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1

Eigenvalue distribution of large sample covariance matrices of linear processes

Pfaffel O., Schlemm E.

Probability and Mathematical Statistics

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2011

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Vol. 31, Fasc. 2

313--329

EN

We derive the distribution of the eigenvalues of a large sample covariance matrix when the data is dependent in time. More precisely, the dependence for each variable i = 1, . . . , p is modelled as a linear process (Xi;t)t=1...,n = (Σ∞j=0 cjZi;t−j )t=1,...,n, where {Zi;t} are assumed to be independent random variables with finite fourth moments. If the sample size n and the number of variables p = pn both converge to infinity such that y = limn→∞ n/pn > 0, then the empirical spectral distribution of p−1XXT converges to a non-random distribution which only depends on y and the spectral density of (X1;t)t∈Z. In particular, our results apply to (fractionally integrated) ARMA processes, which will be illustrated by some examples.

2

A limit theorem for sums of bounded functionals of linear processes without finite mean

Honda T.

Probability and Mathematical Statistics

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2009

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Vol. 29, Fasc. 2

337--351

EN

We consider the partial sum process of a bounded functional of a linear process and the linear process has no finite mean. We assume the innovations of the linear process are independent and identically distributed and that the distribution of the innovations belongs to the domain of attraction of an α-stable law and satisfies some additional assumptions. Then we establish the finite-dimensional convergence in distribution of the partial sum process to a stable Lévy motion.

3

Bounds for E׀Sn׀Q for subordinated linear processes with application to M-estimation

Furmańczyk K.

Probability and Mathematical Statistics

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2008

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Vol. 28, Fasc. 1

129--141

EN

Let Xj =Σ∞r=0 ArZj−r be a one-sided m-dimensional linear process, where (Zn) is a sequence of i.i.d. random vectors with zero mean and finite covariance matrix. The aim of this paper is to prove the moment inequalities of the form [formula] where G is a real function defined on Rm: The form of the constant C in (0.1) plays an important role in applications concerning the problems of M-estimation, especially the Ghosh representation.

4

Some remarks on the central limit theorem for functionals of linear processes under short-range dependence

Furmańczyk K.

Probability and Mathematical Statistics

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2007

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Vol. 27, Fasc. 2

235--245

EN

In this paper we consider the central limit theorems for functionals G: Rm-> R of one-sided m-dimensional linear processes Xt=∑∞r=0 where Ar is a nonrandom matrix mxm and Zt’s are i.i.d. random vectors in Rm.

5

Ekonomiczne aspekty zrównania kosztów z zyskiem we współczesnym zakładzie odlewniczym

Binczyk F., Szymszal J., Piątkowski J.

Archiwum Odlewnictwa

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2003

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R. 3, nr 8

15--22

PL

Tematem pracy jest wyznaczenie tzw. punktu break-even, który określa zrównanie kosztów z zyskiem, czyli które wyroby i w jakich ilościach należy wytworzyć, aby nie przekroczyć posiadanych zasobów środków produkcji oraz spełnić ewentualne ograniczenia, dotyczące struktury możliwości wykonawczych, przy uzyskaniu maksymalnego zysku z ich sprzedaży.

EN

The contents of this work are determination so-called balance point between costs. The break-even point, it is means which articles and in which quantities is necessary to produce, so that to the crossed possessed supplies of resources of production as well as to fulfil possible limitations, relate structure of executive possibility, at obtaining of maximum profit from them of sale.

6

Note on asymptotic normality of kernel density estimator for linear process under short-range dependence

Furmańczyk K.

Probability and Mathematical Statistics

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2001

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Vol. 21, Fasc. 2

253--263

EN

We consider the problem of density estimation for a one-sided linear process [formula] with i.i.d. square integrable innovations [formula]. We prove that under weak conditions on [formula], which imply short-range dependence of the linear process, finite-dimensional distributions of kernel density estimate area symptotically multivariate normal. In particular, the result holds for |an|=θ(n−a) with a >2, which is much weaker than previously known sufficient conditions for asymptotic normality. No conditions on bandwidths bn are assumed besides bn→0 and nbn→ ∞.The proof uses Chanda’s [1], [2] conditioning technique as well as Bernstein’s “large block-small block” argument.