Non-Uniform Speed of Convergence to Normality for Some Stationary m-Dependent Processes

1992 ◽  
Vol 42 (3-4) ◽  
pp. 149-162 ◽  
Author(s):  
Ratan Dasgupta
Keyword(s):  
Finite Order ◽  
Generating Function ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Moment Generating Function ◽  
Speed Of Convergence ◽  
Finite Moment ◽  
The Individual ◽  
Dependent Processes ◽  
Sample Sum

Nonuniform rates of convergence to normality are studied for standardised sample sum for some stationary m dependent processes when i) some finite moment of order⩾ 2 exists, ii) when all the finite order moments exist but moment generating function of the individual random variables may not exist. These rates are then used to compute the probabilities of deviations and certain moment type convergences. Possible extensions of the results to non-stationary m-dependent processes are indicated.

Probabilistic Proofs of Euler Identities

2013 ◽  
Vol 50 (04) ◽  
pp. 1206-1212 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Generating Function ◽  
Local Limit Theorem ◽  
Infinite Product ◽  
Random Variables ◽  
Local Limit ◽  
Moment Generating Function ◽  
Basel Problem

Formulae for ζ(2n) andLχ4(2n+ 1) involving Euler and tangent numbers are derived using the hyperbolic secant probability distribution and its moment generating function. In particular, the Basel problem, where ζ(2) = π2/ 6, is considered. Euler's infinite product for the sine is also proved using the distribution of sums of independent hyperbolic secant random variables and a local limit theorem.

Application to the Uniform Laws of Large Numbers for Dependent Processes

2019 ◽  
pp. 438-447
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Random Variables ◽  
Rates Of Convergence ◽  
Partial Sums ◽  
Laws Of Large Numbers ◽  
Large Numbers ◽  
Single Function ◽  
Strongly Mixing ◽  
Mixing Coefficients ◽  
Mixing Sequences ◽  
Dependent Processes

As mentioned in Chapter 5, one of the most powerful techniques to derive limit theorems for partial sums associated with a sequence of random variables which is mixing in some sense is the coupling of the initial sequence by an independent one having the same marginal. In this chapter, we shall see how the coupling results mentioned in Section 5.1.3 are very useful to derive uniform laws of large numbers for mixing sequences. The uniform laws of large numbers extend the classical laws of large numbers from a single function to a collection of such functions. We shall address this question for sequences of random variables that are either absolutely regular, or ϕ‎-mixing, or strongly mixing. In all the obtained results, no condition is imposed on the rates of convergence to zero of the mixing coefficients.

scholarly journals On measure concentration in graph products

2009 ◽  
Vol 50 ◽  
Author(s):  
Matas Šileikis
Keyword(s):  
Generating Function ◽  
Random Variables ◽  
Moment Generating Function ◽  
Graph Products ◽  
Grid Graph ◽  
Connected Graphs ◽  
Measure Concentration ◽  
Product Graph ◽  
The Moment ◽  
The One

Bollobás and Leader [1] showed that among the n-fold products of connected graphs of order k the one with minimal t-boundary is the grid graph. Given any product graph G and a set A of its vertices that contains at least half of V (G), the number of vertices at a distance at least t from A decays (as t grows) at a rate dominated by P(X1 + . . . + Xn  \geq   t) where Xi are some simple i.i.d. random variables. Bollobás and Leader used the moment generating function to get an exponentialbound for this probability. We insert a missing factor in the estimate by using a somewhat more subtle technique (cf. [3]).

scholarly journals Probabilistic Proofs of Euler Identities

2013 ◽  
Vol 50 (4) ◽  
pp. 1206-1212 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Limit Theorem ◽  
Probability Distribution ◽  
Generating Function ◽  
Local Limit Theorem ◽  
Infinite Product ◽  
Random Variables ◽  
Local Limit ◽  
Moment Generating Function ◽  
Basel Problem

Formulae for ζ(2n) andLχ4(2n+ 1) involving Euler and tangent numbers are derived using the hyperbolic secant probability distribution and its moment generating function. In particular, the Basel problem, where ζ(2) = π2/ 6, is considered. Euler's infinite product for the sine is also proved using the distribution of sums of independent hyperbolic secant random variables and a local limit theorem.

Sign in / Sign up

Export Citation Format

Share Document