scholarly journals Functional Limit Theorem for Products of Sums of Independent and Nonidentically Distributed Random Variables

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Przemysław Matuła ◽  
Iwona Stępień
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Limit Theorems ◽  
Random Variables ◽  
Functional Limit Theorem ◽  
Functional Limit ◽  
Generalize Limit ◽  
Products Of Sums

We study the weak convergence in the spaceD[0,1]of processes constructed from products of sums of independent but not necessarily identically distributed random variables. The presented results extend and generalize limit theorems known so far for i.i.d. sequences.

Limit theorems for dissociated random variables

1976 ◽  
Vol 8 (04) ◽  
pp. 806-819 ◽  
Author(s):  
B. W. Silverman
Keyword(s):  
Data Analysis ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Limit Theorems ◽  
Normal Approximation ◽  
Empirical Distribution ◽  
Random Variables ◽  
Distribution Process ◽  
Suitable Space

Families of exchangeably dissociated random variables are defined and discussed. These include families of the form g(Yi, Yj , …, Yz ) for some function g of m arguments and some sequence Yn of i.i.d. random variables on any suitable space. A central limit theorem for exchangeably dissociated random variables is proved and some remarks on the closeness of the normal approximation are made. The weak convergence of the empirical distribution process to a Gaussian process is proved. Some applications to data analysis are discussed.

Limit theorems for dissociated random variables

1976 ◽  
Vol 8 (4) ◽  
pp. 806-819 ◽  
Author(s):  
B. W. Silverman
Keyword(s):  
Data Analysis ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Limit Theorems ◽  
Normal Approximation ◽  
Empirical Distribution ◽  
Random Variables ◽  
Distribution Process ◽  
Suitable Space

Families of exchangeably dissociated random variables are defined and discussed. These include families of the form g(Yi, Yj, …, Yz) for some function g of m arguments and some sequence Yn of i.i.d. random variables on any suitable space. A central limit theorem for exchangeably dissociated random variables is proved and some remarks on the closeness of the normal approximation are made. The weak convergence of the empirical distribution process to a Gaussian process is proved. Some applications to data analysis are discussed.

A central limit theorem by remainder term for renewal processes

1992 ◽  
Vol 24 (2) ◽  
pp. 267-287 ◽  
Author(s):  
Allen L. Roginsky
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Remainder Term ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Renewal Processes

Three different definitions of the renewal processes are considered. For each of them, a central limit theorem with a remainder term is proved. The random variables that form the renewal processes are independent but not necessarily identically distributed and do not have to be positive. The results obtained in this paper improve and extend the central limit theorems obtained by Ahmad (1981) and Niculescu and Omey (1985).

Limit theorems for extremes with random sample size

1998 ◽  
Vol 30 (03) ◽  
pp. 777-806 ◽  
Author(s):  
Dmitrii S. Silvestrov ◽  
Jozef L. Teugels
Keyword(s):  
Weak Convergence ◽  
Sample Size ◽  
Random Sample ◽  
Limit Theorems ◽  
Functional Limit ◽  
Functional Limit Theorems ◽  
Random Sample Size ◽  
Extremal Processes ◽  
Convergence Type

This paper is devoted to the investigation of limit theorems for extremes with random sample size under general dependence-independence conditions for samples and random sample size indexes. Limit theorems of weak convergence type are obtained as well as functional limit theorems for extremal processes with random sample size indexes.

Some Functional Stable Limit Theorems

1973 ◽  
Vol 16 (2) ◽  
pp. 173-177 ◽  
Author(s):  
D. R. Beuerman
Keyword(s):  
Weak Convergence ◽  
Limit Theorems ◽  
Stable Process ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Stable Limit ◽  
Stable Law ◽  
Partial Sum Process ◽  
Image Position

Let Xl,X2,X3, … be a sequence of independent and identically distributed (i.i.d.) random variables which belong to the domain of attraction of a stable law of index α≠1. That is,1whereandwhere L(n) is a function of slow variation; also take S0=0, B0=l.In §2, we are concerned with the weak convergence of the partial sum process to a stable process and the question of centering for stable laws and drift for stable processes.

Sign in / Sign up

Export Citation Format

Share Document