On Asymptotic Expansions for Distribution Functions of Sums of Independent Random Variables

1971 ◽  
Vol 16 (2) ◽  
pp. 333-343 ◽  
Author(s):  
L. V. Osipov
Keyword(s):  
Asymptotic Expansions ◽  
Random Variables ◽  
Distribution Functions ◽  
Independent Random Variables

scholarly journals Asymptotic expansions for distribution of sums quasi-lattice random variables

2011 ◽  
Vol 52 ◽  
pp. 359-364
Author(s):  
Algimantas Bikelis ◽  
Kazimieras Padvelskis ◽  
Pranas Vaitkus
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Independent Random Variables

Althoug Chebyshev [3] and Edeworth [5] had conceived of the formal expansions for distribution of sums of independent random variables, but only in Cramer’s work [4] was laid a proper foundation of this problem. In the case when random variables are lattice Esseen get the asymptotic expansion in a new different form. Here we extend this problem for quasi-lattice random variables.  

scholarly journals Asymptotic Expansions for Distributions of Compound Sums of Random Variables with Rapidly Varying Subexponential Distribution

2007 ◽  
Vol 44 (3) ◽  
pp. 670-684 ◽  
Author(s):  
Ph. Barbe ◽  
W. P. McCormick ◽  
C. Zhang
Keyword(s):  
Asymptotic Expansion ◽  
Weibull Distribution ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Subexponential Distribution ◽  
Independent Random Variables ◽  
Sums Of Random Variables

We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions.

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.

scholarly journals An Asymptotic Formula for a Class of Distribution Functions

1935 ◽  
Vol 4 (3) ◽  
pp. 138-143
Author(s):  
E. W. Cannon ◽  
Aurel Wintner ◽  
A. C. Aitken
Keyword(s):  
Asymptotic Formula ◽  
Asymptotic Behaviour ◽  
Random Variables ◽  
Distribution Functions ◽  
Independent Random Variables ◽  
Distribution Law ◽  
General Conditions

If x1, x2, …., xk, …. are independent random variables each of which is subjected to a distribution law σ = σ(x) independent of k and having a finite positive dispersion, then x1 + x2 + …. + xn is known to obey the Gauss law as n→ + ∞, no matter how σ (x) be chosen. There arises, however, the question whether it is nevertheless possible to determine the elementary law σ (x) from the asymptotic behaviour of the distribution law of x1 + x2 + …. + xn for very large but finite values of n. It will be shown that the answer is affirmative under very general conditions.

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