (1) If f n f a.s., then f n P f . (2) If 0 < p < and f n L p f , then f n P f . (3) If f n P f , then f n d f . (4) One has that f n d f if and only if F f n ( x ) F f ( x ) at each point x of continuity of F f ( x ) , where F f n and F f are the distribution-functions of f n and f , respectively. (5) If f n P f , then there is a subsequence 1 n 1 < n 2 < n 3 < · · · such that f n k f a.s. as k → ∞ . Proof . See [4].
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3.8. MODES OF CONVERGENCE 73 Example 3.8.4 Assume ([0 , 1] , B ([0 , 1]) , λ ) where λ is the Lebesgue mea- sure. We take f 1 = 1I [0 , 1 2 ) , f 2 = 1I [ 1 2 , 1] , f 3 = 1I [0 , 1 4 ) , f 4 = 1I [ 1 4 , 1 2 ] , f 5 = 1I [ 1 2 , 3 4 ) , f 6 = 1I [ 3 4 , 1] , f 7 = 1I [0 , 1 8 ) , . . . This implies lim n →∞ f n ( x ) 0 for all x [0 , 1] . But it holds convergence in probability f n λ 0: choosing 0 < ε < 1 we get λ ( { x [0 , 1] : | f n ( x ) | > ε } ) = λ ( { x [0 , 1] : f n ( x ) = 0 } ) = 1 2 if n = 1 , 2 1 4 if n = 3 , 4 , . . . , 6 1 8 if n = 7 , . . . . . . As a preview on the next probability course we give some examples for the above concepts of convergence. We start with the weak law of large numbers as an example of the convergence in probability: Proposition 3.8.5 [Weak law of large numbers] Let ( f n ) n =1 be a sequence of independent random variables with E f k = m and E ( f k - m ) 2 = σ 2 for all k = 1 , 2 , . . . . Then f 1 + · · · + f n n P -→ m as n → ∞ , that means, for each ε > 0 , lim n P ω : | f 1 + · · · + f n n - m | > ε 0 . Proof . By Chebyshev’s inequality (Corollary 3.6.9) we have that P ω : f 1 + · · · + f n - nm n > ε E | f 1 + · · · + f n - nm | 2 n 2 ε 2 = E ( n k =1 ( f k - m )) 2 n 2 ε 2 = 2 n 2 ε 2 0 as n → ∞ . Using a stronger condition, we get easily more: the almost sure convergence instead of the convergence in probability. This gives a form of the strong law of large numbers .
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74 CHAPTER 3. INTEGRATION Proposition 3.8.6 [Strong law of large numbers] Let ( f n ) n =1 be a sequence of independent random variables with E f k = 0 , k = 1 , 2 , . . . , and c := sup n E f 4 n < . Then f 1 + · · · + f n n a.s. 0 . Proof . Let S n := n k =1 f k . It holds E S 4 n = E n k =1 f k 4 = E n i,j,k,l, =1 f i f j f k f l = n k =1 E f 4 k + 3 n k,l =1 k = l E f 2 k E f 2 l , because for distinct { i, j, k, l } it holds E f i f 3 j = E f i f 2 j f k = E f i f j f k f l = 0 by independence. For example, E f i f 3 j = E f i E f 3 j = 0 · E f 3 j = 0, where one gets that f 3 j is integrable by E | f j | 3 ( E | f j | 4 ) 3 4 c 3 4 . Moreover, by Jensen’s inequality, ( E f 2 k ) 2 E f 4 k c.
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