Samorodnitsky, G. & Taqqu, M. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with In nite Variance, Chapman & Hall, New York.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Center for Mathematical Sciences, Munich University of Technology, D 80290 Munich, Germany, e mail: fcklu, kuehng mathematik.tu muenchen.de, http: www m4.ma.tum.de m4 researchers, cf. e.g. Hu and ksendal [14] and the references therein. For an introduction to FBM see Samorodnitsky and Taqqu [22]. Certain nancial time series show long memory properties as observed since the 1980s; see Granger [11] resp. Granger and Joyeux [12] and Mandelbrot [18] Such observation has led to an ongoing debate among econometricians and statisticians. It is obvious that any deterministic component like a ....

G. Samorodnitsky and M.S. Taqqu. Stable non-Gaussian Random Processes: Stochastic Models with In nite Variance. Chapman and Hall, New York, 1994.

.... [E (FA; v N (16) As it is well own from probability theory, the N th power of the Laplace transform of a nondin generate distribution function F converges to the nondengenerate limiting transform, N tends to infinity, if and only if F belongs to the domain of attrition of the L vy stable law (see [15,16]) Hence, lira E FA; exp (At) a c (17) N where A is a positive constant and 0 a 1. The range (0, 1] for the index of stability a follows from the nonnegativity of the random riables Ai, i = 1, 2, N. The ce a = 1 corresponds to a degenerate limiting distribution of A = limN = A) rN) ....

G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman & Hall, London, (1994).

....processes, but none of them play the same prominent role as the fBm, which has been used successfully to model a variety of natural phenomena, such as terrains, coast lines, and clouds, and, of course, network tra#c data. Note that, if H = 1 2, the fBm coincides with the classical Brownian motion [38]. Since fractional Brownian motion has stationary increments, its sampled increments, YH (k) BH (k) BH (k 1) k Z, form a stationary sequence. The sequence (k) k#Z is called fractional Gaussian noise (FGN) Note that, as # the autocovariance function of the FGN is #(#) 0 ....

....model is provided by FARIMA with stable innovations. Basically the model is defined as in the second order case (cf. 8) except that the innovation sequence is now an i.i.d. # stable sequence. It can be shown that the existence of the process is then guaranteed provided that d 1 1 # [38]. LRD arises when d 0, which is possible only if # 1. FARIMA models have been used with some success for tra#c modeling, notably for variable bitrate (VBR) video tra#c following [7] see also [16] Simulation and prediction of FARIMA processes is a non trivial task and nothing exact can be ....

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G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, 1994

.... sources are known to exhibit many properties of self similarity such as long range dependence in time, and high degrees of correlation between arrivals [1 3] In this paper, we use self similar traffic generated using the fractional autoregressive integrated moving average (FARIMA) model [4], to evaluate output buffering strategies in switching elements typically used in multimedia server networks. Buffering strategies in switches that selectively allow packets to be discarded when a buffer is full, have been extensively analyzed in the research literature [5] However, buffering ....

....only if there is space available in the next queue in the path of the packet. Each input port of our switching element is fed by N selfsimilar traffic sources, each of which generates packets headed to different outputs. The self similar traffic model used in the traffic sources is FARIMA model [4], a generalization of the popular ARMA model. The FARIMA(0, d, 0) is defined as, X n = j=0 b j ( d)# n j , where b 0 ( d) 1, and, b j ( d) #(j d) #(d)#(j 1) j = 1, 2, If # n is # stable with a distribution S# (1, 0, 0) then d = H 1 #, where H is the Hurst parameter. ....

S. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman and Hall, NY, 1994.

....can be estimated from noisy observations. A review of the state of the art on stable processes from a statistical point of view is provided by a collection of papers edited by Cambanis, Samorodnitsky and Taqqu [20] while textbooks in the area have been written by Samorodnitsky and Taqqu [21], and by Nikias and Shao [22] 2.1 Basic Properties of the Alpha Stable Family The appeal of symmetric alpha stable (S#S) distributions as a statistical model for signals derives from some important theoretical and empirical reasons. First, stable random variables satisfy the stability property ....

