Convergence rates of approximate sums of Riemann integrals

We represent the convergence rates of the Riemann sums and the trapezoidal sums with respect to regular divisions and optimal divisions of a bounded closed interval to the Riemann integrals as some limits of their expanded error terms.     

On the limiting behavior of randomly weighted partial sums

We study the almost sure limiting behavior and convergence in probability of weighted partial sums of the form where {W_nj, 1jn, n1} and {X_nj, 1jn, n1} are triangular arrays of random variables. The results obtain irrespective of the joint distributions of the random variables within each array. Applications concerning the Efron bootstrap and queueing theory are discussed.     

Stochastic convergence of weighted sums in normed linear spaces

Let X be a (real) separable Banach space, let {V_k} be a sequence of random elements in X, and let {a_nk} be a double array of real numbers such that lim_n→∞ a_nk = 0 for all k and Σ^∞_k=1 |a_nk| ≤ 1 for all n. Define S_n = Σⁿ_k=1 a_nk(V_k − EV_k). The convergence of {S_n} to zero in probability is proved under conditions on the coordinates of a Schauder basis or on the dual space of X and conditions on the distributions of {V_k}. Convergence with probability one for {S_n} is proved for separable normed linear spaces which satisfy Beck's convexity condition with additional restrictions on {a_nk} but without distribution conditions for the random elements {V_k}. Finally, examples of arrays {a_nk}, spaces, and applications of these results are considered.     

随机变量阵列加权和最大值的收敛性

本文研究了形如maxun≤j≤un｜∑ji=un aniXni｜的弱大数律和Lr收敛性,其中0＜r≤p,0＜p≤2,{ani,un≤i≤vn,n≥1}是实数阵列,{Xni,un≤i≤vn,n≥1}当0＜p＜1时是任意随机变量阵列,当1≤p≤2时是均值为零的行为NA的随机变量阵列.所得结果丰富和推广了许多已知的结果.     

A strong law for weighted sums of i.i.d. random variables

A strong law is proved for weighted sumsS _n=a _in X _i whereX _i are i.i.d. and {a _in} is an array of constants. When sup(n ^–1|a _in | ^q )^1/q <, 1<q andX _i are mean zero, we showE|X| ^p <,p ^l+q ^–1=1 impliesS _n /n 0. Whenq= this reduces to a result of Choi and Sung who showed that when the {a _in} are uniformly bounded,EX=0 andE|X|< impliesS _n /n 0. The result is also true whenq=1 under the additional assumption that lim sup |a _in |n ^–1 logn=0. Extensions to more general normalizing sequences are also given. In particular we show that when the {a _in} are uniformly bounded,E|X|^1/< impliesS _n /n 0 for >1, but this is not true in general for 1/2<<1, even when theX _i are symmetric. In that case the additional assumption that (x ^1/ log^1/–1 x)P(|X|x)0 asx provides necessary and sufficient conditions for this to hold for all (fixed) uniformly bounded arrays {a _in}.     

Convergence rates in the law of logarithm of random elements

1. IntroductionLet {Xu, n 2 1} be a sequence of r.v.IS in the same probability space and put Sa =nZ Xi, n 2 1; L(x) = mad (1, logx).i=1Since the definition of complete convergence is illtroduced by Hsu and Robbins[6], therehave been many authors who devote themselves to the study of the complete convergence forsums of i.i.d. real-valued r.v.'s, and obtain a series of elegys results, see [3,7]. Meanwhile,the convergence rates in the law of logarithm of i.i.d. real-vained r.v.'s have also be…     

Weak type inequality for noncommutative differentially subordinated martingales

In the paper we focus on self-adjoint noncommutative martingales. We provide an extension of the notion of differential subordination, which is due to Burkholder in the commutative case. Then we show that there is a noncommutative analogue of the Burkholder method of proving martingale inequalities, which allows us to establish the weak type (1,1) inequality for differentially subordinated martingales. Moreover, a related sharp maximal weak type (1,1) inequality is proved. Research supported by MEN Grant 1 PO3A 012 29.     

