Estimate of the rate of convergence in limit theorems for sums of a random number of independent random vectors

1973 ◽  
Vol 13 (3) ◽  
pp. 385-391 ◽  
Author(s):  
L. Berzhnitskas ◽  
V. Bernotas ◽  
V. I. Paulauskas
Keyword(s):  
Rate Of Convergence ◽  
Limit Theorems ◽  
Random Number ◽  
Random Vectors

Asymptotic approximations for coupon collectors

2009 ◽  
Vol 46 (1) ◽  
pp. 61-96
Author(s):  
Anna Pósfai ◽  
Sándor Csörgő
Keyword(s):  
Distribution Function ◽  
Asymptotic Expansion ◽  
Rate Of Convergence ◽  
Limit Theorems ◽  
Random Number ◽  
Limiting Distribution ◽  
Asymptotic Approximations ◽  
The One ◽  
First Time

A collector samples with replacement a set of n ≧ 2 distinct coupons until he has n − m , 0 ≦ m < n , distinct coupons for the first time. We refine the limit theorems concerning the standardized random number of necessary draws if n → ∞ and m is fixed: we give a one-term asymptotic expansion of the distribution function in question, providing a better approximation of it, than the one given by the limiting distribution function, and proving in particular that the rate of convergence in these limiting theorems is of order (log n )/ n .

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.

scholarly journals Non-central limit theorems for functionals of random fields on hypersurfaces

2020 ◽  
Vol 24 ◽  
pp. 315-340
Author(s):  
Andriy Olenko ◽  
Volodymyr Vaskovych
Keyword(s):  
Rate Of Convergence ◽  
Central Limit ◽  
Random Fields ◽  
Limit Theorems ◽  
Gaussian Random Fields ◽  
Central Limit Theorems ◽  
Integral Functionals ◽  
Asymptotic Results ◽  
Non Linear ◽  
Linear Integral

This paper derives non-central asymptotic results for non-linear integral functionals of homogeneous isotropic Gaussian random fields defined on hypersurfaces in ℝd. We obtain the rate of convergence for these functionals. The results extend recent findings for solid figures. We apply the obtained results to the case of sojourn measures and demonstrate different limit situations.

Limit theorems for the minimal position in a branching random walk with independent logconcave displacements

2000 ◽  
Vol 32 (01) ◽  
pp. 159-176 ◽  
Author(s):  
Markus Bachmann
Keyword(s):  
Random Walk ◽  
Limit Theorems ◽  
Random Number ◽  
Time Lag ◽  
Extreme Value Distribution ◽  
Extreme Value ◽  
Limiting Distribution ◽  
Branching Random Walk ◽  
Minimal Position ◽  
Conditional Limit

Consider a branching random walk in which each particle has a random number (one or more) of offspring particles that are displaced independently of each other according to a logconcave density. Under mild additional assumptions, we obtain the following results: the minimal position in the nth generation, adjusted by its α-quantile, converges weakly to a non-degenerate limiting distribution. There also exists a ‘conditional limit’ of the adjusted minimal position, which has a (Gumbel) extreme value distribution delayed by a random time-lag. Consequently, the unconditional limiting distribution is a mixture of extreme value distributions.

scholarly journals Almost Sure Limit Theorems for Multivariate General Standard Normal Sequences and Applications

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Zhicheng Chen ◽  
Xinsheng Liu
Keyword(s):  
Central Limit ◽  
Limit Theorems ◽  
Almost Sure Convergence ◽  
Central Limit Theorems ◽  
Random Vectors ◽  
Standard Normal ◽  
Almost Sure Limit Theorems ◽  
General Standard

Under suitable conditions, the almost sure central limit theorems for the maximum of general standard normal sequences of random vectors are proved. The simulation of the almost sure convergence for the maximum is firstly performed, which helps to visually understand the theorems by applying to two new examples.

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