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  • 1
    Electronic Resource
    Electronic Resource
    Weak ergodicity and products of random matrices (1993)
    Springer
    Journal of theoretical probability 6 (1993), S. 389-405 
    Details
    ISSN: 1572-9230
    Keywords: Random matrix ; stochastic matrix ; weakly ergodic ; space-time harmonic ; stationary ; ergodic theorem ; central limit theorem ; strong mixing ; ρ-mixing
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A representation for a weakly ergodic sequence of (nonstochastic) matrices allows products of nonnegative matrices which eventually become strictly positive to be expressed via products of some associated stochastic matrices and ratios of values of a certain function. This formula used in a random setup leads to a representation for the logarithm of a random matrix product. If the sequence of random matrices is in addition stationary then automatically almost all sequences are weakly ergodic, and the representation is expressed in terms of an one-dimensional stationary process. This permits properties of products of random matrices to be deduced from the latter. Second moment assumptions guarantee that central limit theorems and laws of the iterated logarithm hold for the random matrix products if and only if they hold for the corresponding stationary process. Finally, a central limit theorem for some classes of weakly dependent stationary random matrices is derived doing away with the restriction of boundedness of the ratios of colum entries assumed by previous studies. Extensions beyond stationarity are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01047581
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  • 2
    Electronic Resource
    Electronic Resource
    Sharp Conditions for the CLT of Linear Processes in a Hilbert Space (1997)
    Springer
    Journal of theoretical probability 10 (1997), S. 681-693 
    Details
    ISSN: 1572-9230
    Keywords: Central limit theorem ; linear process in Hilbert space
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we study the behavior of sums of a linear process $$X_k = \sum {_{j = - \infty }^\infty } a_j (\xi _{k - j} )$$ associated to a strictly stationary sequence $$\{ \xi _k \} _{k \in \mathbb{Z}} $$ with values in a real separable Hilbert space and $$\{ a_k \} _{k \in \mathbb{Z}} $$ are linear operators from H to H. One of the results is that $$\sum {_{i = 1}^n } X_i /\sqrt n $$ satisfies the CLT provided $$\{ \xi _k \} _{k \in \mathbb{Z}} $$ are i.i.d. centered having finite second moments and $$\sum {_{j = - \infty }^\infty } \left\| {a_j } \right\|_{L(H)} 〈 \infty $$ . We shall provide an example which shows that the condition on the operators is essentially sharp. Extensions of this result are given for sequences of weak dependent random variables $$\{ \xi _k \} _{k \in \mathbb{Z}} $$ under minimal conditions.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1022653728014
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    Paper (German National Licenses)
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  • 3
    Electronic Resource
    Electronic Resource
    Almost-Sure Results for a Class of Dependent Random Variables (1999)
    Springer
    Journal of theoretical probability 12 (1999), S. 87-104 
    Details
    ISSN: 1572-9230
    Keywords: Random field ; moment inequality ; strong law ; identically distributed random variables ; maximal coefficient of correlation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The aim of this note is to establish almost-sure Marcinkiewicz-Zygmund type results for a class of random variables indexed by ℤ d + —the positive d-dimensional lattice points—and having maximal coefficient of correlation strictly smaller than 1. The class of applications include filters of certain Gaussian sequences and Markov processes.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1021744626773
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  • 4
    Electronic Resource
    Electronic Resource
    On the asymptotic normality of sequences of weak dependent random variables (1996)
    Springer
    Journal of theoretical probability 9 (1996), S. 703-715 
    Details
    ISSN: 1572-9230
    Keywords: Central limit theorem ; mixing sequences of random variables
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The aim of this paper is to investigate the asymptotic normality for strong mixing sequences of random variables in the absense of stationarity or strong mixing rates. An additional condition is imposed to the coefficients of interlaced mixing. The results are applied to linear processes of strongly mixing sequences. The class of applications include filters of certain Gaussian sequences.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02214083
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  • 5
    Electronic Resource
    Electronic Resource
    Self-normalized central limit theorem for sums of weakly dependent random variables (1994)
    Springer
    Journal of theoretical probability 7 (1994), S. 309-338 
    Details
    ISSN: 1572-9230
    Keywords: Central limit theorem ; weakly dependent random variables
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Let {X n,n≥1} be a strictly stationary sequence of weakly dependent random variables satisfyingEX n=μ,EX n 2 〈∞,Var S n /n→σ2 and the central limit theorem. This paper presents two estimators of σ2. Their weak and strong consistence as well as their rate of convergence are obtained for α-mixing, ρ-mixing and associated sequences.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02214272
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    Paper (German National Licenses)
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  • 6
    Electronic Resource
    Electronic Resource
    Convergence rates of the strong law for stationary mixing sequences (1985)
    Springer
    Probability theory and related fields 70 (1985), S. 307-314 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary In this note we estimate the rate of convergence in Marcinkiewicz-Zygmung strong law, for partial sumsS n of strong stationary mixing sequences of random variables. The results improve the corresponding ones obtained by Tze Leung Lai (1977) and Christian Hipp (1979).
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02451434
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  • 7
    Electronic Resource
    Electronic Resource
    An invariance principle for dependent random variables (1981)
    Springer
    Probability theory and related fields 57 (1981), S. 495-507 
    Details
    ISSN: 1432-2064
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01025871
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    Paper (German National Licenses)
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  • 8
    Electronic Resource
    Electronic Resource
    A criterion for tightness for a class of dependent random variables (1982)
    Springer
    Acta mathematica hungarica 39 (1982), S. 311-314 
    Details
    ISSN: 1588-2632
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01896693
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    Paper (German National Licenses)
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