Precise asymptotics in self-normalized sums of iterated logarithm for multidimensionally indexed random variables

In the case of Zd (d ≥ 2)-the positive d-dimensional lattice points with partial ordering ≤, {Xk,k ∈ Zd } i.i.d. random variables with mean 0, Sn = ∑k≤nXk and Vn2 = ∑j≤nX2j, the precise asymptotics for ∑n1/|n|(log|n|)dP(|Sn/vn|≥ ε√loglog|n|) and ∑n(logn|)δ/|n|(log|n|)d-1 P(|Sn/Vn| ≥ ε√log n), as ε ↘ 0, is established.     

Moderate deviation principles for trajectories of sums of independent Banach space valued random variables

Let be a sequence of i.i.d. random vectors with values in a separable Banach space. Moderate deviation principles for trajectories of sums of are proved, which generalize related results of Borovkov and Mogulskii (1980) and Deshayes and Picard (1979). As an application, functional laws of the iterated logarithm are given. The paper also contains concluding remarks, with examples, on extending results for partial sums to corresponding ones for trajectory setting.

     

Precise asymptotics in the self-normalized law of the iterated logarithm

Let X,X₁,X₂,… be i.i.d. nondegenerate random variables with zero means, and . We investigate the precise asymptotics in the law of the iterated logarithm for self-normalized sums, S_n/V_n, also for the maximum of self-normalized sums, max_1kn|S_k|/V_n, when X belongs to the domain of attraction of the normal law.     

The Law of the Logarithm for Weighted Sums of Independent Random Variables

Let X,X _n ;n1 be a sequence of real-valued i.i.d. random variables with E(X)=0. Assume B(u) is positive, strictly increasing and regularly-varying at infinity with index 1/2<1. Set b _n =B(n),n1. If

正相协列乘积和的重对数律与强大数律

证明了强平稳正相协列乘积和的重对数律与不同分布正相协列乘积和的强大数律,指出了部分和服从强大数律但乘积和未必服从强大数律这一事实,并讨论了定理2中一个条件的必要性．     

An almost sure central limit theorem for self-normalized weighted sums

Let X, X1 , X2 , ··· be a sequence of nondegenerate i.i.d. random variables with zero means, which is in the domain of attraction of the normal law. Let {a ni , 1≤i≤n, n≥1} be an array of real numbers with some suitable conditions. In this paper, we show that a central limit theorem for self-normalized weighted sums holds. We also deduce a version of ASCLT for self-normalized weighted sums.     

A self-normalized law of the iterated logarithm for the geometrically weighted random series

Let {X, X_n; n ≥ 0} be a sequence of independent and identically distributed random variables with EX=0, and assume that EX~2I(|X| ≤ x) is slowly varying as x →∞, i.e., X is in the domain of attraction of the normal law. In this paper, a self-normalized law of the iterated logarithm for the geometrically weighted random series Σ~∞_(n=0)β~nX_n(0 β 1) is obtained, under some minimal conditions.     

How big are the CsörgŐ-Révész increments of two-parameter Wiener processes?

In this paper, we prove a theorem on the set of limit points of the increments of a two-parameter Wiener process via establishing a large deviation principle on the increments of the two-parameter Wiener process.     

How big are the Cs(o)rg(o)-Révész increments of two-parameter Wiener processes?

In this paper, we prove a theorem on the set of limit points of the increments of a two-parameter Wiener process via establishing a large deviation principle on the increments of the two-parameter Wiener process.     

Limsup results and LIL for partial sum processes of a Gaussian random field

Let {ξ _j ; j ∈ ℤ₊ ^d be a centered stationary Gaussian random field, where ℤ₊ ^d is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝ^d, having nonnegative integer coordinates. For each j = (j ₁ , ..., jd) in ℤ₊ ^d , we denote |j| = j ₁ ... j _d and for m, n ∈ ℤ₊ ^d , define S(m, n] = Σ _m<j≤n ζ _j , σ²(|n−m|) = ES ² (m, n], S _n = S(0, n] and S ₀ = 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t > 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 < α < 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes. Research supported by NSERC Canada grants at Carleton University, Ottawa     

How big are the Csrg"非汉字符号"-Révész increments of two-parameter Wiener processes?

