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54
Small deviations for fractional stable processes
, 2003
"... Let {Rt, 0 ≤ t ≤ 1} be a symmetric αstable RiemannLiouville process with Hurst parameter H> 0. Consider a translation invariant, βselfsimilar, and ppseudoadditive functional seminorm .. We show that if H> β + 1/p and γ = (H − β − 1/p) −1, then lim ε ε↓0 γ log P [R  ≤ ε] = −K ∈ [−∞, ..."
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Cited by 15 (7 self)
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Let {Rt, 0 ≤ t ≤ 1} be a symmetric αstable RiemannLiouville process with Hurst parameter H> 0. Consider a translation invariant, βselfsimilar, and ppseudoadditive functional seminorm .. We show that if H> β + 1/p and γ = (H − β − 1/p) −1, then lim ε ε↓0 γ log P [R  ≤ ε] = −K ∈ [−∞, 0), with K finite in the Gaussian case α = 2. If α < 2, we prove that K is finite when R is continuous and H> β + 1/p + 1/α. We also show that under the above assumptions, lim ε ε↓0 γ log P [X  ≤ ε] = −K ∈ (−∞, 0), where X is the linear αstable fractional motion with Hurst parameter H ∈ (0, 1) (if α = 2, then X is the classical fractional Brownian motion). These general results cover many cases previously studied in the literature, and also prove the existence of new small deviation constants, both in Gaussian and NonGaussian frameworks.
Weak convergence of positive self similar Markov processes and overshoots of Lévy processes
, 2004
"... We give necessary and sufficient conditions for the law of a positive selfsimilar Markov process to converge weakly as its initial state tends to 0 and we describe the limit law. Our proof is based on Lamperti's representation which relates any positive selfsimilar process to a unique Levy process ..."
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Cited by 14 (2 self)
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We give necessary and sufficient conditions for the law of a positive selfsimilar Markov process to converge weakly as its initial state tends to 0 and we describe the limit law. Our proof is based on Lamperti's representation which relates any positive selfsimilar process to a unique Levy process. Then we show that the convergence mentioned above holds if and only if the process of the overshoots of the underlying Levy process # in the Lamperti's representation converges weakly at infinity and E T1 where T 1 = inf{t : # t 1}. Under these conditions, we give a pathwise construction of the limit law.
2007): Analysis of the Rosenblatt process
 ESAIMPS
"... We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears as limit in the socalled Non Central Limit Theorem (Dobrushin and Major (1979), Taqqu (1979)). This process is nonGaussian and it lives in the second Wiener chaos. We give its representati ..."
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Cited by 13 (8 self)
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We analyze the Rosenblatt process which is a selfsimilar process with stationary increments and which appears as limit in the socalled Non Central Limit Theorem (Dobrushin and Major (1979), Taqqu (1979)). This process is nonGaussian and it lives in the second Wiener chaos. We give its representation as a WienerItô 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.
Variations and estimators for the selfsimilarity order through Malliavin calculus
, 2007
"... Using multiple stochastic integrals, we analyze the asymptotic behavior of quadratic variations for Gaussian and nonGaussian selfsimilar process. We apply our results to the study of statistical estimators for the selfsimilarity index. ..."
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Cited by 11 (3 self)
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Using multiple stochastic integrals, we analyze the asymptotic behavior of quadratic variations for Gaussian and nonGaussian selfsimilar process. We apply our results to the study of statistical estimators for the selfsimilarity index.
A law of iterated logarithm for increasing selfsimilar Markov processes
, 2002
"... We consider increasing selfsimilar Markov processes (X t , t 0) on ]0, #[. ..."
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Cited by 11 (4 self)
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We consider increasing selfsimilar Markov processes (X t , t 0) on ]0, #[.
Random rewards, fractional Brownian local times and stable selfsimilar processes
 Ann. Appl. Probab
"... We describe a new class of selfsimilar symmetric αstable processes with stationary increments arising as a large time scale limit in a situation where many users are earning random rewards or incurring random costs. The resulting models are different from the ones studied earlier both in their mem ..."
