Martingales II Randomized Algorithms. Sariel Har-Peled. September 25, 2002

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1 Martigales II Radomized Algorithms Sariel Har-Peled September 25, 2002 The lectric Mok was a labor-savig device, like a dishwasher or a video recorder Dishwashers washed tedious dishes for you, thus savig you the bother of washig them yourself, video recorders watched tedious televisio for you, thus savig you the bother of lookig at it yourself; lectric Moks believed thigs for you, thus savig you what was becomig acreasigly oerous task, that of believig all the thigs the world epected you to believe Dirk Getly s Holistic Detective Agecy, Douglas Adams 1 Filters ad Martigales Defiitio 11 Give a σ-field Ω, F with F 2 Ω, a filter also filtratio is a ested sequece F 0 F 1 F of subsets of 2 Ω such that 1 F 0 {, Ω} 2 F 2 Ω 3 For 0 i, Ω, F i is a σ-field Ituitively, each F i defie a partitio of Ω ito blocks This partitios gettig more ad more refied as we progress with the filter ample 12 Cosider a algorithm A that uses radom bits, ad let F i be the σ-field geerated by the partitio of Ω ito the blocks B w, where w {0, 1} i The F 0, F 1,, F form a filter Defiitio 13 A radom variable X is said to be F i -measurable if for each R, the evet {X } is cotaied i F i ample 14 Let F 0,, F be the filter defied i ample 12 Let X be the parity of the bits Clearly, X is a valid evet oly i F why? Namely, it is oly measurable i F, but ot i F i, for i < Namely, a radom variable X is F i -measurable, oly if it is a costat o the blocks of F i Defiitio 15 Let Ω, F be ay σ-field, ad Y ay radom variable that takes o distict values o the elemetary elemets i F The X F X Y 1

2 2 Martigales Defiitio 21 A sequece of radom variables Y 1, Y 2,, is said to be a martigale differece sequece if for all i 0, Y i Y1,, Y i 1 0 Clearly, X 1,, is a martigale sequece iff Y 1, Y 2,, is a martigale differece sequece where Y i X i X i 1 Defiitio 22 A sequece of radom variables Y 1, Y 2,, is said to be a super martigale sequece if for all i, Y i Y1,, Y i 1 Y i 1, ad a sub martigale sequece if Y i Y1,, Y i 1 Y i 1 ample 23 Let U be a ur with b black balls, ad w white balls We repeatedly select a ball ad replace it by c balls havig the same color Let X i be the fractio of black balls after the first i trials This sequece is a martigale Ideed, let b + w + ic 1 be the umber of balls i the ur after the i-th trial Clearly, X i Xi 1,, X 0 X i 1 c 1 + X i 1 1 X i 1c 1 + X i Martigales, a alterative defiitio + 1 X i 1 Xi 1 1 X i 1 c X i 1 X i 1 Defiitio 24 Let Ω, F, Pr be a probability space with a filter F 0, F 1, Suppose that X 0, X 1,, are radom variables such that for all i 0, X i is F i -measurable The sequece X 0,, kx s a martigale provided, for all i 0, X i+1 Fi X i Lemma 25 Let Ω, F ad Ω, G be two σ-fields such that F G The, for ay radom variable X, X G F X F 2

3 Proof: X G F g G X G g F f PrX G g PrG g PrX Gg PrGg PrG g F f PrF f PrX Gg PrGg PrG g F f PrF f PrX Gg PrG g PrGg PrF f PrX G g PrF f PrX G g PrF f PrX F f PrF f F X F f Theorem 26 Let Ω, F, Pr be a probability space, ad let F 0,, F be a filter with respect to it Let X be ay radom variable over this probability space ad defie X i X F i the, the sequece X 0,, X s a martigale Proof: We eed to show that X i+1 Fi X i Namely, X i+1 Fi X F i+1 Fi X F i X i, by Lemma 25 ad by defiitio of X i Defiitio 27 Let f : D 1 D R be a real-valued fuctio with a argumets from possibly distict domais The fuctio f is said to satisfy the Lipschitz coditio If for ay 1 D 1,, D, ad i {1,, } ad ay y i D i, f 1,, i 1, i, i+1,, f 1,, i 1, y i, i+1,, 1 Defiitio 28 Let X 1,, X be a sequece of radom variables, ad a fuctio fx 1,, X defied over them that such that f satisfies the Lipschitz coditio The Dobb martigale sequece Y 0,, Y m is defied by Y 0 fx 1,, X ad Y i fx 1,, X X 1,, X i, for i 1,, Clearly, Y 0,, Y s a martigale, by Theorem 26 Furthermore, X i X i 1 1, for i 1,, Thus, we ca use Azuma s iequality o such a sequece 3

4 3 Occupacy Revisited We have m balls throwdepedetly ad uiformly ito bis Let Z deote the umber of bis that remais empty Let X i be the bi chose the i-th trial, ad let Z F X 1,, X m Clearly, we have by Azuma s iequality that Pr Z Z > λ m 2e λ2 /2 The followig is a etesio of Azuma s iequality show class We do ot provide a proof Theorem 31 Azuma s Iequality - Stroger Form Let XS 0, X 1,, be a martigale sequece such that for each k, X k X k 1 c k, where c k may deped o k The, for all t 0, ad ay λ > 0, λ 2 Pr X t X 0 λ 2 ep 2 t k1 c2 k Theorem 32 Let r m/, ad Z m be the umber of empty bis whe m balls are throw radomly ito bis The µ Z m 1 1 m e r ad for λ > 0, Pr Z m µ λ 2 ep λ2 1/2 2 µ 2 Proof: Let zy, t be the epected umber of empty bis, i there are Y empty bis i time t Clearly, zy, t Y 1 1 m t I particular, µ z, 0 m 1 1 Let F t be the σ-field geerated by the bis chose i the first t steps Let Z m be the ed of empty balls at time m, ad let Z t Z m Ft Namely, Z t is the epected umber of empty bis after we kow where the first t balls had bee placed The radom variables Z 0, Z 1,, Z m form a martigale Let Y t be the umber of empty bis after t balls where throw We have Z t 1 zy t 1, t 1 Cosider the ball throw the t-step Clearly: 1 With probability 1 Y t 1 / the ball falls ito a o-empty bi The Y t Y t 1, ad Z t zy t 1, t Thus, t Z t Z t 1 zy t 1, t zy t 1, t 1 Y t m t 1 1 m t+1 Y t m t 1 1 m t 4

5 2 Otherwise, with probability Y t 1 / the ball falls ito a empty bi, ad Y t Y t 1 1 Namely, Z t zy t 1, t t Z t Z t 1 zy t 1 1, t zy t 1, t 1 Y t m t Y t m t Y t 1 1 Y t m t 1 + Y t m t m t m t 1 Y t 1 Thus, Z 0,, Z m is a martigale sequece, where Z t Z t 1 t c t, where c t 1 1 m t We have c 2 t t /2m / 2m 1 1 1/ Now, deployig Azuma s iequality, yield the result 2 µ