Nonuniform estimates of convergence rate in local theorems for the maximum of sums of independent random variables

1977 ◽  
Vol 17 (3) ◽  
pp. 305-312
Author(s):  
V. V. Naudžiūnienė
Keyword(s):  
Convergence Rate ◽  
Random Variables ◽  
Independent Random Variables ◽  
Estimates Of Convergence Rate

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.

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 Exponential inequalities under sub-linear expectations with applications to strong law of large numbers

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 2951-2961
Author(s):  
Xufei Tang ◽  
Xuejun Wang ◽  
Yi Wu
Keyword(s):  
Convergence Rate ◽  
Strong Convergence ◽  
Law Of Large Numbers ◽  
Random Variables ◽  
Independent Random Variables ◽  
Exponential Inequalities ◽  
Strong Law ◽  
Large Numbers ◽  
Strong Convergence Rate

In this paper, we give some exponential inequalities for extended independent random variables under sub-linear expectations. As an application, we obtain the strong convergence rate O(n-1/2 ln1/2 n) for the strong law of large numbers under sub-linear expectations, which generalizes some corresponding ones under the classical linear expectations.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

Methodology of Calculation the Terminal Settling Velocity Distribution of Irregular Particles for High Values of the Reynold’s Number/Metodologia Wyliczania Rozkładu Granicznej Prędkości Opadania Ziaren Nieregularnych Dla Wysokich Wartości Liczb Reynoldsa

2014 ◽  
Vol 59 (2) ◽  
pp. 553-562 ◽  
Author(s):  
Agnieszka Surowiak ◽  
Marian Brożek
Keyword(s):  
Velocity Distribution ◽  
Settling Velocity ◽  
Random Variables ◽  
Independent Random Variables ◽  
Calculation Algorithm ◽  
Shape Coefficient ◽  
Geometrical Properties ◽  
Size And Shape ◽  
Physical Density ◽  
Settling Velocity Distribution

Abstract Settling velocity of particles, which is the main parameter of jig separation, is affected by physical (density) and the geometrical properties (size and shape) of particles. The authors worked out a calculation algorithm of particles settling velocity distribution for irregular particles assuming that the density of particles, their size and shape constitute independent random variables of fixed distributions. Applying theorems of probability, concerning distributions function of random variables, the authors present general formula of probability density function of settling velocity irregular particles for the turbulent motion. The distributions of settling velocity of irregular particles were calculated utilizing industrial sample. The measurements were executed and the histograms of distributions of volume and dynamic shape coefficient, were drawn. The separation accuracy was measured by the change of process imperfection of irregular particles in relation to spherical ones, resulting from the distribution of particles settling velocity.

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