General Weak Laws of Large Numbers for Bootstrap Sample Means

Abstract

For bootstrap sample means resulting from a sequence {X _n , n ≥ 1} of random variables, very general weak laws of large numbers are established. The random variables {X _n , n ≥ 1} do not need to be independent or identically distributed or be of any particular dependence structure. In general, no moment conditions are imposed on the {X _n , n ≥ 1}. Examples are provided that illustrate the sharpness of the main results.     

Convergence of weighted sums of tight random elements

Convergence of weighted sums of tight random elements {V_n} (in a separable Banach space) which have zero expected values and uniformly bounded rth moments (r > 1) is obtained. In particular, if {a_nk} is a Toeplitz sequence of real numbers, then | Σ_k=1^∞a_nkf(V_k)| → 0 in probability for each continuous linear functional f if and only if 6Σ_k=1^∞a_nkV_k 6→ 0 in probability. When the random elements are independent and max_{1≤k≤n | ank | = $\text{O}$(n^?8)} for some $\text{0 <}\text{1}\text{s}\text{< r ? 1}$, then |Σ_k=1^∞a_nkV_k 6→ 0 with probability 1. These results yield laws of large numbers without assuming geometric conditions on the Banach space. Finally, these results can be extended to random elements in certain Fréchet spaces.     

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.     

Convergence of weighted sums of random functions in D[0, 1]

Conditions are investigated which imply the tightness of certain weighted sums Σ_{i = 1}^k_n a_niX_i of random functions (X_n) taking values in D([0, 1]; E), where E is a separable Banach space. Improved weak laws of large numbers result as corollaries. Examples are presented to clarify the relative strengths of the moment conditions and their relationship to tightness and the strong law of large numbers. A tightness condition is defined using a certain class of sets measurable in the Skorokhod J₁-topology, which yields J₁-tightness of sequences of weighted sums. As a consequence, tightness of a sequence (X_n) in the Skorokhod M₁-topology is used to obtain J₁-tightness of a sequence ( ) of averages and a strong law of large numbers in D(R₊).     

Mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables

Under some conditions of uniform integrability and appropriate conditions, mean convergence theorems and weak laws of large numbers for weighted sums of dependent random variables are obtained. Our results extend and improve the results of [H.S. Sung, S. Lisawadi, A. Volodin, Weak laws of large numbers for arrays under a condition of uniform integrability, J. Korean Math. Soc. 45 (2008) 289-300] and [M. Ordóñez Cabrera, A. Volodin, Mean convergence theorems and weak laws of large numbers for weighted sums of random variables under a condition of weighted integrability, J. Math. Anal. Appl. 305 (2005) 644-658].     

双随机狄里克莱级数收敛性

该文研究特点是;用强大数定律,中心极限定理研究随机系数{a_n}部分和及随机指数λ_n极限性质, 研究结果是;(i)在易满足条件下,(ii)在a_n独立同分布,方差存在条件下;(iii)在{a_n}独立,Ea_n=0,及附加适当条件下,得出收敛横坐标σ_c简洁公式。     

Exact laws for sums of ratios of order statistics from the Pareto distribution

Consider independent and identically distributed random variables {X _nk, 1 ≤ k ≤ m, n ≤ 1} from the Pareto distribution. We select two order statistics from each row, X _n(i) ≤ X _n(j), for 1 ≤ i < j ≤ = m. Then we test to see whether or not Laws of Large Numbers with nonzero limits exist for weighted sums of the random variables R _ij = X _n(j)/X _n(i).     

A weak law for normed weighted sums of random elements in rademacher type p banach spaces

For weighted sums Σ_{j = 1}ⁿa_jV_j of independent random elements {V_n, n ≥ 1} in real separable, Rademacher type p (1 ≤ p ≤ 2) Banach spaces, a general weak law of large numbers of the form (Σ_{j = 1}ⁿa_jV_j − v_n)/b_n →^p 0 is established, where {v_n, n ≥ 1} and b_n → ∞ are suitable sequences. It is assumed that {V_n, n ≥ 1} is stochastically dominated by a random element V, and the hypotheses involve both the behavior of the tail of the distribution of |V| and the growth behaviors of the constants {a_n, n ≥ 1} and {b_n, n ≥ 1}. No assumption is made concerning the existence of expected values or absolute moments of the {V_n, n >- 1}.     

Weak law of large numbers for almost periodically correlated processes

This note contains two simple observations concerning the weak law of large numbers for almost periodically correlated processes.