scholarly journals Convergence Rates in the Law of Large Numbers for Arrays of Banach Valued Martingale Differences

2013 ◽  
Vol 2013 ◽  
pp. 1-26 ◽  
Author(s):  
Shunli Hao
Keyword(s):  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Sufficient Conditions ◽  
Random Variables ◽  
General Setting ◽  
Weighted Sums ◽  
Moment Condition ◽  
Martingale Differences ◽  
Large Numbers ◽  
The Law

We study the convergence rates in the law of large numbers for arrays of Banach valued martingale differences. Under a simple moment condition, we show sufficient conditions about the complete convergence for arrays of Banach valued martingale differences; we also give a criterion about the convergence for arrays of Banach valued martingale differences. In the special case where the array of Banach valued martingale differences is the sequence of independent and identically distributed real valued random variables, our result contains the theorems of Hsu-Robbins-Erdös (1947, 1949, and 1950), Spitzer (1956), and Baum and Katz (1965). In the real valued single martingale case, it generalizes the results of Alsmeyer (1990). The consideration of Banach valued martingale arrays (rather than a Banach valued single martingale) makes the results very adapted in the study of weighted sums of identically distributed Banach valued random variables, for which we prove new theorems about the rates of convergence in the law of large numbers. The results are established in a more general setting for sums of infinite many Banach valued martingale differences. The obtained results improve and extend those of Ghosal and Chandra (1998).

Convergence rates in the weak law of large numbers for weighted sums of i.i.d. random variables and applications in errors-in-variables models

2019 ◽  
Vol 19 (06) ◽  
pp. 1950041 ◽  
Author(s):  
Mingyang Zhang ◽  
Pingyan Chen ◽  
Soo Hak Sung
Keyword(s):  
Regression Models ◽  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Weighted Sums ◽  
Errors In Variables ◽  
Error In Variables ◽  
Large Numbers ◽  
Necessary And Sufficient

We prove necessary and sufficient conditions for the convergence rates in the weak law of large numbers for weighted sums of independent and identically distributed (i.i.d.) random variables. This result is applied to simple linear error-in-variables regression models.

REFINEMENT OF CONVERGENCE RATES FOR WEIGHTED SUMS OF INDEPENDENT RANDOM VARIABLES

2012 ◽  
Vol 05 (01) ◽  
pp. 1250007
Author(s):  
Si-Li Niu ◽  
Jong-Il Baek
Keyword(s):  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Random Variables ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Counting Processes ◽  
Precise Asymptotics ◽  
Strong Law ◽  
Large Numbers ◽  
Passage Times

In this paper, we establish one general result on precise asymptotics of weighted sums for i.i.d. random variables. As a corollary, we have the results of Lanzinger and Stadtmüller [Refined Baum–Katz laws for weighted sums of iid random variables, Statist. Probab. Lett. 69 (2004) 357–368], Gut and Spătaru [Precise asymptotics in the law of the iterated logarithm, Ann. Probab. 28 (2000) 1870–1883; Precise asymptotics in the Baum–Katz and Davis laws of large numbers, J. Math. Anal. Appl. 248 (2000) 233–246], Gut and Steinebach [Convergence rates and precise asymptotics for renewal counting processes and some first passage times, Fields Inst. Comm. 44 (2004) 205–227] and Heyde [A supplement to the strong law of large numbers, J. Appl. Probab. 12 (1975) 173–175]. Meanwhile, we provide an answer for the possible conclusion pointed out by Lanzinger and Stadtmüller [Refined Baum–Katz laws for weighted sums of iid random variables, Statist. Probab. Lett. 69 (2004) 357–368].

A pair of complementary theorems on convergence rates in the law of large numbers

1967 ◽  
Vol 63 (1) ◽  
pp. 73-82 ◽  
Author(s):  
C. C. Heyde ◽  
V. K. Rohatgi
Keyword(s):  
Law Of Large Numbers ◽  
Convergence Rates ◽  
Random Variables ◽  
Strong Law ◽  
Large Numbers ◽  
The Law ◽  
Independent And Identically Distributed

Introduction. Let Xi (i= 1, 2, 3,…) be a sequence of independent and identically distributed random variables with law ℒ(X) and write The Kolmogorov-Marcinkiewicz strong law of large numbers (Loève(6), p. 243) has the following statement:If E|X|r < ∞, then with cr = 0 or EX according as r 1 or r ≥ 1.

scholarly journals On Strong Convergence for Weighted Sums of a Class of Random Variables

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Aiting Shen
Keyword(s):  
Strong Convergence ◽  
Complete Convergence ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Maximal Inequality ◽  
Weighted Sums ◽  
Moment Condition ◽  
Strong Law ◽  
Linear Statistics ◽  
Large Numbers

Let{Xn,n≥1}be a sequence of random variables satisfying the Rosenthal-type maximal inequality. Complete convergence is studied for linear statistics that are weighted sums of identically distributed random variables under a suitable moment condition. As an application, the Marcinkiewicz-Zygmund-type strong law of large numbers is obtained. Our result generalizes the corresponding one of Zhou et al. (2011) and improves the corresponding one of Wang et al. (2011, 2012).

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