On read-once transformations of random variables over finite fields

2015 ◽  
Vol 25 (5) ◽  
Author(s):  
Aleksey D. Yashunsky
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Probability Distributions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Independent Identically Distributed ◽  
Family Of Distributions

AbstractTransformations of independent random variables over a finite field by read-once formulas are considered. Subsets of probability distributions that are preserved by read-once transformations are constructed. Also we construct a family of distributions that may be arbitrarily closely approximated by a read-once combination of independent identically distributed random variables, whose distributions have no zero components.

On asymptotic independence of the partial sums of positive and negative parts of independent random variables

1971 ◽  
Vol 3 (02) ◽  
pp. 404-425
Author(s):  
Howard G. Tucker
Keyword(s):  
Stable Distribution ◽  
Characteristic Exponent ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Distribution Functions ◽  
Independent Random Variables ◽  
Asymptotic Independence ◽  
Partial Sums ◽  
Common Distribution Function ◽  
Independent Identically Distributed

The aim of this study is an investigation of the joint limiting distribution of the sequence of partial sums of the positive parts and negative parts of a sequence of independent identically distributed random variables. In particular, let {Xn} be a sequence of independent identically distributed random variables with common distribution functionF, assumeFis in the domain of attraction of a stable distribution with characteristic exponent α, 0 < α ≦ 2, and let {Bn} be normalizing coefficients forF. Let us denoteXn+=XnI[Xn> 0]andXn−= −XnI[Xn<0], so thatXn=Xn+-Xn−. LetF+andF−denote the distribution functions ofX1+andX1−respectively, and letSn(+)=X1++ · · · +Xn+,Sn(-)=X1−+ · · · +Xn−. The problem considered here is to find under what conditions there exist sequences of real numbers {an} and {bn} such that the joint distribution of (Bn-1Sn(+)+an,Bn-1Sn(-)+bn) converges to that of two independent random variables (U, V). As might be expected, different results are obtained when α < 2 and when α = 2. When α < 2, there always exist such sequences so that the above is true, and in this case bothUandVare stable with characteristic exponent a, or one has such a stable distribution and the other is constant. When α = 2, and if 0 < ∫x2dF(x) < ∞, then there always exist such sequences such that the above convergence takes place; bothUandVare normal (possibly one is a constant), and if neither is a constant, thenUandVarenotindependent. If α = 2 and ∫x2dF(x) = ∞, then at least one ofF+,F−is in the domain of partial attraction of the normal distribution, and a modified statement on the independence ofUandVholds. Various specialized results are obtained for α = 2.

On asymptotic independence of the partial sums of positive and negative parts of independent random variables

1971 ◽  
Vol 3 (2) ◽  
pp. 404-425 ◽  
Author(s):  
Howard G. Tucker
Keyword(s):  
Stable Distribution ◽  
Characteristic Exponent ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Distribution Functions ◽  
Independent Random Variables ◽  
Asymptotic Independence ◽  
Partial Sums ◽  
Common Distribution Function ◽  
Independent Identically Distributed

The aim of this study is an investigation of the joint limiting distribution of the sequence of partial sums of the positive parts and negative parts of a sequence of independent identically distributed random variables. In particular, let {Xn} be a sequence of independent identically distributed random variables with common distribution function F, assume F is in the domain of attraction of a stable distribution with characteristic exponent α, 0 < α ≦ 2, and let {Bn} be normalizing coefficients for F. Let us denote Xn+ = XnI[Xn > 0] and Xn− = − XnI[Xn<0], so that Xn = Xn+ - Xn−. Let F+ and F− denote the distribution functions of X1+ and X1− respectively, and let Sn(+) = X1+ + · · · + Xn+, Sn(-) = X1− + · · · + Xn−. The problem considered here is to find under what conditions there exist sequences of real numbers {an} and {bn} such that the joint distribution of (Bn-1Sn(+) + an, Bn-1Sn(-) + bn) converges to that of two independent random variables (U, V). As might be expected, different results are obtained when α < 2 and when α = 2. When α < 2, there always exist such sequences so that the above is true, and in this case both U and V are stable with characteristic exponent a, or one has such a stable distribution and the other is constant. When α = 2, and if 0 < ∫ x2dF(x) < ∞, then there always exist such sequences such that the above convergence takes place; both U and V are normal (possibly one is a constant), and if neither is a constant, then U and V are not independent. If α = 2 and ∫ x2dF(x) = ∞, then at least one of F+, F− is in the domain of partial attraction of the normal distribution, and a modified statement on the independence of U and V holds. Various specialized results are obtained for α = 2.

