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1.
The product of mN independent random square matrices whose elements are independent identically distributed random variables with zero mean and unit variance is considered. It is known that, as the size of the matrices increases to infinity, the empirical spectral measure of the normalized eigenvalues of the product converges with probability 1 to the distribution of the mth power of the random variable uniformly distributed on the unit disk of the complex plane. In particular, in the case of m = 1, the circular law holds. The purpose of this paper is to prove the validity of the local circular law (as well as its generalization to the case of any fixed m) in the case where the distribution of the matrix elements has finite absolute moment of order 4 + δ,δ > 0,. Recent results of Bourgade, Yau, and Yin, of Tao and Vu, and of Nemish are generalized.  相似文献   

2.
Etemadi (in Z. Wahrscheinlichkeitstheor. Verw. Geb. 55, 119–122, 1981) proved that the Kolmogorov strong law of large numbers holds for pairwise independent identically distributed (pairwise i.i.d.) random variables. However, it is not known yet whether the Marcinkiewicz–Zygmund strong law of large numbers holds for pairwise i.i.d. random variables. In this paper, we obtain the Marcinkiewicz–Zygmund type strong law of large numbers for pairwise i.i.d. random variables {X n ,n≥1} under the moment condition E|X 1| p (loglog|X 1|)2(p?1)<∞, where 1<p<2.  相似文献   

3.
陈平炎 《数学学报》2006,49(5):1061-106
设{X,Xn,n≥0}是两两独立同分布的随机变量序列,11.为了证明这一结论而获得到的两两负相关随机变量序列的Cesaro强大数定律收敛速度的结果本身也是有意义的.此结果对于同分布的两两NQD序列也是对的.  相似文献   

4.
The object of the present investigation is to show that the elegant asymptotic almost-sure representation of a sample quantile for independent and identically distributed random variables, established by Bahadur [1] holds for a stationary sequence of φ-mixing random variables. Two different orders of the remainder term, under different φ-mixing conditions, are obtained and used for proving two functional central limit theorems for sample quantiles. It is also shown that the law of iterated logarithm holds for quantiles in stationary φ-mixing processes.  相似文献   

5.
The solutions of the discrete Safronov–Dubovski coagulation equation are investigated. We prove global existence for a class of unbounded coagulation kernels. We also show that for sub-linear unbounded kernels, the mass conservation law holds. Finally, we show that for bounded kernels, this equation has a unique global solution that is continuously dependent on the initial data.  相似文献   

6.
Probability Theory and Related Fields - We show that the law of iterated logarithm holds for a sequence of independent random variables (X n ) provided (i) $$\sum\limits_{n = 1}^\infty {(s_n^2...  相似文献   

7.
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   

8.
In this paper,we prove that,under certain conditions,a strong law of large number holds for a class of branching particle systems X corresponding to the parameters(Y,β,ψ),where Y is a Hunt process andψis the generating function for the offspring.The main tool of this paper is the spine decomposition and we only need an L log L condition.  相似文献   

9.
We consider a partial-sum process generated by a sequence of nonidentically distributed independent random variables. Assuming that this process is available for observation along an arbitrary time sequence, we fill the gaps by linear interpolation and prove the functional law of the iterated logarithm (FLIL) for sample paths obtained in this way. Assuming that the V. A. Egorov condition holds, we show that FLIL is valid, while under other conditions sufficient for the usual law of the iterated logarithm FLIL may fail. Bibliography: 16 titles. Translated, fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 73–95.  相似文献   

10.
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