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Clear All  # A note on asymptotics of linear combinations of iid random variables

Periodica Mathematica Hungarica
Author: Péter Kevei

## Abstract

Let X1,X2, ... be iid random variables, and let an = (a1,n, ..., an,n) be an arbitrary sequence of weights. We investigate the asymptotic distribution of the linear combination

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$S_{a_n }$$ \end{document}
= a1,nX1 + ... + an,nXn under the natural negligibility condition limn→∞ max{|ak,n|: k = 1, ..., n} = 0. We prove that if
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$S_{a_n }$$ \end{document}
is asymptotically normal for a weight sequence an, in which the components are of the same magnitude, then the common distribution belongs to
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{D}$$ \end{document}
(2).

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# On asymptotics of large Haar distributed unitary matrices

Periodica Mathematica Hungarica
Authors: Dénes Petz and Júlia Réffy

## Abstract

Let U n be an n n Haar unitary matrix. In this paper, the asymptotic normality and independence of Tr U n, Tr U n 2 ,..., Tr U n k are shown by using elementary methods. More generally, it is shown that the renormalized truncated Haar unitaries converge to a Gaussian random matrix in distribution.

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# A strong approximation for logarithmic averages of partial sums of random variables

Periodica Mathematica Hungarica
Author: G. Hurelbaatar

LetS n be the partial sums of ?-mixing stationary random variables and letf(x) be a real function. In this note we give sufficient conditions under which the logarithmic average off(S n/sn) converges almost surely to ?-8 8 f(x)dF(x). We also obtain strong approximation forH(n)=?k=1 n k -1 f(S k/sk)=logn ?-8 8 f(x)dF(x) which will imply the asymptotic normality ofH(n)/log1/2 n. But for partial sums of i.i.d. random variables our results will be proved under weaker moment condition than assumed for ?-mixing random variables.

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# An almost sure limit theorem for mixtures of domains in random allocation

Studia Scientiarum Mathematicarum Hungarica
Author: Peter Becker-Kern

The problem of random allocation is that of placing n balls independently with equal probability to N boxes. For several domains of increasing numbers of balls and boxes, the final number of empty boxes is known to be asymptotically either normally or Poissonian distributed. In this paper we first derive a certain two-index transfer theorem for mixtures of the domains by considering random numbers of balls and boxes. As a consequence of a well known invariance principle this enables us to prove a corresponding general almost sure limit theorem. Both theorems inherit a mixture of normal and Poisson distributions in the limit. Applications of the general almost sure limit theorem for logarithmic weights complement and extend results of Fazekas and Chuprunov  and show that asymptotic normality dominates.

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# Limsup theorems and a uniform LIL for asymptotically negatively associated random sequences

Studia Scientiarum Mathematicarum Hungarica
Authors: Yong-Kab Choi and Kyo-Shin Hwang

23 127 136 Yang, Y. and Wang, Y. B. , Asymptotical normality of the renewal process generated by strictly stationary LPQD sequences

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# Modelling the asymmetric volatility of anti-pollution patents in the USA

Scientometrics
Authors: Felix Chan, Dora Marinova, and Michael McAleer

Asymptotic normality for the quasi-maximum likelihood estimator of a GARCH model Comptes Rendus de l'Académie des Sciences 331 81 – 84

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# A survey of product posterior distributions

Studia Scientiarum Mathematicarum Hungarica

, C. Y. , Asymptotic normality of poly-t densities with Bayesian applications , Communications in Statistics –Theory and Methods , 17 ( 1988 ), 1613 – 1627 . 173

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# Assessing the value of patent portfolios: an international country comparison

Scientometrics
Author: Torben Schubert

individual observations (see Davison and Hinkley 1997 , p. 71). In any case, because asymptotic normality still holds, we can rely on a normal approximation adjusted with a bootstrapped variance. Step 3: Recovering the unconditional

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