We thus conclude that 1 pdn oj pdn 1 2 410 fact 3 the

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We thus conclude that 1 P[dn =OJ= P[dn = 1] = -. 2 (4.10) Fact 3. The sequence {dn, n ;::: 1} is iid. The previous fact proved in (4.10) that {dn} is identically distributed and thus we only have to prove {dn} is independent. For this, it suffices to pick n ;::: 1 and prove { dt. ... , dn} is independent. For (u1, ... , Un) E {0, 1}n, we have n n[d; = u;] = (.UtUz ... Un000 ... , .UtU2 ••• Un111 ••. ); i=1 Again, the left end of the interval is open due to our convention decreeing that we take non-terminating expansions when a number has two expansions. Since the probability of an interval is its length, we get n n oo 1 n P(n[d; = U; 1) = L 2; + L 2i - L 2; i=1 i=1 i=n+1 i=l 2-(n+1) 1 =- 1-! zn n = n P[d; =u;] i=1 where the last step used (4.10). So the joint mass function of d1, ... , dn factors into a product of individual mass functions and we have proved independence of the finite collection, and hence of {dn, n ;::: 1}. 4.4 More on Independence: Groupings It is possible to group independent events or random variables according to dis- joint subsets of the index set to achieve independent groupings. This is a useful property of independence.
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4.4 More on Independence : Groupings 101 Lemma 4.4.1 (Grouping Lemma) Let {8, t E T} be an independent family of a-fields. LetS be an index set and suppose for s E S that T 5 C T and {T 5 , s E S} is pairwise disjoint. Now define Then 8r, = V8,. tETs {8r,. s e S} is an independent family of a -fields. Remember that Vrer,8r is the smallest a-field containing all the 8(s. Before discussing the proof, we consider two examples. For these and other purposes, it is convenient to write when X and Y are independent random variables. Similarly, we write 81 JL 82 when the two a-fields 81 and 82 are independent. (a) Let {Xn, n 1} be independent random variables. Then a(Xj, j:::; n) JL a(Xj, j > n), n n+k LX; JL LX;, i=l i=n+l n n+k vx; JL v Xj. i=l j=n+l (b) Let {An} be independent events. Then Aj and Aj are inde- pendent. Proof. Without loss of generality we may suppose S is finite. Define Cr, := { n Ba : Ba E 8a, K C T 5 , K is finite.} aeK Then Cr, is a 1r-system for each s, and {Cr,, s E S} are independent classes. So by the Basic Criterion 4.1.1 we are done, provided you believe a(Cr.) = 8r,. Certainly it is the case that Cr, c 8r, and hence a(Crs> C 8r,.
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102 4. Independence Also, Cr. :::> Ba, 'Ia E Ts (we can take K ={a}) and hence a(Cr.) :::> Ba. 'Ia E T 5 • It follows that and hence a(Cr.) :::>a ( U Ba) = : V Ba. aeTs aeTs 4.5 Independence, Zero-One Laws, Borel-Cantelli Lemma 0 There are several common zero-one laws which identify the possible range of a random variable to be trivial. There are also several zero-one laws which provide the basis for all proofs of almost sure convergence . We take these up in turn . 4.5.1 Borel-Cantelli Lemma The Borel-Cantelli Lemma is very simple but still is the basic tool for proving almost sure convergence . Proposition 4.5 .1 (Borei-Cantelli Lemma.) Let {An} be any events. If LP(An) < 00, n then P([An i.o. ]) = P(limsupAn) = 0.
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