....result when # = 1. In fact, no closed form expressions for the general S#S PDF are known except for the Gaussian and the Cauchy members. Although the S#S density behaves approximately like a Gaussian density near the origin, its tails decay at a lower rate than the Gaussian density tails [21]. Indeed, let X be a non Gaussian S#S random variable. Then, as x P (X x) c # x # (2) where c # = #(#) sin ) #, #(x) t x 1 e t dt is the Gamma function, and the statement h(x) g(x) as x # # means that lim x## h(x) g(x) 1. Hence, the tail probabilities are ....

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G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. New York: Chapman and Hall, 1994.

....burstiness as observed in [2, 6] This paper presents a simulation study of the impact of a switching network on the self similar properties of the traffic, and investigates the causes underlying the observed phenomena. We use self similar traffic generated using the fractional ARIMA model [7], and a baseline Banyan topology for the switching network. Section 2 discusses the network and the traffic model in greater detail. Our simulation study reveals that switching networks tend to reduce the self similarity of highly self similar traffic. This is because of the truncation of long ....

....dropped. No packets, however, are dropped at any other point within the switching element, i.e. packets are forwarded to the shared buffer, or to an output buffer only if there is room available. 2.2. Traffic Model We use the fractional autoregressive integrated moving average (FARIMA) model [7] to synthesize self similar traffic. FARIMA(p, d, q) is defined as #(B)X n = #(B)# d # n , where B is the backward operator, i.e. Bx n = x n 1 . The definition above can be also expressed as X i = # d # i # 1 # d # i 1 # q # d # i q # 1 X i 1 # p X i p . 1) In equation ....

G. Samorodnitsky and M. S. Taqqu. Stable NonGaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, NY, 1994.

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Samorodnitsky, G. & Taqqu, M. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with In nite Variance, Chapman & Hall, New York.

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G. Samorodnitsky and M S Taqqu. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall, New York, 1994.

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G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. NewYork: Chapman & Hall, 1994.

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G. Samorodnitsky and M.S. Taqqu. Stable non-Gaussian Random Processes: Stochastic Models with In nite Variance. Chapman and Hall, New York, 1994.

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Samorodnitsky G. and Taqqu M.S. (1994), Stable Non-Gaussian Random Processes: Stochastic Models with Infinte Variance,Chambridge University Press.

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Samorodnitsky G. and Taqqu M.S. (1994), Stable Non-Gaussian Random Processes: Stochastic Models with Infinte Variance,Chambrdige University Press.

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G. Samorodnitsky and M.S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models With Infinite Variance, Chapman and Hal, New York (NY), 1994.

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G. Samorodnitsky and M. S. Taqqu, "Stable Non-Gaussian Random Processes: Stochastic Models with infinite Variance", New York:Chapman & Hall, 1994

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G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random processes: Stochastic Models with Infinite Variance, New York: Chapman and Hall, 1994.

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G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random processes: Stochastic Models with Infinite Variance, New York: Chapman and Hall, 1994.

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G. Samorodnitsky and M. Taqqu. Stable Non-Gaussian Random Processes: Stochastic Models with Innite Variance. Chapman & Hall, 1994. 204

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G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random processes: Stochastic Models with Infinite Variance, New York: Chapman and Hall, 1994.

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G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random processes: Stochastic Models with Infinite Variance, New York: Chapman and Hall, 1994.

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G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, New York: Chapman and Hall, 1994.

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G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random processes: Stochastic Models with In nite Variance, New York: Chapman and Hall, 1994.

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G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models With Infinite Variance. NewYork: Chapman and Hall, 1994.

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Samorodnitsky G. &M.S. Taqqu (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, London.

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Samorodnitsky G. and Taqqu, M.S. : Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance (Chapman & Hall, London 1994).

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G. Samorodnitsky and M. S. Taqqu, Stable Non-Gaussian Random Processes: Stochastic Models with In nite Variance, Chapman & Hall, New York, 1994.

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