Rates of convergence in the strong law of large numbers for degenerate U-statistics

1987 ◽  
Vol 5 (5) ◽  
pp. 371-374 ◽  
Author(s):  
P.N. Kokic
Keyword(s):  
Law Of Large Numbers ◽  
Rates Of Convergence ◽  
Strong Law ◽  
U Statistics ◽  
Large Numbers

A remark on strong law of large numbers for weighted U-statistics

2014 ◽  
Vol 30 (9) ◽  
pp. 1595-1605
Author(s):  
Hyung-Tae Ha ◽  
Mei Ling Huang ◽  
De Li Li
Keyword(s):  
Law Of Large Numbers ◽  
Strong Law ◽  
U Statistics ◽  
Large Numbers

Two-parameter strong laws and maximal inequalities for U-statistics

1987 ◽  
Vol 107 (1-2) ◽  
pp. 133-151 ◽  
Author(s):  
Terry R. McConnell
Keyword(s):  
Law Of Large Numbers ◽  
Weak Type ◽  
Sufficient Conditions ◽  
Strong Law ◽  
Strong Laws ◽  
Maximal Inequalities ◽  
U Statistics ◽  
Large Numbers ◽  
Necessary And Sufficient ◽  
Two Parameter

SynopsisWe provide necessary and sufficient conditions for two-parameter convergence in the strong law of large numbers for U-statistics. We also obtain weak-type (1,1) inequalities for one and two-sample U-statistics of order 2 which are, in a sense, best possible.

Obtaining Prescribed Rates of Convergence for the Ergodic Theorem

1983 ◽  
Vol 35 (6) ◽  
pp. 1129-1146 ◽  
Author(s):  
G. L. O'Brien
Keyword(s):  
Ergodic Theorem ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Stationary Sequence ◽  
Strong Law ◽  
Iterated Logarithm ◽  
Large Numbers ◽  
The Law ◽  
Image Position

Let {Yn, n ∊ Z} be an ergodic strictly stationary sequence of random variables with mean zero, where Z denotes the set of integers. For n ∊ N = {1, 2, …}, let Sn = Y1 + Y2 + … + Yn. The ergodic theorem, alias the strong law of large numbers, says that n–lSn → 0 as n → ∞ a.s. If the Yn's are independent and have variance one, the law of the iterated logarithm tells us that this convergence takes place at the rate in the sense that1It is our purpose here to investigate what other rates of convergence are possible for the ergodic theorem, that is to say, what sequences {bn, n ≧ 1} have the property that2for some ergodic stationary sequence {Yn, n ∊ Z}.

Sign in / Sign up

Export Citation Format

Share Document