scholarly journals Interval type local limit theorems for lattice type random variables and distributions

Stochastics ◽  
2021 ◽  
pp. 1-12
Author(s):  
M. Fleermann ◽  
W. Kirsch ◽  
G. Toth
Keyword(s):  
Limit Theorems ◽  
Random Variables ◽  
Local Limit ◽  
Lattice Type ◽  
Interval Type ◽  
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).

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.

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