scholarly journals A central limit theorem for numbers satisfying a class of triangular arrays associated with Hermite polynomials

2021 ◽  
Vol 61 ◽  
pp. 1-7
Author(s):  
Igoris Belovas

The paper extends the investigations of limit theorems for numbers satisfying a class of triangular arrays. We obtain analytical expressions for the semiexponential generating function the numbers, associated with Hermite polynomials. We apply the results to prove the asymptotic normality of the numbers and specify the convergence rate to the limiting distribution.    

2021 ◽  
Vol 56 (2) ◽  
pp. 195-223
Author(s):  
Igoris Belovas ◽  

The paper extends the investigations of limit theorems for numbers satisfying a class of triangular arrays, defined by a bivariate linear recurrence with bivariate linear coefficients. We obtain the partial differential equation and special analytical expressions for the numbers using a semi-exponential generating function. We apply the results to prove the asymptotic normality of special classes of the numbers and specify the convergence rate to the limiting distribution. We demonstrate that the limiting distribution is not always Gaussian.


1976 ◽  
Vol 13 (1) ◽  
pp. 148-154 ◽  
Author(s):  
D. L. McLeish

A functional central limit theorem extending the central limit theorem of Chung (1954) for the Robbins–Munro procedure is proved. It is shown that the asymptotic normality is preserved under certain random stopping rules.


1976 ◽  
Vol 13 (01) ◽  
pp. 148-154
Author(s):  
D. L. McLeish

A functional central limit theorem extending the central limit theorem of Chung (1954) for the Robbins–Munro procedure is proved. It is shown that the asymptotic normality is preserved under certain random stopping rules.


2019 ◽  
Vol 178 (1) ◽  
pp. 1-37 ◽  
Author(s):  
Olli Hella ◽  
Mikko Stenlund

AbstractWe study random compositions of transformations having certain uniform fiberwise properties and prove bounds which in combination with other results yield a quenched central limit theorem equipped with a convergence rate, also in the multivariate case, assuming fiberwise centering. For the most part we work with non-stationary randomness and non-invariant, non-product measures. Independently, we believe our work sheds light on the mechanisms that make quenched central limit theorems work, by dissecting the problem into three separate parts.


1992 ◽  
Vol 24 (2) ◽  
pp. 267-287 ◽  
Author(s):  
Allen L. Roginsky

Three different definitions of the renewal processes are considered. For each of them, a central limit theorem with a remainder term is proved. The random variables that form the renewal processes are independent but not necessarily identically distributed and do not have to be positive. The results obtained in this paper improve and extend the central limit theorems obtained by Ahmad (1981) and Niculescu and Omey (1985).


1978 ◽  
Vol 10 (04) ◽  
pp. 852-866
Author(s):  
A. J. Stam

Let be a family of random walks with For ε↓0 under certain conditions the random walk U (∊) n converges to an oscillating random walk. The ladder point distributions and expectations converge correspondingly. Let M ∊ = max {U (∊) n , n ≧ 0}, v 0 = min {n : U (∊) n = M ∊}, v 1 = max {n : U (∊) n = M ∊}. The joint limiting distribution of ∊2σ∊ –2 v 0 and ∊σ∊ –2 M ∊ is determined. It is the same as for ∊2σ∊ –2 v 1 and ∊σ–2 ∊ M ∊. The marginal ∊σ–2 ∊ M ∊ gives Kingman's heavy traffic theorem. Also lim ∊–1 P(M ∊ = 0) and lim ∊–1 P(M ∊ < x) are determined. Proofs are by direct comparison of corresponding probabilities for U (∊) n and for a special family of random walks related to MI/M/1 queues, using the central limit theorem.


2011 ◽  
Vol 48 (04) ◽  
pp. 1189-1196 ◽  
Author(s):  
Qunqiang Feng ◽  
Zhishui Hu

We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments ofZn, the Zagreb index of a random recursive tree of sizen, are obtained. We also show that the random process {Zn− E[Zn],n≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by an application of the martingale central limit theorem. Finally, two other topological indices are also discussed in passing.


Sign in / Sign up

Export Citation Format

Share Document