Some renewal theorems for random walks in multidimensional time

1984 ◽  
Vol 95 (1) ◽  
pp. 149-154 ◽  
Author(s):  
Makoto Maejima ◽  
Toshio Mori
Keyword(s):  
Random Walks ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sum ◽  
Positive Integers ◽  
Independent Identically Distributed

Let Kr denote the set of r-tuples n = (n1n2, …, nr) with positive integers as coordinates (r ≥ 1) and {X, Xn, n ε Kr} be a family of independent, identically distributed random variables with positive mean 0 < EX ≡ μ < ∞ and finite positive variance 0 < var X ≡ σ2 ∞. The notation m ≤ n, where m = (mi) and n = (ni), means that mi ≤ ni (i = 1, 2,…, r) and |n| = n1n2 … nr. Denote Sn =Σj ≤ nXj (j, n ε Kr). When r = 1, {Xn, n ε Kr) reduces to the sequence {Xj, j ε 1} of independent random variables each distributed as X, and Sn becomes the ordinary partial sum .

scholarly journals Asymptotic expansions for distributions of sums of a double sequence of quasilattice random variables

2011 ◽  
Vol 52 ◽  
pp. 353-358
Author(s):  
Algimantas Bikelis ◽  
Juozas Augutis ◽  
Kazimieras Padvelskis
Keyword(s):  
Asymptotic Expansion ◽  
Probability Distribution ◽  
Asymptotic Expansions ◽  
Probability Distributions ◽  
Double Sequence ◽  
Random Variables ◽  
Independent Random Variables ◽  
Infinitely Divisible ◽  
Formal Asymptotic Expansion

We consider the formal asymptotic expansion of probability distribution of the sums of independent random variables. The approximation was made by using infinitely divisible probability distributions.  

Order statistics of partial sums of mutually independent random variables

1975 ◽  
Vol 12 (02) ◽  
pp. 390-395 ◽  
Author(s):  
Felix Pollaczek
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Characteristic Functions ◽  
Analytic Proof ◽  
Mutually Independent ◽  
Independent Identically Distributed

Herein is exposed a simplified analytic proof of formulas for the characteristic functions of ordered partial sums of mutually independent identically distributed random variables. This problem which we had raised and solved in 1952 by another method, has since been treated by several authors (see Wendel [6]), and recently by de Smit [4], who made use of a kind of Wiener-Hopf decomposition†. On the contrary our present as well as our previous proof essentially uses the explicit solution of a certain singular integral equation in a complex domain.

scholarly journals Max Weibull-G Power Series Distributions

2021 ◽  
pp. 15-29
Author(s):  
Munteanu Bogdan Gheorghe
Keyword(s):  
Probability Distribution ◽  
Power Series ◽  
Probability Distributions ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
New Family ◽  
The Family ◽  
Distribution Family ◽  
Power Series Distributions

Based on the Weibull-G Power probability distribution family, we have proposed a new family of probability distributions, named by us the Max Weibull-G power series distributions, which may be applied in order to solve some reliability problems. This implies the fact that the Max Weibull-G power series is the distribution of a random variable max (X1 ,X2 ,...XN) where X1 ,X2 ,... are Weibull-G distributed independent random variables and N is a natural random variable the distribution of which belongs to the family of power series distribution. The main characteristics and properties of this distribution are analyzed.

Order statistics of partial sums of mutually independent random variables

1975 ◽  
Vol 12 (2) ◽  
pp. 390-395 ◽  
Author(s):  
Felix Pollaczek
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Characteristic Functions ◽  
Analytic Proof ◽  
Mutually Independent ◽  
Independent Identically Distributed

Herein is exposed a simplified analytic proof of formulas for the characteristic functions of ordered partial sums of mutually independent identically distributed random variables. This problem which we had raised and solved in 1952 by another method, has since been treated by several authors (see Wendel [6]), and recently by de Smit [4], who made use of a kind of Wiener-Hopf decomposition†. On the contrary our present as well as our previous proof essentially uses the explicit solution of a certain singular integral equation in a complex domain.

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