Weak convergence of the adjusted range of cumulative sums of exchangeable random variables

1983 ◽  
Vol 20 (2) ◽  
pp. 297-304 ◽  
Author(s):  
Brent M. Troutman
Keyword(s):  
Weak Convergence ◽  
Asymptotic Distribution ◽  
Random Variables ◽  
Dependent Random Variables ◽  
Random Functions ◽  
Exchangeable Random Variables ◽  
Convergence Results ◽  
Cumulative Sums

Let be the adjusted range of the cumulative sums of a sequence , where . Weak convergence results for random functions constructed from cumulative sums of {Xs} are used to obtain the asymptotic distribution and moments of when {Xs} are exchangeable, or symmetrically dependent, random variables.

Weak convergence of the adjusted range of cumulative sums of exchangeable random variables

1983 ◽  
Vol 20 (02) ◽  
pp. 297-304 ◽  
Author(s):  
Brent M. Troutman
Keyword(s):  
Weak Convergence ◽  
Asymptotic Distribution ◽  
Random Variables ◽  
Dependent Random Variables ◽  
Random Functions ◽  
Exchangeable Random Variables ◽  
Convergence Results ◽  
Cumulative Sums

Let be the adjusted range of the cumulative sums of a sequence , where . Weak convergence results for random functions constructed from cumulative sums of {Xs } are used to obtain the asymptotic distribution and moments of when {Xs } are exchangeable, or symmetrically dependent, random variables.

A note on Spitzer's lemma and its application to the maximum of partial sums of dependent random variables

1976 ◽  
Vol 13 (2) ◽  
pp. 361-364 ◽  
Author(s):  
M. E. Solari ◽  
J. E. A. Dunnage
Keyword(s):  
Infinite Sequence ◽  
Random Variables ◽  
Finite Sequence ◽  
Partial Sums ◽  
Dependent Random Variables ◽  
Finite Segment ◽  
Exchangeable Random Variables ◽  
Partial Sum

We give an expression for the expectation of max (0, S1, …, Sn) where Sk is the kth partial sum of a finite sequence of exchangeable random variables X1, …, Xn. When the Xk are also independent, the formula we give has already been obtained by Spitzer; and when the sequence is a finite segment of an infinite sequence of exchangeable random variables, it is a consequence of a theorem of Hewitt.

scholarly journals Complete convergence for weighted sums of widely orthant-dependent random variables

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Pingyan Chen ◽  
Soo Hak Sung
Keyword(s):  
Complete Convergence ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Weighted Sums ◽  
Dependent Random Variables ◽  
Strong Law ◽  
Link Type ◽  
Convergence Results ◽  
Large Numbers ◽  
Widely Orthant Dependent

AbstractThe complete convergence results for weighted sums of widely orthant-dependent random variables are obtained. A strong law of large numbers for weighted sums of widely orthant-dependent random variables is also obtained. Our results extend and generalize some results of Chen and Sung (J. Inequal. Appl. 2018:121, 2018), Zhang et al. (J. Math. Inequal. 12:1063–1074, 2018), Chen and Sung (Stat. Probab. Lett. 154:108544, 2019), Lang et al. (Rev. Mat. Complut., 2020, 10.1007/s13163-020-00369-5), and Liang (Stat. Probab. Lett. 48:317–325, 2000).

Asymptotic Distribution of Sample Quantiles for Exchangeable Random Variables

1971 ◽  
Vol 20 (4) ◽  
pp. 135-142
Author(s):  
K. C. Chanda
Keyword(s):  
Normal Distribution ◽  
Asymptotic Distribution ◽  
Order Statistic ◽  
Multivariate Normal Distribution ◽  
Random Variables ◽  
Multivariate Normal ◽  
Sample Quantile ◽  
Exchangeable Random Variables ◽  
Sample Quantiles ◽  
Special Case

Summary The purpose of this article is to investigate the ‘large sample’ properties of sample quantiles when we assume that the basic random variables are exchangeable (ref. Loève (1960) p. 365). It is shown that under different conditions (to be specified below) on the nature of these exchangeable random variables the distribution of the sample quantile Xr : n where Xr : n is the rth order statistic for the first n exchangeable random variables [Formula: see text] tends, as n → ∞, to different nondegenerate forms. As an example, the special case of random variables with equicor-related multivariate normal distribution is discussed.

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