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322 PRUITT, E.W., General one-sided laws of the iterated logarithm, Ann. Probab. 9 (1981), 1-48. MR 82k :60066 General one-sided laws of the iterated logarithm

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Abstract  

Takahashi [15] gave a concrete upper bound estimate of the law of the iterated logarithm for ∑ f(n k x). We extend this result and prove the best possibility of this bound.

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Abstract  

Takahashi [4] gave a concrete upper bound estimate of the law of the iterated logarithm for Σf(n k x). We extend this result and prove the best possibility of this bound.

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Abstract  

An upper bound estimate in the law of the iterated logarithm for Σf(n k ω) where nk+1∫nk≧ 1 + ck (α≧0) is investigated. In the case α<1/2, an upper bound had been given by Takahashi [15], and the sharpness of the bound was proved in our previous paper [8]. In this paper it is proved that the upper bound is still valid in case α≧1/2 if some additional condition on {n k} is assumed. As an application, the law of the iterated logarithm is proved when {n k} is the arrangement in increasing order of the set B(τ)={1 i 1...qτ i τ|i1,...,iτN 0}, where τ≧ 2, N 0=NU{0}, and q 1,...,q τ are integers greater than 1 and relatively prime to each others.

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Abstract  

An exponential inequality for the tail of the conditional expectation of sums of centered independent random variables is obtained. This inequality is applied to prove analogues of the Law of the Iterated Logarithm and the Strong Law of Large Numbers for conditional expectations. As corollaries we obtain certain strong theorems for the generalized allocation scheme and for the nonuniformly distributed allocation scheme.

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Let { B H;K ( t ), t ≧ 0} be a bifractional Brownian motion with indexes 0 < H < 1 and 0 < K ≦ 1 and define the statistic
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$V_T = \mathop {\sup }\limits_{0 \leqq s \leqq T - a_T } \beta _T \left| {B_{H,K} (s + a_T ) - B_{H,K} (s)} \right|$$ \end{document}
where β T and α T are suitably chosen functions of T ≧ 0. We establish some laws of the iterated logarithm for V T .
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Abstract  

Given a field of independent identically distributed (i.i.d.) random variables
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ {X_{\bar n} ;\bar n \in \aleph ^d } \right\}$$ \end{document}
indexed by d-tuples of positive integers and taking values in a separable Banach space B, let
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$X_{\bar n}^{(r)} = X_{\bar m}$$ \end{document}
is the r-th maximum of
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ {\left\| {X_{\bar k} } \right\|;\bar k \leqq \bar n} \right\}$$ \end{document}
and let
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$^{(r)} S_{\bar n} = S_{\bar n} - \left( {X_{\bar n}^{(1)} + \cdots + X_{\bar n}^{(r)} } \right)$$ \end{document}
be the trimmed sums, where
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$S_{\bar n} = \sum\nolimits_{\bar k \leqq \bar n} {X_{\bar k} }$$ \end{document}
. This paper aims to obtain a general law of the iterated logarithm (LIL) for the trimmed sums which improves previous works.
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Abstract  

Let (n k)k≧1 be a lacunary sequence of positive integers, i.e. a sequence satisfying n k+1/n k > q > 1, k ≧ 1, and let f be a “nice” 1-periodic function with ∝0 1 f(x) dx = 0. Then the probabilistic behavior of the system (f(n k x))k≧1 is very similar to the behavior of sequences of i.i.d. random variables. For example, Erdős and Gál proved in 1955 the following law of the iterated logarithm (LIL) for f(x) = cos 2πx and lacunary
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(n_k )_{k \geqq 1}$$ \end{document}
:
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathop {\lim \sup }\limits_{N \to \infty } (2N\log \log N)^{1/2} \sum\limits_{k = 1}^N {f(n_k x)} = \left\| f \right\|_2$$ \end{document}
((1))
for almost all x ∈ (0, 1), where ‖f2 = (∝0 1 f(x)2 dx)1/2 is the standard deviation of the random variables f(n k x). If (n k)k≧1 has certain number-theoretic properties (e.g. n k+1/n k → ∞), a similar LIL holds for a large class of functions f, and the constant on the right-hand side is always ‖f2. For general lacunary (n k)k≧1 this is not necessarily true: Erdős and Fortet constructed an example of a trigonometric polynomial f and a lacunary sequence (n k)k≧1, such that the lim sup in the LIL (1) is not equal to ‖f2 and not even a constant a.e. In this paper we show that the class of possible functions on the right-hand side of (1) can be very large: we give an example of a trigonometric polynomial f such that for any function g(x) with sufficiently small Fourier coefficients there exists a lacunary sequence (n k)k≧1 such that (1) holds with √‖f2 2 + g(x) instead of ‖f2 on the right-hand side.
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Abstract  

We establish strong limsup theorems related to the law of the iterated logarithm (LIL) for finite dimensional Gaussian random fields by using the second Borel-Cantelli lemma.

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