scholarly journals Limit theorems for randomly selected adjacent order statistics from a Pareto distribution

2005 ◽  
Vol 2005 (21) ◽  
pp. 3427-3441 ◽  
Author(s):  
André Adler
Keyword(s):  
Order Statistics ◽  
Prior Distribution ◽  
Limit Theorems ◽  
Pareto Distribution ◽  
Random Variables ◽  
Weighted Sums ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Weak Laws ◽  
Selection Of

Consider independent and identically distributed random variables{Xnk,  1≤k≤m, n≥1}from the Pareto distribution. We randomly select two adjacent order statistics from each row,Xn(i)andXn(i+1), where1≤i≤m−1. Then, we test to see whether or not strong and weak laws of large numbers with nonzero limits for weighted sums of the random variablesXn(i+1)/Xn(i)exist, where we place a prior distribution on the selection of each of these possible pairs of order statistics.

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

2006 ◽  
Vol 4 (1) ◽  
pp. 1-4 ◽  
Author(s):  
André Adler
Keyword(s):  
Order Statistics ◽  
Pareto Distribution ◽  
Random Variables ◽  
Weighted Sums ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Independent And Identically Distributed

AbstractConsider 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).

scholarly journals A note on convergence of weighted sums of random variables

1985 ◽  
Vol 8 (4) ◽  
pp. 805-812 ◽  
Author(s):  
Xiang Chen Wang ◽  
M. Bhaskara Rao
Keyword(s):  
Integrability Condition ◽  
Random Variables ◽  
Independent Random Variables ◽  
Weighted Sums ◽  
Uniform Integrability ◽  
Laws Of Large Numbers ◽  
Sums Of Random Variables ◽  
Large Numbers ◽  
Pairwise Independent Random Variables ◽  
Weak Laws

Under uniform integrability condition, some Weak Laws of large numbers are established for weighted sums of random variables generalizing results of Rohatgi, Pruitt and Khintchine. Some Strong Laws of Large Numbers are proved for weighted sums of pairwise independent random variables generalizing results of Jamison, Orey and Pruitt and Etemadi.

scholarly journals Limit theorems for multivariate Brownian semistationary processes and feasible results

2019 ◽  
Vol 51 (03) ◽  
pp. 667-716
Author(s):  
Riccardo Passeggeri ◽  
Almut E. D. Veraart
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Behaviour ◽  
Central Limit ◽  
Gaussian Processes ◽  
Limit Theorems ◽  
Laws Of Large Numbers ◽  
Stationary Increments ◽  
Large Numbers ◽  
Weak Laws

AbstractIn this paper we introduce the multivariate Brownian semistationary (BSS) process and study the joint asymptotic behaviour of its realised covariation using in-fill asymptotics. First, we present a central limit theorem for general multivariate Gaussian processes with stationary increments, which are not necessarily semimartingales. Then, we show weak laws of large numbers, central limit theorems, and feasible results for BSS processes. An explicit example based on the so-called gamma kernels is also provided.

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