On a Uniform Estimate of the Rate of Convergence in the Multidimensional Local Limit Theorem for Densities

1971 ◽  
Vol 16 (4) ◽  
pp. 741-743 ◽  
Author(s):  
T. L. Shervashidze
Keyword(s):  
Limit Theorem ◽  
Rate Of Convergence ◽  
Local Limit Theorem ◽  
Uniform Estimate ◽  
Local Limit

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.

A non-uniform rate of convergence in a local limit theorem

1978 ◽  
Vol 84 (2) ◽  
pp. 351-359 ◽  
Author(s):  
Sujit K. Basu
Keyword(s):  
Limit Theorem ◽  
Rate Of Convergence ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Standard Normal ◽  
The Common ◽  
Necessary And Sufficient

AbstractLet {Xn} be a sequence of iid random variables. If the common charac-teristic function is absolutely integrable in mth power for some integer m ≥ 1, then Zn = n−½(X1 + … + Xn) has a pdf fn for all n ≥ m. Here we give a necessary and sufficient condition for sup as n. → ∞, where φ (x) is the standard normal pdf and M(x) is a non-decreasing function of x ≥ 0 such that M(0) > 0 and M(x)/xδ is non-increasing for 0 < δ ≤ 1.

On the rate of convergence in a local limit theorem

1974 ◽  
Vol 76 (1) ◽  
pp. 307-312 ◽  
Author(s):  
Sujit K. Basu
Keyword(s):  
Distribution Function ◽  
Limit Theorem ◽  
Probability Density Function ◽  
Characteristic Function ◽  
Rate Of Convergence ◽  
Inversion Formula ◽  
Local Limit Theorem ◽  
Local Limit ◽  
Common Distribution ◽  
Common Distribution Function

Let Zn = n−½(X1 + X2 + … + Xn), where {Xn} is a sequence of independent and identically distributed random variables with EX1 = 0, and a common distribution function F and characteristic function ω. Suppose |ω|r is integrable for some integer r ≥ 1. For all n ≥ r, then Zn has a probability density function fn obtained by using the inversion formula.

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