On extremal theory for self-similar processes

Albin, J.M.P.

doi:10.1214/aop/1022855649

Cited by 41 publications

(70 citation statements)

References 27 publications

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“…Albin (1990) studied the exact asymptotics of Eq. 1.7 for a centered stationary generalized chi-process using Berman's approach (see Berman 1992 andAlbin 1998 for self-similar chi-processes), whereas Piterbarg (1994a) obtained a generalization of Albin's result by resorting to the double sum method. In Piterbarg (1994b), the author investigated the exact asymptotics of Eq.…”

Section: Introductionmentioning

confidence: 99%

Piterbarg theorems for chi-processes with trend

Hashorva

2014

Extremes

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Let χ n (t) = ( n i=1 X 2 i (t)) 1/2 , t ≥ 0 be a chi-process with n degrees of freedom where X i 's are independent copies of some generic centered Gaussian process X. This paper derives the exact asymptotic behaviour ofwhere T is a given positive constant, and g(·) is some non-negative bounded measurable function. The case g(t) ≡ 0 has been investigated in numerous contributions by V.I. Piterbarg. Our novel asymptotic results, for both stationary and non-stationary X, are referred to as Piterbarg theorems for chi-processes with trend.

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Section: Introductionmentioning

confidence: 99%

Piterbarg theorems for chi-processes with trend

Hashorva

2014

Extremes

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“…It was introduced by Rosenblatt in [19] and it has been called in this way by Taqqu in [20]. The Rosenblatt process has also practical applications and different aspects of this process like wavelet type expansion or extremal properties have been studied in [1], [2], [15] and [16].…”

Section: Introductionmentioning

confidence: 99%

“…This is an extension of the fractional Ornstein-Uhlenbeck process in [6]. Of central interest is for us a Non-Central Limit Theorem that has as limit the Wiener integral with respect to the Hermite process (Z k H (t)) t≥0 in (2). Recall that it has been proved in [17] that, if f is a deterministic function in a suitable space, then the sequence…”

Section: Introductionmentioning

confidence: 99%

Wiener Integrals with Respect to the Hermite Process and a Non-Central Limit Theorem

Maejima

Tudor

2007

Stochastic Analysis and Applications

101

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“…For example, extremal properties of the Rosenblatt distribution have been studied by J.M. Albin in [2] and [3]. The rate of convergence to the Rosenblatt process in the Non Central Limit Theorem has been given by Leonenko and Ahn [23].…”

Section: Introductionmentioning

confidence: 99%

Analysis of the Rosenblatt process

2008

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Abstract. We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears as limit in the so-called Non Central Limit Theorem (Dobrushin and Majòr (1979), Taqqu (1979)). This process is non-Gaussian and it lives in the second Wiener chaos. We give its representation as a Wiener-Itô multiple integral with respect to the Brownian motion on a finite interval and we develop a stochastic calculus with respect to it by using both pathwise type calculus and Malliavin calculus.Mathematics Subject Classification. 60G12, 60G15, 60H05, 60H07.

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On extremal theory for self-similar processes

Cited by 41 publications

References 27 publications

Piterbarg theorems for chi-processes with trend

Piterbarg theorems for chi-processes with trend

Wiener Integrals with Respect to the Hermite Process and a Non-Central Limit Theorem

Analysis of the Rosenblatt process

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