Asymptotic normality of numbers of non-occurring values of m-dependent random variables

2014 ◽  
Vol 24 (5) ◽  
Author(s):  
Margarita I. Tikhomirova ◽  
Vladimir P. Chistyakov
Keyword(s):  
Asymptotic Normality ◽  
Random Variables ◽  
Stationary Sequence ◽  
Order Of Growth ◽  
Dependent Random Variables ◽  
Asymptotic Order ◽  
Normally Distributed

AbstractThe paper is concerned with the numbers of non-occurring values in intervals of a stationary sequence of m-dependent random variables with N outcomes. These numbers are shown to be asymptotically joint normally distributed, provided that the number N and the lengths of intervals tend to infinity with the same asymptotic order of growth.

On the asymptotic normality of some sums of dependent random variables

2017 ◽  
Vol 27 (2) ◽  
Author(s):  
Margarita I. Tikhomirova ◽  
Vladimir P. Chistjakov
Keyword(s):  
Asymptotic Normality ◽  
Random Sequence ◽  
Random Variables ◽  
Stationary Sequence ◽  
Dependency Graph ◽  
Strong Mixing ◽  
Mixing Condition ◽  
Dependent Random Variables ◽  
Sum Of Functions ◽  
Strong Mixing Condition

AbstractA theorem on the asymptotic normality of the sum of dependent random variables is stated and proved. Conditions of the theorem are formulated in terms of a dependency graph which characterizes the relationships between random variables. This theorem is used to prove the asymptotic normality of the sum of functions defined on subsets of elements of the stationary sequence satisfying the strong mixing condition. As an illustration of possible applications of these theorems we give a theorem on the asymptotic normality of the number of empty cells if the random sequence of cells occupied by particles is a stationary sequence satisfying the uniform strong mixing condition.

A remark on the central limit question for dependent random variables

1980 ◽  
Vol 17 (1) ◽  
pp. 94-101 ◽  
Author(s):  
Richard C. Bradley
Keyword(s):  
Central Limit ◽  
Random Variables ◽  
Stationary Sequence ◽  
Stable Limit ◽  
Limit Laws ◽  
Dependent Random Variables ◽  
Asymptotically Normal ◽  
Second Moments

Given a strictly stationary sequence {Xk, k = …, −1,0,1, …} of r.v.'s one defines for n = 1, 2, 3 …, . Here an example of {Xk} is given with finite second moments, for which Var(X1 + … + Xn)→∞ and ρ n → 0 as n→∞, but (X1 + … + Xn) fails to be asymptotically normal; instead there is partial attraction to non-stable limit laws.

A remark on the central limit question for dependent random variables

1980 ◽  
Vol 17 (01) ◽  
pp. 94-101 ◽  
Author(s):  
Richard C. Bradley
Keyword(s):  
Central Limit ◽  
Random Variables ◽  
Stationary Sequence ◽  
Stable Limit ◽  
Limit Laws ◽  
Dependent Random Variables ◽  
Asymptotically Normal ◽  
Second Moments

Given a strictly stationary sequence {Xk, k= …, −1,0,1, …} of r.v.'s one defines forn= 1, 2, 3 …,. Here an example of {Xk} is given with finite second moments, for which Var(X1+ … +Xn)→∞ andρn→ 0 asn→∞, but (X1+ … +Xn) fails to be asymptotically normal; instead there is partial attraction to non-stable limit laws.

Obtaining remainder term estimates by an inversion technique

1984 ◽  
Vol 16 (01) ◽  
pp. 88-110
Author(s):  
Gunnar Englund
Keyword(s):  
Asymptotic Normality ◽  
Remainder Term ◽  
General Theorem ◽  
Double Sequence ◽  
Random Variables ◽  
Independent Random Variables ◽  
Inversion Technique ◽  
Capture Recapture ◽  
Dual Sequence ◽  
Normally Distributed

Let be a double sequence of random variables such that there exists a ‘dual' sequence satisfying , and that can be expressed as a sum of independent random variables. If (suitably centered and rescaled) is approximately normally distributed as k and N → ∞ in some fashion, we can use this fact to obtain a remainder-term estimate for the asymptotic normality of as n and N → ∞ in some prescribed manner. The result in the general theorem is used in two specific situations: (i) classical occupancy where the balls can fall through the boxes, (ii) a capture-recapture problem where tagging affects catchability.

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