In this paper, we prove a theorem on the set of limit points of the increments of a two-parameter Wiener process via establishing a large deviation principle on the increments of the two-parameter Wiener process.     

Some Limit Results for Moving Sums of Stable Random Variables

We discuss the integral test for a moving sum of an iid symmetric stable sequence with exponent α (0 < α < 2), and the Chover-type LIL is given as a corollary. We also obtain the law of a single logarithm for moving sums of a normal sequence.     

NA序列重对数律的几个极限定理

设{X_n;n≥1}均值为零、方差有限的NA平稳序列。记S_n=∑_(k=1)~n X_k,M_n=maxk≤n|S_k|,n≥1.假设σ~2=EX_1~2+2∑_(k=2)~∞EX_1X_k>0。本文讨论了:当ε 0时,P{M_n≥εσ(2nloglogn)~(1/2)的一类加权级数的精确渐近性质,以及当ε∞时,P{M_n≤εσ(π~2n/(8loglogn))~(1/2)}的一类加权级数的精确渐近性质。这些性质与重对数律和Chung重对数律的速度有关。     

LIL and the Approximation of Rectangular Sums of B-valued Random Variables when Extreme Terms are Excluded

Let be a field of i.i.d. random variables indexed by d-tuples of positive integers and taking values in a Banach space ℬ and let if is the r-th maximum of . Let and . We approximate the trimmed sums by a Brownian sheet and obtain sufficient and necessary conditions for to satisfy the compact and functional laws of the iterated logarithm. These results improve the previous works by Morrow (1981), Li and Wu (1989) and Ledoux and Talagrand (1990). Received March 22, 1999, Accepted October 13, 2000     

Laws of the iterated logarithm for iterated Wiener processes

The recent interest in iterated Wiener processes was motivated by apparently quite unrelated studies in probability theory and mathematical statistics. Laws of the iterated logarithm (LIL) were independently obtained by Burdzy⁽²⁾ and Révész⁽¹⁷⁾. In this work, we present a functional version of LIL for a standard iterated Wiener process, in the spirit of functional asymptotic results of an ²-valued Gaussian process given by Deheuvels and Mason⁽⁹⁾ in view of Bahadur-Kiefer-type theorems. Chung's liminf sup LIL is established as well, thus providing further insight into the asymptotic behavior of iterated Wiener processes.     

Convergence rates for probabilities of moderate deviations for moving average processes

The present paper first shows that, without any dependent structure assumptions for a sequence of random variables, the refined results of the complete convergence for the sequence is equivalent to the corresponding complete moment convergence of the sequence. Then this paper investigates the convergence rates and refined convergence rates （or complete moment convergence） for probabilities of moderate deviations of moving average processes. The results in this paper extend and generalize some well-known results.     

A weak invariance principle for self-normalized products of sums of mixing sequences

Let variables in the {X, Xn, n ≥ 1} be a sequence of strictly stationary φ-mixing positive random domain of attraction of the normal law. Under some suitable conditions the principle for self-normalized products of partial sums is obtained.     

顶点着色随机图边数的中偏差

本文讨论了对顶点按照一定比列着色的随机图,利用泰勒展式和斯特灵公式,得到了随机图边数的中偏差和重对数律.     

Generalized One-Sided Laws for the Larger Observations of a Triangular Array

Consider independent and identically distributed random variables {X,X _nj , 1jn,n1} with density f(x)=px ^–p–1 I(x1), where p>0. We show that there exist unusual generalized Laws of the Iterated Logarithm involving the larger order statistics from our array.     

Refinement of Convergence Rates for Tail Probabilities

Let X₁, X₂,... be, i.i.d. random variables, and put . We find necessary and sufficient moment conditions for , where δ≥ 0 and q>0, and with a_n>0 and b_n is either or The series f(x) we deal with are classical series studied by Hsu and Robbins, Erdős, Spitzer, Baum and Katz, Davis, Lai, Gut, etc