An Arithmetic Method for Obtaining Local Limit Theorems for Lattice Random Variables

1970 ◽  
Vol 15 (1) ◽  
pp. 87-97 ◽  
Author(s):  
D. A. Moskvin ◽  
L. P. Postnikova ◽  
A. A. Yudin
Keyword(s):  
Limit Theorems ◽  
Random Variables ◽  
Local Limit ◽  
Arithmetic Method ◽  
Local Limit Theorems

Local limit theorems for the maxima of discrete random variables

1980 ◽  
Vol 88 (1) ◽  
pp. 161-165 ◽  
Author(s):  
C. W. Anderson
Keyword(s):  
Limit Theorems ◽  
Random Variables ◽  
Local Limit ◽  
Value Theory ◽  
Extreme Value ◽  
Extreme Value Distributions ◽  
Discrete Random Variables ◽  
Image Position ◽  
Independent Identically Distributed ◽  
Local Limit Theorems

Let , where the Xi, i = 1, 2, … are independent identically distributed random variables. Classical extreme value theory, described for example in the books of do Haan(6) and Galambos(3) gives conditions under which there exist constants an > 0 and bn such thatwhere G(x) is taken to be one of the extreme value distributions G1(x) = exp (− e−x), G2(x) = exp (− x−a) (x > 0, α > 0) and G3(x) = exp (−(− x)α) (x < 0, α > 0).

Structure Theory of Set Addition and Local Limit Theorems for Independent Lattice Random Variables

1974 ◽  
Vol 19 (1) ◽  
pp. 53-64 ◽  
Author(s):  
D. A. Moskvin ◽  
G. A. Frejman ◽  
A. A. Yudin
Keyword(s):  
Limit Theorems ◽  
Random Variables ◽  
Local Limit ◽  
Structure Theory ◽  
Set Addition ◽  
Local Limit Theorems ◽  
Independent Lattice

scholarly journals Large Deviation Local Limit Theorems for Arbitrary Sequences of Random Variables

1985 ◽  
Vol 13 (1) ◽  
pp. 97-114 ◽  
Author(s):  
Narasinga R. Chaganty ◽  
J. Sethuraman
Keyword(s):  
Limit Theorems ◽  
Large Deviation ◽  
Random Variables ◽  
Local Limit ◽  
Local Limit Theorems

Local limit theorems for one class of distributions in probabilistic combinatorics

2018 ◽  
Vol 28 (6) ◽  
pp. 405-420
Author(s):  
Aleksandr N. Timashev
Keyword(s):  
Power Series ◽  
Limit Theorems ◽  
Random Variables ◽  
Local Limit ◽  
Random Variable ◽  
Asymptotical Normality ◽  
Probabilistic Combinatorics ◽  
Nonnegative Coefficients ◽  
Positive Coefficients ◽  
Local Limit Theorems

Abstract Let a function f(z) be decomposed into a power series with nonnegative coefficients which converges in a circle of positive radius R. Let the distribution of the random variable ξn, n ∈ {1, 2, …}, be defined by the formula $$\begin{array}{} \displaystyle P\{\xi_n=N\}=\frac{\mathrm{coeff}_{z^n}\left(\frac{\left(f(z)\right)^N}{N!}\right)}{\mathrm{coeff}_{z^n}\left(\exp(f(z))\right)},\,N=0,1,\ldots \end{array} $$ for some ∣z∣ < R (if the denominator is positive). Examples of appearance of such distributions in probabilistic combinatorics are given. Local theorems on asymptotical normality for distributions of ξn are proved in two cases: a) if f(z) = (1 − z)−λ, λ = const ∈ (0, 1] for ∣z∣ < 1, and b) if all positive coefficients of expansion f (z) in a power series are equal to 1 and the set A of their numbers has the form $$\begin{array}{} \displaystyle A = \{m^r \, | \, m \in \mathbb{N} \}, \, \, r = \mathrm {const},\; r \in \{2,3,\ldots\}. \end{array} $$ A hypothetical general local limit normal theorem for random variables ξn is stated. Some examples of validity of the statement of this theorem are given.

scholarly journals The discounted local limit theorems for large deviations

2008 ◽  
Vol 48 ◽  
Author(s):  
Leonas Saulis ◽  
Dovilė Deltuvienė
Keyword(s):  
Large Deviations ◽  
Density Function ◽  
Limit Theorems ◽  
Distribution Density ◽  
Normal Approximation ◽  
Random Variables ◽  
Local Limit ◽  
Distribution Density Function ◽  
Local Limit Theorems

Theorems of large deviations, both in the Cramer zone and the Linnik power zones, for the normal approximation of the distribution density function of normalized sum Sv = \sum∞ k=0 vkXk, 0 < v < 1, of i.i.d. random variables (r.v.) X0, X1, . . . satisfying the generalized Bernstein’s condition are obtained.

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