law of iterated logarithm proof The law of the iterated logarithm is stated as below Let X n be i i d random variables with mean zero and unit variance Define S n sum i 1 n X i Then almost surely limsup n rightarrow infty frac S n sqrt 2 n log log n 1
1 1 Outline of Proof Proofs of laws of the iterated logarithm are rather technical The rst and a central tool are the Kolmogorov upper and lower exponential bounds for tail probabilities The proof of the upper bound is manageable and will be presented but we omit the proof of In this paper I seek to present a proof for the Hartman Wintner law of iterated logarithm The law states that for any random walk Sn with the increment of zero mean and nite variance 2 the following holds almost surely limsup n Sn 2 2nloglog n 1 The proof presented in this paper requires the use of Skorokhod embedding
law of iterated logarithm proof
law of iterated logarithm proof
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PDF An Elementary Proof Of The Law Of Iterated Logarithm For Minima
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The Law Of Iterated Expectations Introduction To Nested Form YouTube
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The following equivalent formulation of the Law of the Iterated Logarithm illustrates how the Borel Cantelli lemmas will be used in the proof of the theorem To simplify notation de ne the function n p 2np 1 p loglogn Theorem 2 5 Khinchin For any 0 let A n be the event that on the nth ip 2 1 S n np n 1 In this lecture we prove the law of iterated logarithm First we prove the following lemma i LCLT local CLT If k o n3 4 and n kis even then P S n k s n 1 q 2 n e k 2 2n ii orF all n k 0 P S n k e k2 2n not tight by a polynomial factor Then by using the Borel Cantelli lemma we show that limsup n 1 p Sn 2nloglogn 1 a s
Chung s law of the iterated logarithm We recommend the Ref 1 for an extensive survey on both limsup and liminf laws of the iterated logarithm In this short note we establish the limit law of the iterated logarithm Theorem 1 1 Under the assumption 1 1 lim n 2loglogn 1 2 max 1 The next proposition which is known as the law of iterated logarithm shows in particular that Brownian paths are not latex frac 1 2 H lder continuous Theorem Let latex B t t ge 0 be a Brownian
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Law Of Iterated Logarithm Simulated Download Scientific Diagram
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The following is the statement due to Strassen of the law of the iterated logarithm given in Bauer 1 The proof indeed involves a lot of machinery but the machinery is laid out cleanly in Bauer s presentation We write 1 log x e L x log log x log x e Theorem 1 Law of the iterated logarithm Suppose that Xn F P The results obtained on the law of the iterated logarithm for sequences of independent random variables have served as a starting point for numerous researches on the applicability of this law to sequences of dependent random variables and vectors and to
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Irrational Logarithm Proof 1 Of 2 By Cases YouTube
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law of iterated logarithm proof - In this lecture we prove the law of iterated logarithm First we prove the following lemma i LCLT local CLT If k o n3 4 and n kis even then P S n k s n 1 q 2 n e k 2 2n ii orF all n k 0 P S n k e k2 2n not tight by a polynomial factor Then by using the Borel Cantelli lemma we show that limsup n 1 p Sn 2nloglogn 1 a s