scholarly journals On the asymptotic distribution of sums of independent identically distributed random variables

1962 ◽  
Vol 4 (4) ◽  
pp. 323-332 ◽  
Author(s):  
Bengt Rosén
Keyword(s):  
Asymptotic Distribution ◽  
Random Variables ◽  
Independent Identically Distributed

Limit distributions for compounded sums of extreme order statistics

1992 ◽  
Vol 29 (03) ◽  
pp. 557-574 ◽  
Author(s):  
Jan Beirlant ◽  
Jozef L. Teugels
Keyword(s):  
Order Statistics ◽  
Asymptotic Distribution ◽  
Counting Process ◽  
Extreme Values ◽  
Random Variables ◽  
Limit Distributions ◽  
Extreme Order Statistics ◽  
Renewal Counting Process ◽  
Independent Identically Distributed

Let X (1) ≦ X (2) ≦ ·· ·≦ X (N(t)) be the order statistics of the first N(t) elements from a sequence of independent identically distributed random variables, where {N(t); t ≧ 0} is a renewal counting process independent of the sequence of X's. We give a complete description of the asymptotic distribution of sums made from the top kt extreme values, for any sequence kt such that kt → ∞, kt /t → 0 as t → ∞. We discuss applications to reinsurance policies based on large claims.

On road traffic with free overtaking

1969 ◽  
Vol 6 (3) ◽  
pp. 524-549 ◽  
Author(s):  
Torbjörn Thedéen
Keyword(s):  
Asymptotic Distribution ◽  
Road Traffic ◽  
Random Variables ◽  
Time Axis ◽  
Independent Identically Distributed

The cars are considered as points on an infinite road with no intersections. They can overtake each other without any delay and they travel at constant speeds. These are independent identically distributed random variables also independent of the initial positions of the cars. The main purpose of the paper is the study of the asymptotic distribution for the number of overtakings (and/or meetings) in increasing rectangles in the time-road plane. Under the assumption of (weighted) Poisson distributed cars along the time-axis we deduce the asymptotic distribution of the standardized number of overtakings (and/or meetings) for large rectangles in the time-road plane. Lastly we shall indicate an application of the results.

Limit distributions for compounded sums of extreme order statistics

1992 ◽  
Vol 29 (3) ◽  
pp. 557-574 ◽  
Author(s):  
Jan Beirlant ◽  
Jozef L. Teugels
Keyword(s):  
Order Statistics ◽  
Asymptotic Distribution ◽  
Counting Process ◽  
Extreme Values ◽  
Random Variables ◽  
Limit Distributions ◽  
Extreme Order Statistics ◽  
Renewal Counting Process ◽  
Independent Identically Distributed

LetX(1)≦X(2)≦ ·· ·≦X(N(t))be the order statistics of the firstN(t) elements from a sequence of independent identically distributed random variables, where {N(t);t≧ 0} is a renewal counting process independent of the sequence ofX's. We give a complete description of the asymptotic distribution of sums made from the topktextreme values, for any sequencektsuch thatkt→ ∞,kt/t→ 0 ast→ ∞. We discuss applications to reinsurance policies based on large claims.

On road traffic with free overtaking

1969 ◽  
Vol 6 (03) ◽  
pp. 524-549 ◽  
Author(s):  
Torbjörn Thedéen
Keyword(s):  
Asymptotic Distribution ◽  
Road Traffic ◽  
Random Variables ◽  
Time Axis ◽  
Independent Identically Distributed

The cars are considered as points on an infinite road with no intersections. They can overtake each other without any delay and they travel at constant speeds. These are independent identically distributed random variables also independent of the initial positions of the cars. The main purpose of the paper is the study of the asymptotic distribution for the number of overtakings (and/or meetings) in increasing rectangles in the time-road plane. Under the assumption of (weighted) Poisson distributed cars along the time-axis we deduce the asymptotic distribution of the standardized number of overtakings (and/or meetings) for large rectangles in the time-road plane. Lastly we shall indicate an application of the results.

A Note on the Accuracy of Normal Approximation of Random Quantities

2021 ◽  
Vol 73 (1) ◽  
pp. 62-67
Author(s):  
Ibrahim A. Ahmad ◽  
A. R. Mugdadi
Keyword(s):  
Sample Size ◽  
Order Statistic ◽  
Normal Approximation ◽  
Order Approximation ◽  
Random Variables ◽  
Random Variable ◽  
Limiting Distribution ◽  
Exact Order ◽  
Fixed Sample ◽  
Independent Identically Distributed

For a sequence of independent, identically distributed random variable (iid rv's) [Formula: see text] and a sequence of integer-valued random variables [Formula: see text], define the random quantiles as [Formula: see text], where [Formula: see text] denote the largest integer less than or equal to [Formula: see text], and [Formula: see text] the [Formula: see text]th order statistic in a sample [Formula: see text] and [Formula: see text]. In this note, the limiting distribution and its exact order approximation are obtained for [Formula: see text]. The limiting distribution result we obtain extends the work of several including Wretman[Formula: see text]. The exact order of normal approximation generalizes the fixed sample size results of Reiss[Formula: see text]. AMS 2000 subject classification: 60F12; 60F05; 62G30.

scholarly journals A counterexample to the existence of a general central limit theorem for pairwise independent identically distributed random variables

2021 ◽  
Vol 499 (1) ◽  
pp. 124982
Author(s):  
Benjamin Avanzi ◽  
Guillaume Boglioni Beaulieu ◽  
Pierre Lafaye de Micheaux ◽  
Frédéric Ouimet ◽  
Bernard Wong
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variables ◽  
Independent Identically Distributed
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