Local Limit Theorems for Sums of Power Series Distributed Random Variables and for the Number of Components in Labelled Relational Structures

1992 ◽  
Vol 3 (4) ◽  
pp. 403-426 ◽  
Author(s):  
Lyuben R. Mutafchiev
Keyword(s):  
Power Series ◽  
Limit Theorems ◽  
Random Variables ◽  
Local Limit ◽  
Number Of Components ◽  
Relational Structures ◽  
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.

Local limit theorems for generalized scheme of allocation of particles into ordered cells

2021 ◽  
Vol 31 (4) ◽  
pp. 293-307
Author(s):  
Aleksandr N. Timashev
Keyword(s):  
Weak Convergence ◽  
Negative Binomial Distribution ◽  
Limit Theorems ◽  
Negative Binomial ◽  
Sufficient Conditions ◽  
Local Limit ◽  
Number Of Components ◽  
Normal Law ◽  
Local Limit Theorems ◽  
Generalized Scheme

Abstract A generalized scheme of allocation of n particles into ordered cells (components). Some statements containing sufficient conditions for the weak convergence of the number of components with given cardinality and of the total number of components to the negative binomial distribution as n → ∞ are presented as hypotheses. Examples supporting the validity of these statements in particular cases are considered. For some examples we prove local limit theorems for the total number of components which partially generalize known results on the convergence of this distribution to the normal law.

Limit theorems for some classes of power series type distributions

2020 ◽  
Vol 30 (1) ◽  
pp. 69-78
Author(s):  
Aleksandr N. Timashev
Keyword(s):  
Power Series ◽  
Limit Theorems ◽  
Local Limit ◽  
Random Variable ◽  
Descent Method ◽  
Steepest Descent Method ◽  
Convergence Radius ◽  
Series Type ◽  
The Right ◽  
Local Limit Theorems

AbstractSeveral classes of distributions of power series type with finite and infinite radii of convergence are considered. For such distributions local limit theorems are obtained as the parameter of distribution tends to the right end of the interval of convergence. For the case when the convergence radius equals to 1, we prove an integral limit theorem on the convergence of distributions of random variables (1 − x)ξxas x → 1− to the gamma-distribution (ξx is a random variable with corresponding distribution of the power series type). The proofs are based on the steepest descent method.

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).

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