A note on Rosenblatt distributions

Albin, J.M.P.

doi:10.1016/s0167-7152(98)00109-6

Cited by 33 publications

(48 citation statements)

References 9 publications

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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%

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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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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“…Let f 1 (x) = [0,1] (x), f 2 (x) = [0,2/3] (x), and f 3 (x) = [1/3,1] (x). Since rk M φ ≤ 2, there exist numbers α 1 , α 2 , and α 3 , not all being zero, such that α 1 …”

Section: Georgiȋ Shevchenkomentioning

confidence: 99%

“…The most studied among these processes is the Hermite process of rank 2 defined in the paper by Rosenblatt [8] (the latter is also known as the Rosenblatt process). Among publications devoted to the Rosenblatt process, we mention papers by Pipiras [7], where a wavelet expansion is constructed for this process; Tudor [13], where stochastic analysis with respect to the Rosenblatt process is developed; Albin [1], where the distribution of the maximum of this process is found; and Tudor and Torres [12], where an application of the Rosenblatt process in finance mathematics is considered (namely, the Rosenblatt process is considered in [12] as a model of price evolutions).…”

Section: Introductionmentioning

confidence: 99%

Properties of trajectories of a multifractional Rosenblatt process

Shevchenko

2011

Theor. Probability and Math. Statist.

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Abstract. A Rosenblatt process and its multifractional counterpart are considered. For a multifractional Rosenblatt process, we investigate the local properties of its trajectories, namely the continuity and localizability. We prove the existence of square integrable local times for both processes. IntroductionStochastic processes with long memory (in other words, with long range dependence) remain an extensively developing topic over more than a half of a century because of their numerous applications in modelling various natural phenomena, transferring information in computer nets, dynamics of prices of financial assets, etc.The long range dependence phenomena is modelled most often with the help of a fractional Brownian motion. In the monograph by Mishura [5], a detailed survey of the literature devoted to the fractional Brownian motion as well as main results of this topic are given. A disadvantage of a fractional Brownian motion that restricts an area of its possible applications is that this process has light tails of the normal distribution. There is a number of studies devoted to processes with long range dependence that have heavier tails. These are, in particular, stable processes (see the book by Samorodnitsky and Taqqu [9]).An interesting class of processes with long range dependence that have "moderate" tails ("moderate" means that the tails are heavier than those of the normal distribution but lighter than those of a stable distribution), known as Hermite processes, appears in the so-called noncentral limit theorem for strongly dependent random variables proved in the papers by Dobrushin and Major [3] and Taqqu [10,11]. The most studied among these processes is the Hermite process of rank 2 defined in the paper by Rosenblatt [8] (the latter is also known as the Rosenblatt process). Among publications devoted to the Rosenblatt process, we mention papers by Pipiras [7], where a wavelet expansion is constructed for this process; Tudor [13], where stochastic analysis with respect to the Rosenblatt process is developed; Albin [1], where the distribution of the maximum of this process is found; and Tudor and Torres [12], where an application of the Rosenblatt process in finance mathematics is considered (namely, the Rosenblatt process is considered in [12] as a model of price evolutions).2010 Mathematics Subject Classification. Primary 60G22; Secondary 60J55, 60B10.

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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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A note on Rosenblatt distributions

Cited by 33 publications

References 9 publications

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

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

Properties of trajectories of a multifractional Rosenblatt process

Analysis of the Rosenblatt process

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