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1.
A. N. Frolov 《Journal of Mathematical Sciences》2006,137(1):4575-4582
We obtain the law of the iterated logarithm for increments of sums of independent random variables. Our results generalize
the Kolmogorov theorem and the Hartman—Wintner theorem on the law of the iterated logarithm. Bibliography: 17 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 174–186. 相似文献
2.
We investigate necessary and sufficient conditions for the almost-sure boundedness of normalized solutions of linear stochastic differential equations in R
dand their almost-sure convergence to zero. We establish an analog of the bounded law of iterated logarithm. 相似文献
3.
O. Sh. Sharipov 《Lithuanian Mathematical Journal》2009,49(2):203-215
Under optimal moment conditions, we prove the compact law of the iterated logarithm and the almost sure invariance principle
for ψ-mixing random variables with values in type 2 Banach spaces. These results, together with the bounded law of the iterated
logarithm proved earlier by author, allow us to prove the same kind of results for the Banach space valued autoregressive
processes with ψ-mixing innovations. The results for autoregressive processes can be considered as asymptotic properties of the estimator
of mean. 相似文献
4.
A. N. Frolov 《Journal of Mathematical Sciences》2006,133(3):1356-1370
We derive universal strong laws for increments of sums of independent, nonidentically distributed, random variables. These
results generalize universal results of the author for the i.i.d. case which include the strong law of large numbers, law
of the iterated logarithm, Erdos-Renyi law, and Csorgo-Revesz laws. Bibliography: 27 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 311, 2004, pp. 260–285. 相似文献
5.
D. Deng 《Journal of Theoretical Probability》2004,17(2):367-385
We present an analogue of Wittmann's law of iterated logarithm (LIL) for tail sums of independent B-valued random variables by using the isoperimetric method and give the precise value of the upper limit for the LIL for tail sums. 相似文献
6.
The Chung–Smirnov law of the iterated logarithm and the Finkelstein functional law of the iterated logarithm for empirical processes are used to establish new results on the central limit theorem, the law of the iterated logarithm, and the strong law of large numbers for L-statistics with certain bounded and smooth weight functions. These results are used to obtain necessary and sufficient conditions for almost sure convergence and for convergence in distribution of some well-known L-statistics and U-statistics, including Gini's mean difference statistic. A law of the logarithm for weighted sums of order statistics is also presented. 相似文献
7.
V. A. Egorov 《Journal of Mathematical Sciences》2007,147(4):6912-6917
Some optimal asymptotic estimates of constants for the right-hand inequalities of Marcinkiewicz and Rosenthal are derived.
These estimates imply some new inequalities for the rate of increase of sums and optimal right-hand estimates for the law
of the iterated logarithm. Similar estimates are derived for self-normalized sums. Bibliography: 12 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 115–123. 相似文献
8.
9.
We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for U-statistics in Hilbert spaces. As a tool we also develop moment and tail estimates for canonical Hilbert-space valued U-statistics of arbitrary order, which are of independent interest.
R. Adamczak’s research partially supported by MEiN Grant 2 PO3A 019 30.
R. Latała’s research partially supported by MEiN Grant 1 PO3A 012 29. 相似文献
10.
Let X
1, X
2,... be independent, but not necessarily identically distributed random variables in the domain of attraction of a normal law or a stable law with index 0 < α < 2. Using suitable self-normalizing (or Studentizing) factors, laws of the iterated logarithm for self-normalized Hanson–Russo type increments are discussed. Also, some analogous results for self-normalized weighted sums of i.i.d. random variables are given. 相似文献
11.
Pingyan Chen 《随机分析与应用》2013,31(1):89-103
Abstract This article considers the partial sums from a sequence of independent and identically distributed random variables. It is well-known that the Hartman-Wintner law of the iterated logarithm holds if and only if the second moment exists. This article studies the generalized law of the iterated logarithm for the partial sums when they are normalized by a sequence of constants that are regularly varying with index 1/2. As a result, two equivalent conditions for the law are obtained. 相似文献
12.
New conditions for the functional law of the iterated logarithm for trimmed sums of symmetric independent identically distributed
random variables are obtained. Bibliography: 7 titles.
Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 119–125.
Translated by N. B. Lebedinskaya. 相似文献
13.
T. Mikosch 《Monatshefte für Mathematik》1988,106(1):25-40
In this paper we prove bounded laws of the iterated logarithm for Gaussian quadratic forms. The underlying sequence of Gaussian variables is assumed to satisfy quite general conditions on its covariance structure. Basic tools are maximal inequalities of exponential type for sums of dependent random variables which may be of own interest. Several examples illustrate the sharpness of the results. In a particular section the bounded law of the iterated logarithm is shown for quadratic variation of Brownian motion. 相似文献
14.
Michael Lacey 《Journal of Theoretical Probability》1989,2(3):377-398
We establish a bounded and a compact law of the iterated logarithm for partial sum processes indexed by classes of functions. We assume a growth condition on the metric entropy under bracketing. Examples show that our results are sharp. As a corollary we obtain new results for weighted sums of independent identically distributed random variables. 相似文献
15.
Uwe Einmahl 《Probability Theory and Related Fields》1988,77(1):65-85
Summary We obtain a strong approximation theorem for partial sums of i.i.d.d-dimensional r.v.'s with possibly infinite second moments. Using this result, we can extend Philipp's strong invariance principle for partial sums of i.i.d.B-valued r.v.'s satisfying the central limit theorem toB-valued r.v.'s which are only in the domain of attraction of a Gaussian law. This new strong invariance principle implies a compact as well as a functional law of the iterated logarithm which improve some recent results of Kuelbs (1985). 相似文献
16.
Erika Péter 《Acta Mathematica Hungarica》2000,87(1-2):23-31
We prove a law of the iterated logarithm for sums ∑
k
≦
N
f(n
k
x) where f is a periodic measurable function and (n
k
) is a rapidly increasing sequence of integers. Our result applies also in the sub-Hadamard case and extends and improves
earlier results in the field.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
17.
For a sequence of independent random elements in a Banach space, we obtain an upper bound for moments of the supremum of normalized
sums in the law of the iterated logarithm by using an estimate for moments in the law of large numbers. An example of their
application to the law of the iterated logarithm in Banach lattices is given.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 653–665, May, 2006. 相似文献
18.
C. Aistleitner I. Berkes R. Tichy 《Proceedings of the Steklov Institute of Mathematics》2012,276(1):3-20
It is known that for any smooth periodic function f the sequence (f(2
k
x))
k≥1 behaves like a sequence of i.i.d. random variables; for example, it satisfies the central limit theorem and the law of the
iterated logarithm. Recently Fukuyama showed that permuting (f(2
k
x))
k≥1 can ruin the validity of the law of the iterated logarithm, a very surprising result. In this paper we present an optimal
condition on (n
k
)
k≥1, formulated in terms of the number of solutions of certain Diophantine equations, which ensures the validity of the law of
the iterated logarithm for any permutation of the sequence (f(n
k
x))
k≥1. A similar result is proved for the discrepancy of the sequence ({n
k
x})
k≥1, where {·} denotes the fractional part. 相似文献
19.
Allan Gut 《Probability Theory and Related Fields》1979,46(2):205-220
Summary For a set of i.i.d. random variables indexed by the positive integer d-dimensional lattice points we give conditions for the existence of moments of the supremum of normed partial sums, thereby obtaining results related to the Kolmogorov-Marcinkiewicz strong law of large numbers and the law of the iterated logarithm. 相似文献
20.
Wen Jiwei Yan Yunliang 《高校应用数学学报(英文版)》2006,21(1):87-95
Let X,X1,X2 be i. i. d. random variables with EX^2+δ〈∞ (for some δ〉0). Consider a one dimensional random walk S={Sn}n≥0, starting from S0 =0. Let ζ* (n)=supx∈zζ(x,n),ζ(x,n) =#{0≤k≤n:[Sk]=x}. A strong approximation of ζ(n) by the local time for Wiener process is presented and the limsup type and liminf-type laws of iterated logarithm of the maximum local time ζ*(n) are obtained. Furthermore,the precise asymptoties in the law of iterated logarithm of ζ*(n) is proved. 相似文献