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Cited by 7 (1 self)
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We describe a new class of selfsimilar symmetric αstable processes with stationary increments arising as a large time scale limit in a situation where many users are earning random rewards or incurring random costs. The resulting models are different from the ones studied earlier both in their memory properties and smoothness of the sample paths. 1. Introduction. With
Absolute moments of generalized hyperbolic distributions and approximate scaling of normal inverse Gaussian Lévy processes
 SCANDINAVIAN JOURNAL OF STATISTICS
, 2005
"... Expressions for (absolute) moments of generalized hyperbolic and normal inverse Gaussian (NIG) laws are given in terms of moments of the corresponding symmetric laws. For the (absolute) moments centred at the location parameter l explicit expressions as series containing Bessel functions are provid ..."
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Cited by 6 (2 self)
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Expressions for (absolute) moments of generalized hyperbolic and normal inverse Gaussian (NIG) laws are given in terms of moments of the corresponding symmetric laws. For the (absolute) moments centred at the location parameter l explicit expressions as series containing Bessel functions are provided. Furthermore, the derivatives of the logarithms of absolute lcentred moments with respect to the logarithm of time are calculated explicitly for NIG Lévy processes. Computer implementation of the formulae obtained is briefly discussed. Finally, some further insight into the apparent scaling behaviour of NIG Lévy processes is gained.
Fractional Brownian motion in finance and queueing
, 2003
"... e consider briefly the (early) history of the fractional Brownian motion. In sections 2 and 3 we study some of its basic properties and provide some proofs. Regarding the proofs the author claims no originality. Indeed, they are mostly gathered from the existing literature. In sections 4 to 7 we rec ..."
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Cited by 5 (2 self)
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e consider briefly the (early) history of the fractional Brownian motion. In sections 2 and 3 we study some of its basic properties and provide some proofs. Regarding the proofs the author claims no originality. Indeed, they are mostly gathered from the existing literature. In sections 4 to 7 we recall some less elementary facts about the fractional Brownian motion that serve as background to the articles [a], [c] and [d]. The included articles are summarised in Section 8. Finally, Section 9 contains an errata of the articles. Part II consists of the articles themselves: [a] Sottinen, T. (2001) Fractional Brownian motion, random walks and binary market models. Finance Stoch. 5, no. 3, 343355. [b] Kozachenko, Yu., Vasylyk, O. and Sottinen, T. (2002) Path Space Large Deviations of a Large Bu#er with Gaussian Input Tra#c. Queueing Systems 42, no. 2, 113129. [c] Sottinen, T. (2002) On Gaussian processes equivalent in law to fractional Brownian motion. University of Helsinki, Depart
A wavelet analysis of the Rosenblatt process: chaos expansion and estimation of the selfsimilarity parameter
, 2008
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An Experimental Study of New and Known Online Packet Buffering Algorithms
"... We present the first experimental study of online packet buffering algorithms for network switches. The design and analysis of such strategies has received considerable research attention in the theory community recently. We consider a basic scenario in which m queues of size B have to be maintained ..."
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Cited by 4 (0 self)
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We present the first experimental study of online packet buffering algorithms for network switches. The design and analysis of such strategies has received considerable research attention in the theory community recently. We consider a basic scenario in which m queues of size B have to be maintained so as to maximize the packet throughput. A Greedy strategy, which always serves the most populated queue, achieves a competitive ratio of only 2. Therefore, various online algorithms with improved competitive factors were developed in the literature. In this paper we first develop a new online algorithm, called HSFOD, which is especially designed to perform well under realworld conditions. We prove that its competitive ratio is equal to 2. The major part of this paper is devoted to the experimental study in which we have implemented all the proposed algorithms, including HSFOD, and tested them on packet traces from benchmark libraries. We have evaluated the experimentally observed competitivess, the running times, memory requirements and actual packet throughput of the strategies. The tests were performed for varying values of m and B as well as varying switch speeds. The extensive experiments demonstrate that despite a relatively high theoretical competitive ratio, heuristic and greedylike strategies are the methods of choice in a practical environment. In particular, HSFOD has the best experimentally observed competitiveness.