On the Law of the Iterated Logarithm

Hartman, Philip; Wintner, Aurel

doi:10.2307/2371287

Cited by 330 publications

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“…The existence of a2 is sufficient to assert that lim sup (Sk -kß) {2&o-2log log k}~ll2=l with probability 1 (Hartman and Wintner [7]). Using (5.1) it is easily seen that this implies the following theorem.…”

Section: Theoremmentioning

confidence: 99%

Fluctuation Theory of Recurrent Events

Feller¹

1949

Transactions of the American Mathematical Society

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Section: Theoremmentioning

confidence: 99%

Fluctuation Theory of Recurrent Events

Feller¹

1949

Transactions of the American Mathematical Society

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“…From this, we can also deduce the classical HartmannWintner law of the iterated logarithm [7] under a stronger form.…”

Section: Introductionmentioning

confidence: 88%

The law of the iterated logarithm on subsequences-characterizations

Weber

1990

Nagoya Mathematical Journal

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Let §> be any increasing sequence of integers and M > 1 we connect to them in a very simply way, an increasing unbounded function φ: 3 -> R + . Let also X u X 2 , be a sequence of i.i.d. random vectors with value in euclidian space R m . We prove that the cluster set of the sequence {(XΊ + + X n )l<\/~nφ(ri), n e 3} almost surely coincides with the unit ball of R m , if, and only if, the covariance matrix of X x is the identity matrix of R m and EX λ is the zero vector of R m . We define a functional A on the set of increasing sequences of integers as follows:We prove that P{limsup β9Λ _ M (-Xi + • + X n )N%n loglog n > 0} > 0, for at least one sequence X u X, of i.i.d. real r.v.'s with EX X = 0 and E(X x f < oo, if, and only if Λ{%) > 0; further the definition of A( ) does not depend of the value of M. Different characterizations are also established. Further, the law of the iterated logarithm in the sense of Strassen is considered. We finally show a functional law of the iterated logarithm on subsequences for lipschitzian random functions. § 1. Introduction This work is a natural continuation of our previous results obtained in [16]. Let X u X 2 , be a sequence of independent, identically distributed (i.i.d), real random variables (r.v.'s). Set Vn, > 1, S n = Xi + + X n .We investigate the asymptotic behavior of the sequence {S n } when n runs on arbitrary subsequences of integers. This is a quite natural

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“…This is slightly stronger than II. Finally II generalizes the law of the iterated logarithm of [3] to Gaussian random variables in B.…”

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The Law of the Iterated Logarithm for Brownian Motion in a Banach Space

Kuelbs¹,

Lepage²

1973

Transactions of the American Mathematical Society

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ABSTRACT. Strassen's version of the law of the iterated logarithm is proved for Brownian motion in a real separable Banach space. We apply this result to obtain the law of the iterated logarithm for a sequence of independent Gaussian random variables with values in a Banach space and to obtain Strassen's result.Introduction. Let H denote a real separable Hilbert space with norm ||-||w and assume ||-||B is a measurable norm on H in the sense of [2]. Then there exists a constant M > 0 such that ||x||¿ < M||x||w for all x G H, and if B is the completion of H in ||-|la it follows that B is a real separable Banach space. We will view H as a subspace of B and since ||-||a is weaker than \\-\\H on H it follows that B*, the topological dual of B, can be continuously injected into //*, the topological dual of H. We call (H, B) an abstract Wiener space.For t > 0, let m, denote the canonical Gaussian cylinder set measure on H with variance parameter t and let ft» (t > 0) denote the Borel probability measure on B induced by m, (t > 0). We call ft, the Wiener measure on B generated by H with variance parameter t.Let flB denote the space of continuous functions w from [0, oo) into B such that w(0) = 0, and let D be the o-field of fiB generated by the functions w -» w(t). Then there is a unique probability measure P on D such that if 0 = t0 < tx <•••<'" then w(tj) -w(tj_x) (j = l,...,n) are independent and w(tj) -w(tj_x) has distribution ft,y_, | on B. In particular, the stochastic process W¡ defined on (fifl,0,F) by W¡(w) = w(t) has stationary independent Gaussian

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On the Law of the Iterated Logarithm

Cited by 330 publications

References 0 publications

Fluctuation Theory of Recurrent Events

Fluctuation Theory of Recurrent Events

The law of the iterated logarithm on subsequences-characterizations

The Law of the Iterated Logarithm for Brownian Motion in a Banach Space

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