lectr13 - 13. The Weak Law and the Strong Law of Large...

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1 13. The Weak Law and the Strong Law of Large Numbers James Bernoulli proved the weak law of large numbers (WLLN) around 1700 which was published posthumously in 1713 in his treatise Ars Conjectandi. Poisson generalized Bernoulli’s theorem around 1800, and in 1866 Tchebychev discovered the method bearing his name. Later on one of his students, Markov observed that Tchebychev’s reasoning can be used to extend Bernoulli’s theorem to dependent random variables as well. In 1909 the French mathematician Emile Borel proved a deeper theorem known as the strong law of large numbers that further generalizes Bernoulli’s theorem. In 1926 Kolmogorov derived conditions that were necessary and sufficient for a set of mutually independent random variables to obey the law of large numbers. PILLAI
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2 Let be independent, identically distributed Bernoulli random Variables such that and let represent the number of “successes” in n trials. Then the weak law due to Bernoulli states that [see Theorem 3-1, page 58, Text] i.e., the ratio “total number of successes to the total number of trials” tends to p in probability as n increases . A stronger version of this result due to Borel and Cantelli states that the above ratio k/n tends to p not only in probability , but with probability 1. This is the strong law of large numbers (SLLN). i X , 1 ) 0 ( , ) ( q p X P p X P i i = = = = n X X X k + + + = 2 1 " {} . 2 ε ε n pq p P n k > (13-1) PILLAI
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3 What is the difference between the weak law and the strong law? The strong law of large numbers states that if is a sequence of positive numbers converging to zero, then From Borel-Cantelli lemma [see (2-69) Text], when (13-2) is satisfied the events can occur only for a finite number of indices n in an infinite sequence, or equivalently, the events occur infinitely often, i.e., the event k/n converges to p almost-surely. Proof: To prove (13-2), we proceed as follows. Since } { n ε {} . 1 < = n n p P n k ε (13-2) { } n p n k ε < 4 4 4 n np k p n k ε ε≥ PILLAI = nn k n Ap ε −≥
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4 we have and hence where By direct computation {} () ε ε ε ε< + = = p P p P n n k p np k n k n k n n k ) ( ) ( 4 4 4 4 0 4 4 4 0 4 ) ( ) ( n k p np k p P n k n n k ε ε = (13-3) k n k n i i n q p k X P k p = = = = k n 1 ) ( } { } { 1 4 4 1 0 4 ) ( ) ( ) ( ) ( ) ( = = = = = n i i n i i n k n p X E np X E k p np k PILLAI
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5 since Substituting (13-4) also (13-3) we obtain Let so that the above integral reads and hence , 3 )] 1 ( 3 [ ) )( 1 ( 3 ) ( ) ( ) ( ) 1 ( 3 ) ( ) ( ) 1 ( 4 ) ( ) ( } ) {( 2 2 3 3 2 11 2 3 1 4 1111
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lectr13 - 13. The Weak Law and the Strong Law of Large...

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