Limit theorems for general size distributions

1981 ◽  
Vol 18 (1) ◽  
pp. 139-147 ◽  
Author(s):  
Wen-Chen Chen
Keyword(s):  
Limit Theorems ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Negative Integer ◽  
Urn Model ◽  
Special Cases ◽  
Mean And Variance ◽  
Bose Einstein ◽  
General Size

Let X1, X2, · ··, Xn, · ·· be independent and identically distributed non-negative integer-valued random variables with finite mean and variance. For any positive integer n and m we consider the random vector i.e., L has the same distribution as the conditional distribution of (X1, · ··, Xm) given the condition It is easy to see that our model includes the classical urn model, the Bose–Einstein urn model and the Pólya urn model as special cases. For any non-negative integer s define G(s) = the number of Li′s such that Li = s, and U = the number of Li′s such that Li is an even number; in this paper we study the asymptotic behaviour of the random variables considered above. Some central limit theorems and a multinormal local limit theorem are proved.

Limit theorems for general size distributions

1981 ◽  
Vol 18 (01) ◽  
pp. 139-147 ◽  
Author(s):  
Wen-Chen Chen
Keyword(s):  
Limit Theorems ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Negative Integer ◽  
Urn Model ◽  
Special Cases ◽  
Mean And Variance ◽  
Bose Einstein ◽  
General Size

Let X 1, X 2, · ··, Xn , · ·· be independent and identically distributed non-negative integer-valued random variables with finite mean and variance. For any positive integer n and m we consider the random vector i.e., L has the same distribution as the conditional distribution of (X 1, · ··, Xm ) given the condition It is easy to see that our model includes the classical urn model, the Bose–Einstein urn model and the Pólya urn model as special cases. For any non-negative integer s define G(s) = the number of Li′s such that Li = s, and U = the number of Li′s such that Li is an even number; in this paper we study the asymptotic behaviour of the random variables considered above. Some central limit theorems and a multinormal local limit theorem are proved.

scholarly journals Central and Local Limit Theorems for Numbers of the Tribonacci Triangle

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 880
Author(s):  
Igoris Belovas
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Normal Distribution ◽  
Rate Of Convergence ◽  
Limit Theorems ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Constructive Proof ◽  
The Central Limit Theorem ◽  
Local Limit Theorems

In this research, we continue studying limit theorems for combinatorial numbers satisfying a class of triangular arrays. Using the general results of Hwang and Bender, we obtain a constructive proof of the central limit theorem, specifying the rate of convergence to the limiting (normal) distribution, as well as a new proof of the local limit theorem for the numbers of the tribonacci triangle.

Limit theorems for first-passage times in linear and non-linear renewal theory

1984 ◽  
Vol 16 (04) ◽  
pp. 766-803 ◽  
Author(s):  
S. P. Lalley
Keyword(s):  
Limit Theorems ◽  
Large Deviation ◽  
Local Limit Theorem ◽  
Renewal Theory ◽  
Local Limit ◽  
Linear Boundary ◽  
First Passage ◽  
Non Linear ◽  
First Time ◽  
Passage Times

A local limit theorem for is obtained, where τ a is the first time a random walk Sn with positive drift exceeds a. Applications to large-deviation probabilities and to the crossing of a non-linear boundary are given.

Probabilistic Proofs of Euler Identities

2013 ◽  
Vol 50 (04) ◽  
pp. 1206-1212 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Generating Function ◽  
Local Limit Theorem ◽  
Infinite Product ◽  
Random Variables ◽  
Local Limit ◽  
Moment Generating Function ◽  
Basel Problem

Formulae for ζ(2n) andLχ4(2n+ 1) involving Euler and tangent numbers are derived using the hyperbolic secant probability distribution and its moment generating function. In particular, the Basel problem, where ζ(2) = π2/ 6, is considered. Euler's infinite product for the sine is also proved using the distribution of sums of independent hyperbolic secant random variables and a local limit theorem.

Interconnected birth and death processes

1968 ◽  
Vol 5 (02) ◽  
pp. 334-349 ◽  
Author(s):  
Prem S. Puri
Keyword(s):  
Limit Theorems ◽  
Probability Generating Function ◽  
Random Variables ◽  
Time Dependent ◽  
Explicit Solutions ◽  
Transition Rates ◽  
Birth And Death Processes ◽  
Death Rates ◽  
Standard Methods ◽  
Special Cases

SummaryTwo cases of multiple linearly interconnected linear birth and death processes are considered. It is found that in general the solution of the Kolmogorov differential equations for the probability generating function (p.g.f)gof the random variables involved is not obtainable by standard methods, although one can obtain moments of the random variables from these equations. A method is considered for obtaining an approximate solution forg.This is based on the introduction of a sequence of stochastic processes such that the sequence {f(n)} of their p.g.f.'s tends togasn → ∞in an appropriate manner. The method is applied to the simple case of two birth and death processes with birth and death rates λiandμi, i =1,2, interconnected linearly with transition rates v andδ(see Figure 2). For this case some limit theorems are established and the probability of ultimate extinction of both the processes is considered. In addition, for the special cases (i) λ1=δ= 0, with the remaining rates time dependent and (ii) λ2=δ= 0, with the remaining rates constant, explicit solutions forgare obtained and studied.

A local limit theorem for attractions under a stable law

1980 ◽  
Vol 87 (1) ◽  
pp. 179-187 ◽  
Author(s):  
Sujit K. Basu ◽  
Makoto Maejima
Keyword(s):  
Limit Theorem ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Independent Random Variables ◽  
Stable Law ◽  
Absolutely Continuous ◽  
Normal Attraction ◽  
Image Position ◽  
Domain Of Normal Attraction

AbstractLet {Xn} be a sequence of independent random variables each having a common d.f. V1. Suppose that V1 belongs to the domain of normal attraction of a stable d.f. V0 of index α 0 ≤ α ≤ 2. Here we prove that, if the c.f. of X1 is absolutely integrable in rth power for some integer r > 1, then for all large n the d.f. of the normalized sum Zn of X1, X2, …, Xn is absolutely continuous with a p.d.f. vn such thatas n → ∞, where v0 is the p.d.f. of Vo.

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