Convergence to normal law in Lp of distribution functions of sums of independent random variables

1977 ◽  
Vol 16 (1) ◽  
pp. 118-127
Author(s):  
L. V. Rozovskii
Keyword(s):  
Random Variables ◽  
Distribution Functions ◽  
Independent Random Variables ◽  
Normal Law

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.

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.

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.

On the asymptotic normality of Hill's estimator

1995 ◽  
Vol 118 (2) ◽  
pp. 375-382 ◽  
Author(s):  
Sándor Csörgő ◽  
László Viharos
Keyword(s):  
Distribution Function ◽  
Asymptotic Normality ◽  
Order Statistics ◽  
Random Variables ◽  
Fixed Number ◽  
Distribution Functions ◽  
Independent Random Variables ◽  
Hill’S Estimator ◽  
Common Distribution ◽  
Common Distribution Function

Let X, X1, X2, …, be independent random variables with a common distribution function F(x) = P {X ≤ x}, x∈ℝ, and for each n∈ℕ, let X1, n ≤ … ≤ Xn, n denote the order statistics pertaining to the sample X1, …, Xn. We assume that 1–F(x) = x−1/cl(x), 0 < x < ∞, where l is some function slowly varying at infinity and c > 0 is any fixed number. The class of all such distribution functions will be denoted by .

scholarly journals On a probability problem connected with Railway traffic

1991 ◽  
Vol 4 (1) ◽  
pp. 1-27 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Empirical Distribution ◽  
Continuous Distribution ◽  
Random Variables ◽  
Distribution Functions ◽  
Continuous Distribution Function ◽  
Independent Random Variables ◽  
Railway Traffic ◽  
Empirical Distribution Functions

Let Fn(x) and Gn(x) be the empirical distribution functions of two independent samples, each of size n, in the case where the elements of the samples are independent random variables, each having the same continuous distribution function V(x) over the interval (0,1). Define a statistic θn by θn/n=∫01[Fn(x)−Gn(x)]dV(x)−min0≤x≤1[Fn(x)−Gn(x)]. In this paper the limits of E{(θn/2n)r}(r=0,1,2,…) and P{θn/2n≤x} are determined for n→∞. The problem of finding the asymptotic behavior of the moments and the distribution of θn as n→∞ has arisen in a study of the fluctuations of the inventory of locomotives in a randomly chosen railway depot.

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