Nonuniform estimate of maximum density convergence rate for independent random variables

1999 ◽  
Vol 4 ◽  
pp. 3-9
Author(s):  
A. Aksomaitis ◽  
A. Jokimaitis
Keyword(s):  
Limit Theorem ◽  
Convergence Rate ◽  
Random Variables ◽  
Maximum Density ◽  
Independent Random Variables ◽  
Nonuniform Estimate ◽  
Density Limit ◽  
Estimate Of Convergence Rate

The nonuniform estimate of convergence rate in the maximum density limit theorem of independent nonidentically distributed random variables is obtained. This result is generalization of the work presented in [1].

scholarly journals An improvement of convergence rate in the local limit theorem for integral-valued random variables

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Tatpon Siripraparat ◽  
Kritsana Neammanee
Keyword(s):  
Limit Theorem ◽  
Convergence Rate ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Independent Random Variables ◽  
Moment Condition ◽  
Formula One ◽  
Explicit Constant ◽  
Independent Integral

AbstractLet $X_{1}, X_{2}, \ldots , X_{n}$ X 1 , X 2 , … , X n be independent integral-valued random variables, and let $S_{n}=\sum_{j=1}^{n}X_{j}$ S n = ∑ j = 1 n X j . One of the interesting probabilities is the probability at a particular point, i.e., the density of $S_{n}$ S n . The theorem that gives the estimation of this probability is called the local limit theorem. This theorem can be useful in finance, biology, etc. Petrov (Sums of Independent Random Variables, 1975) gave the rate $O (\frac{1}{n} )$ O ( 1 n ) of the local limit theorem with finite third moment condition. Most of the bounds of convergence are usually defined with the symbol O. Giuliano Antonini and Weber (Bernoulli 23(4B):3268–3310, 2017) were the first who gave the explicit constant C of error bound $\frac{C}{\sqrt{n}}$ C n . In this paper, we improve the convergence rate and constants of error bounds in local limit theorem for $S_{n}$ S n . Our constants are less complicated than before, and thus easy to use.

scholarly journals Estimation of Convergence Rate in the Transfer Theorem for Maxima

2008 ◽  
Vol 13 (1) ◽  
pp. 3-7
Author(s):  
A. Aksomaitis
Keyword(s):  
Convergence Rate ◽  
Random Variables ◽  
Nonuniform Estimate ◽  
Transfer Theorem ◽  
Independent Identically Distributed ◽  
Estimate Of Convergence Rate

Let ZN be a maximum of independent identically distributed random variables. In this paper, a nonuniform estimate of convergence rate in the transfer theorem max-scheme is obtained. Presented results make the estimates, given in [1] and [2], more precise.

Asymptotics of the Joint Distribution of Multivariate Extrema

2001 ◽  
Vol 6 (1) ◽  
pp. 3-8
Author(s):  
A. Aksomaitis ◽  
A. Jokimaitis
Keyword(s):  
Distribution Function ◽  
Convergence Rate ◽  
Joint Distribution ◽  
Random Variables ◽  
Nonuniform Estimate ◽  
Independent Identically Distributed ◽  
Estimate Of Convergence Rate

Let Wn and Zn be a bivariate extrema of independent identically distributed bivariate random variables with a distribution function F. in this paper the nonuniform estimate of convergence rate of the joint distribution of the normalized and centralized minima and maxima is obtained.

scholarly journals On Feller's criterion for the law of the iterated logarithm

1994 ◽  
Vol 17 (2) ◽  
pp. 323-340 ◽  
Author(s):  
Deli Li ◽  
M. Bhaskara Rao ◽  
Xiangchen Wang
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Uniform Estimate ◽  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
The Central Limit Theorem ◽  
Iterated Logarithm ◽  
The Law

Combining Feller's criterion with a non-uniform estimate result in the context of the Central Limit Theorem for partial sums of independent random variables, we obtain several results on the Law of the Iterated Logarithm. Two of these results refine corresponding results of Wittmann (1985) and Egorov (1971). In addition, these results are compared with the corresponding results of Teicher (1974), Tomkins (1983) and Tomkins (1990)

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