Results 281 to 290 of about 16,691 (298)

On the other law of the iterated logarithm

Probability Theory and Related Fields, 1993
A general integral test is established which refines the Jain-Pruitt Chung LIL for iid random variables. As a corollary we obtain that Chung's integral test for Brownian motion is valid for partial sums of iid random variables satisfyingEX 21{|X|≧t}=O((LLt) −1) ast→∞.
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On functional laws of the iterated logarithm [PDF]

Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 1985
A Skorokhod embedding approach is used to give functional laws of the iterated logarithm which involve the process up to timen in the reverse martingale case and the tail of the process in the martingale case. This complements the more usual versions of the iterated logarithm laws for martingales and reverse martingales.
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The Law of the Iterated Logarithm

1975
In this chapter we shall consider a sequence of independent random variables X n ; n = 1, 2, ... with zero means and finite variances.
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The Law of the Iterated Logarithm

2014
The first law of the iterated logarithm is proved for symmetric Bernoulli random variables, that is, for independent ...
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The law of the iterated logarithm

2013
For B, a standard BMP, we showed in Sec. (5.9) that wp1 \(\frac{{B\left( t \right)}} {t}\mathop { \to 0}\limits^{wp1}\), as t → ∞, that \(\overline {\mathop {\lim }\limits_{t \to \infty } } \frac{{B(t)}} {{\sqrt t }} = \infty\) and \(\mathop {\underline {\lim } }\limits_{t \to \infty } \frac{{B(t)}} {{\sqrt t }} = - \infty\).
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The Limit Law of the Iterated Logarithm

Journal of Theoretical Probability, 2013
For the partial sum $$\{S_n\}$$ of an i.i.d. sequence with zero mean and unit variance, it is pointed out that $$\begin{aligned} \lim _{n\rightarrow \infty }(2\log \log n)^{-1/2}\max _{1\le
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Asymptotics in the Law of the Iterated Logarithm

Theory of Probability & Its Applications, 2009
In this paper the precise asymptotic in the law of the iterated logarithm is considered. The result of A. Gut and A. Spataru [Ann. Probab., 28 (2000), pp. 1870–1883] is generalized for the case of the variables not of the same distribution.
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A generalized law of the iterated logarithm

Statistics & Probability Letters, 1990
Abstract Let {Sn, n ⩾ } denote the partial sums of a sequence of independent random variables, and let (Bn, n ⩾ 1) be a non-decreasing sequence with Bn → ∞. Upper and lower bounds for lim supn → ∞ Sn/(2B2n log log B2n) 1 2 are presented.
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The Functional Law of the Iterated Logarithm

1995
In this section, we shall use a particular problem to demonstrate applications of the general theory developed in Sections 8–12; this example leads, however, to a remarkably beautiful result. We shall deal with the typical form of sample functions of a Wiener process which strongly deviate from the (zero) mean.
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On the law of the iterated logarithm. I

Indagationes Mathematicae (Proceedings), 1955
P. Erdös, I.S. Gál
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