Convergence of random variables, and the Borel-Cantelli lemmas

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1 Stat 205A Setember, 12, 2002 Convergence of ranom variables, an the Borel-Cantelli lemmas Lecturer: James W. Pitman Scribes: Jin Kim 1 Convergence of ranom variables Recall that, given a sequence of ranom variables, almost sure () convergence, convergence in P, an convergence in L sace are true concets in a sense that X. In this lecture, we will efine weak convergence, or convergence in istribution, P Xn P X, which we write, by abuse of notation,. Definition 1.1 (Convergence in istribution) We say if P( x) P(X x) for all x at which the RHS is continuous. This weak convergence aears in the central limit theorem. Theorem 1.2 Proof See Durrett. Ef( ) Ef(X) for all boune an continuous function f. Theorem 1.3 The following roerty hols among the tyes of convergence. L ( ) ( ) Proof ( ) can be roven by Chebychev inequality (with usually = 2): P( X > ɛ) E X ɛ, 1

2 Convergence of ranom variables, an the Borel-Cantelli lemmas 2 an ( ) is roven in Durrett. Exercise. counter examles but not : moving bli but not L : try = n1(0, 1/n). 0 but E 0 = 1, thus X 0 in L 1. L Proosition 1.4 (Inucing a Metric) cannot be metrize, but X n an can be metrize, e.g. using E( X 1). Furthermore, when so metrize, the sace of ranom variables are comlete. Proof See text (uses BCL). Definition 1.5 (Infinitely Often (i.o.) an Eventually (ev.)) Let q n be some statement, e.g., X > ɛ. We say (q n i.o.) if for all n, m n : q m is true, an (q n ev.) if n : for all m n : q m is true. Exercise. Note that the following hols; ɛ > 0, X < ɛ ev. ɛ > 0, X > ɛ i.o. (q n i.o.) = (q n ev.) Similarly, for a sequence of events A n in a rob sace (Ω, F, P), we can say the following; (A n i.o.) = {ω : ω A n i.o.} = n A m (A n ev.) = {ω : ω A n ev.} = n A m (A n i.o.) c = A c n ev. Main alication of the iea of i.o. an ev. is to the roof of convergence. For examle, since we have ( ) = ɛ>0 ( X < ɛ ev.), P( ) = lim ɛ 0 P( X < ɛ ev.). Since the basic criterion for convergence can be written as ( ) ɛ > 0, P( X > ɛ i.o.) = 0, we are intereste in conitions in some sequence of events A n so that P(A n ) i.o. = 0.

3 Convergence of ranom variables, an the Borel-Cantelli lemmas 3 2 Borel-Cantelli Lemma Theorem 2.1 (Borel-Cantelli Lemma). 1. If n P(A n) <, then P(A n i.o.) = If n P(A n) = an A n are ineenent, then P(A n i.o.) = 1. There are many ossible substitutes for ineenence in BCL II, incluing Kochen-Stone Lemma. Before rooving BCL, notice that 1(A n i.o.) = lim su 1(A n ) 1(A n ev.) = lim inf 1(A n ) (A n i.o.) = lim m P( n>m A n ) ( as m, n m A n ) (A n i.o.) = lim m P( n>m A n ) ( as m, n m A n ). Therefore, P(A n ev.) lim inf P(A n) by Fatou s lemma lim su P(A n ) obvious from efinition P(A n i.o.) uel of Fatou s lemma (i.e. aly to ( ) ) Pf of BCL I P(A n i.o.) = lim P( n ma n ) m lim P(A n ) = 0 m n m since P(A n ) <. i=1 Pf of BCL I (Alternative metho) Consier a ranom variable N := 1 (A n), i.e. the number of events that occur. Then E[N] = n=1 P(A n) by the Monotone Convergence Theorem, an P(A n ) < = E[N] < n=1 = P(N < ) = 1 = P(N = ) = 0 = P(A n i.o.) = 0 because (N = ) (A n i.o.).

4 Convergence of ranom variables, an the Borel-Cantelli lemmas 4 Pf of BCL II We will show that P(A c n ev.) = 0. P(A c n ev.) = lim P( A c m) = = lim lim (1 P(A m )) lim = lim ex ( P(A c m)) = 0 P(A c m) (1) For (1) we use the following fact (ue to the ineenence of A n ); P( A c m) = lim P( n m NA c m) = lim N N an 1 x ex( x) was use in (2). n m N ex ( P(A c m)) (2) P(A c m) = P(A c m) As a trivial examle, consier A n = (0, 1/n) in (0, 1). Then, P(A n ) = 1/n, P(A n ) =, but P(A n i.o.) = P( ) = 0. Intuitive examle Consier ranom walk in Z, = 0, 1, S n = X 1 + +,, n = 0, 1, where X i are ineenent in Z. In the simlest case, each X i has uniform istribution on 2 ossible strings. i.e., if = 3, we have 2 3 = 8 neighbors (+1, +1, +1). ( 1, 1, 1) Note that each coorinate of S n oes a simle coin-tossing walk ineenently. We can rove that. n m P(S n = 0 i.o.) = { 1 if = 1 or 2 (recurrent) 0 if 3 (transient). (3) Sketch of Pf of (3) Let us start with = 1, then P(S 2n = 0) = ( ) 2n 2 2n c as n. n n where we use the fact, n! ( n e ) n 2πn. Note ( ) { 1 n = = 1, 2 < = 3, 4, (4) BC II an (4) together gives (3). Because X 0, thus it is enough to unerstan as convergence to 0. Proosition 2.2 The following are equivalent:

5 Convergence of ranom variables, an the Borel-Cantelli lemmas ɛ > 0, P( > 0 i.o.) = 0 3. M n 0 where M n := su n k X k 4. ɛ n 0 : P( > ɛ n i.o.) = 0 If we nee to show but o not know X, then it might be easier to show instea that P( is a Cauchy sequence) = 1. This leas to the following; Lemma 2.3 Let be any sequence of ranom variables, an efine M n := su n m X m. Then X : M n 0 Proof Consier M n := su n m, X m X. Notice M n. Thus M n 0 Mn 0 Combine with the revious result to finish the roof.