On the duality between the behaviour of sums of independent random variables and the sums of their squares

1978 ◽  
Vol 84 (1) ◽  
pp. 117-121 ◽  
Author(s):  
Peter Hall
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Absolute Moment ◽  
Mild Conditions ◽  
Standard Normal Variable ◽  
Normal Variable ◽  
Poisson Law ◽  
Standard Normal ◽  
The Mean ◽  
Normal Law

AbstractLet Xnj, 1 ≤ j ≤ kn, be independent, asymptotically negligible random variables for each n ≥ 1. In certain cases there exists a duality between the behaviour of ΣjXnj and . We extend one of the known forms of this duality, and show that, under mild conditions on the truncated moments of the Xnj, the convergence of to 1 in the mean of order p (p ≥ 1) is equivalent to the convergence of ΣjXnj to the standard normal law, together with the convergence of its 2pth absolute moment to that of a standard normal variable. A similar result holds in the case of convergence to a Poisson law.

Traces of Matrix Products

2014 ◽  
Vol 27 ◽  
Author(s):  
John Greene
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Normal Distributions ◽  
Matrix Products ◽  
Standard Normal

Given two noncommuting matrices, A and B, it is well known that AB and BA have the same trace. This extends to cyclic permutations of products of A’s and B’s. It is shown here that for 2×2 matrices A and B, whose elements are independent random variables with standard normal distributions, the probability that Tr(ABAB) > Tr(AB) is exactly 1/\sqrt{2} .

scholarly journals Lower bound for the number of real roots of a random algebraic polynomial

1983 ◽  
Vol 35 (1) ◽  
pp. 18-27 ◽  
Author(s):  
M. N. Mishra ◽  
N. N. Nayak ◽  
S. Pattanayak
Keyword(s):  
Lower Bound ◽  
Algebraic Polynomial ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Independent Random Variables ◽  
Real Numbers ◽  
Exceptional Set ◽  
Real Roots ◽  
Normal Law ◽  
Number Of Real Roots

AbstractLet X1, X2, …, Xn be identically distributed independent random variables belonging to the domain of attraction of the normal law, have zero means and Pr{Xr ≠ 0} > 0. Suppose a0, a1, …, an are non-zero real numbers and max and εn is such that as n → ∞, εn. If Nn be the number of real roots of the equation then for n > n0, Nn > εn log n outside an exceptional set of measure at most provided limn→∞ (kn/tn) is finite.

scholarly journals Extension of the past lifetime and its connection to the cumulative entropy

2015 ◽  
Vol 52 (04) ◽  
pp. 1156-1174 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Abdolsaeed Toomaj
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Stochastic Orders ◽  
Absolutely Continuous ◽  
The Past ◽  
Probability Densities ◽  
The Mean ◽  
Weighted Probability ◽  
Series Systems ◽  
Cumulative Entropy

Given two absolutely continuous nonnegative independent random variables, we define the reversed relevation transform as dual to the relevation transform. We first apply such transforms to the lifetimes of the components of parallel and series systems under suitably proportionality assumptions on the hazard rates. Furthermore, we prove that the (reversed) relevation transform is commutative if and only if the proportional (reversed) hazard rate model holds. By repeated application of the reversed relevation transform we construct a decreasing sequence of random variables which leads to new weighted probability densities. We obtain various relations involving ageing notions and stochastic orders. We also exploit the connection of such a sequence to the cumulative entropy and to an operator that is dual to the Dickson-Hipp operator. Iterative formulae for computing the mean and the cumulative entropy of the random variables of the sequence are finally investigated.

scholarly journals Extension of the past lifetime and its connection to the cumulative entropy

2015 ◽  
Vol 52 (4) ◽  
pp. 1156-1174 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Abdolsaeed Toomaj
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Stochastic Orders ◽  
Absolutely Continuous ◽  
The Past ◽  
Probability Densities ◽  
The Mean ◽  
Weighted Probability ◽  
Series Systems ◽  
Cumulative Entropy

Given two absolutely continuous nonnegative independent random variables, we define the reversed relevation transform as dual to the relevation transform. We first apply such transforms to the lifetimes of the components of parallel and series systems under suitably proportionality assumptions on the hazard rates. Furthermore, we prove that the (reversed) relevation transform is commutative if and only if the proportional (reversed) hazard rate model holds. By repeated application of the reversed relevation transform we construct a decreasing sequence of random variables which leads to new weighted probability densities. We obtain various relations involving ageing notions and stochastic orders. We also exploit the connection of such a sequence to the cumulative entropy and to an operator that is dual to the Dickson-Hipp operator. Iterative formulae for computing the mean and the cumulative entropy of the random variables of the sequence are finally investigated.

A note on concentration functions

1984 ◽  
Vol 96 (3-4) ◽  
pp. 181-184
Author(s):  
J. E. A. Dunnage
Keyword(s):  
Random Variables ◽  
Concentration Function ◽  
Independent Random Variables ◽  
Absolute Moment ◽  
Third Order ◽  
Concentration Functions

SynopsisWe give an inequality for the concentration function of a sum X1 + … + Xn of independent random variables when Xv has a finite absolute moment of order kv (2 < kv ≦ 3). It is an extension of somewhat similar inequalities found earlier by Offord and by the author in the case of finite third-order absolute moments.

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