Rates of Convergence for Lévy’s Modulus of Continuity and Hinchin’s Law of the Iterated Logarithm

Statistical approximation properties of modified Durrmeyer q-Baskakov type operators with two parameters α and β

Asian-European Journal of Mathematics ◽

10.1142/s1793557117500280 ◽

2016 ◽

Vol 10 (02) ◽

pp. 1750028

Author(s):

Vishnu Narayan Mishra ◽

Preeti Sharma

Keyword(s):

Modulus Of Continuity ◽

Maximal Function ◽

Statistical Convergence ◽

Approximation Theorem ◽

Rates Of Convergence ◽

Approximation Properties ◽

Statistical Approximation ◽

Baskakov Operators ◽

Two Parameters ◽

Lipschitz Type Maximal Function

The main aim of this study is to obtain statistical approximation properties of these operators with the help of the Korovkin type statistical approximation theorem. Rates of statistical convergence by means of the modulus of continuity and the Lipschitz type maximal function are also established. Our results show that rates of convergence of our operators are at least as fast as classical Durrmeyer type modified Baskakov operators.

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An iterated logarithm result for martingales and its application in estimation theory for autoregressive processes

Journal of Applied Probability ◽

10.2307/3212502 ◽

1973 ◽

Vol 10 (1) ◽

pp. 146-157 ◽

Cited By ~ 13

Author(s):

C. C. Heyde

Keyword(s):

Autoregressive Process ◽

Rates Of Convergence ◽

Estimation Theory ◽

Autoregressive Processes ◽

Sample Mean ◽

Iterated Logarithm ◽

Logarithm Law ◽

Iterated Logarithm Law

The paper begins with an iterated logarithm law of classical Hartman-Wintner form for stationary martingales. This is then used to obtain iterated logarithm results giving information on rates of convergence of estimators of the parameters in a stationary autoregressive process. In the case of an autoregression of small order, detailed rate results for each autocorrelation and for the estimators of all parameters can be obtained. A rate result for the convergence of the sample mean is given in the general case.

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An iterated logarithm result for martingales and its application in estimation theory for autoregressive processes

Journal of Applied Probability ◽

10.1017/s0021900200042157 ◽

1973 ◽

Vol 10 (01) ◽

pp. 146-157 ◽

Cited By ~ 1

Author(s):

C. C. Heyde

Keyword(s):

Autoregressive Process ◽

Rates Of Convergence ◽

Estimation Theory ◽

Autoregressive Processes ◽

Sample Mean ◽

Iterated Logarithm ◽

Logarithm Law ◽

Iterated Logarithm Law

The paper begins with an iterated logarithm law of classical Hartman-Wintner form for stationary martingales. This is then used to obtain iterated logarithm results giving information on rates of convergence of estimators of the parameters in a stationary autoregressive process. In the case of an autoregression of small order, detailed rate results for each autocorrelation and for the estimators of all parameters can be obtained. A rate result for the convergence of the sample mean is given in the general case.

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On the rates of convergence of the q-Lupaş-Stancu operators

Filomat ◽

10.2298/fil1605151d ◽

2016 ◽

Vol 30 (5) ◽

pp. 1151-1160

Author(s):

Ogün Doğru ◽

Gürhan İçoz ◽

Kadir Kanat

Keyword(s):

Rate Of Convergence ◽

Modulus Of Continuity ◽

Rates Of Convergence ◽

Lipschitz Class ◽

Approximation Properties ◽

Statistical Approximation ◽

Stancu Operators

We introduce a Stancu type generalization of the Lupa? operators based on the q-integers, rate of convergence of this modification are obtained by means of the modulus of continuity, Lipschitz class functions and Peetre?s K-functional. We will also introduce r-th order generalization of these operators and obtain its statistical approximation properties.

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Obtaining Prescribed Rates of Convergence for the Ergodic Theorem

Canadian Journal of Mathematics ◽

10.4153/cjm-1983-062-3 ◽

1983 ◽

Vol 35 (6) ◽

pp. 1129-1146 ◽

Cited By ~ 1

Author(s):

G. L. O'Brien

Keyword(s):

Ergodic Theorem ◽

Law Of Large Numbers ◽

Random Variables ◽

Rates Of Convergence ◽

Stationary Sequence ◽

Strong Law ◽

Iterated Logarithm ◽

Large Numbers ◽

The Law ◽

Image Position

Let {Yn, n ∊ Z} be an ergodic strictly stationary sequence of random variables with mean zero, where Z denotes the set of integers. For n ∊ N = {1, 2, …}, let Sn = Y1 + Y2 + … + Yn. The ergodic theorem, alias the strong law of large numbers, says that n–lSn → 0 as n → ∞ a.s. If the Yn's are independent and have variance one, the law of the iterated logarithm tells us that this convergence takes place at the rate in the sense that1It is our purpose here to investigate what other rates of convergence are possible for the ergodic theorem, that is to say, what sequences {bn, n ≧ 1} have the property that2for some ergodic stationary sequence {Yn, n ∊ Z}.

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