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.

.... spatio temporal scales [31, 112, 219, 352, 354, 220, 198] Studies of large classes of natural imagery also show characteristic variability at multiple scales [298, 218, 297, 140, 47, 268, 281, 261, 157, 46, 299, 300, 250, 243, 333] as do mathematical models of self similar or fractal processes [288] such as fractional Brownian motion (fBm) 232, 30, 83, 116, 313] motivating examinations of the properties of the wavelet transforms of such signals and images [117, 293, 346, 347, 348, 320, 114, 83, 102, 176, 350, 191, 154, 349, 359, 69, 235, 273] Secondly, whether the phenomenon displays MR ....

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

....of arrivals property. 2. Fractional Brownlan Motion Let us first recall [2] that the definition of autocovariance of a (second order stationary) time series X is fix (k) F, X F,X) X F,X ) Also, we have that the autocorrelation is defined to be Px (k) 0) DEFINITION 1. [6], 1] A stochastic process BH(t) t 0 is said to be a frac tional Brownian motion with Hurst parameter H if 1. BH(t) has stationary increments 2. for t O, BH(t) is normally distributed with mean 0 3. BH(0) a. s. 0o 4. The increments ofBH(t) Z(j) BH(j 1) BH(j) j = O, 1, ....

Samorodnitsky, Gennady and Murad Taqqu (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman ; Hall PETER RABINOVITCH, ALCATEL R;I, 600 MARCH RD., OTTAWA ON K2K 2E6 E-mail address: peter.rabinovitchalcatel. corn

....of sinusoids or cosinusoid is of no particular interest in our case. Then, going to smaller timescales of minutes to several hours, we have the self similar component which we generated via a self similar process with Gaus27 sian increments. Note that we could also have used an stable process [ST94] JW94] KH01] to better match the real behavior but we leave this aspect for further research. We once again divided the sinusoidal seasonality into blocks of size 2048 and each block was added a shift equal to the cumulative sequence of the Gaussian increments. Figure 12 presents the results of ....

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

....RF echoes, rather than the recorded images. 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 [26] while textbooks in the area have been written by Samorodnitsky and Taqqu [27], and by Nikias and Shao [28] A. Basic Properties of the Alpha Stable Family The appeal of symmetric alpha stable ( distributions as a statistical model for signals derives from some important theoretical and empirical reasons. First, stable random variables satisfy the stability property ....

....Cauchy processes result when 1. In fact, no closed form expressions for the general PDF are known except for the Gaussian and the Cauchy members. Although the density behaves approximately like a Gaussian density near the origin, its tails decay at a lower rate than the Gaussian density tails [27]. Indeed, let be a non Gaussian random variable. Then, as (7) where , is the Gamma function, and the statement as means that 1. Hence, the tail probabilities are asymptotically power laws. In other words, while the Gaussian density has exponential tails, the stable densities have algebraic ....

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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.

....sj 2H ) 15) where CH is a constant depending exclusively on H. The increments of fBM have zero mean and are stationary and self similar. Nevertheless, except when H = 0:5, they are not independent, since var(B H (v) Gamma BH (u) CH (v Gamma u) 2H and CH = 0 if and only if H = 0:5. See [11] for further details. If H 0:5, BH exhibits long term dependence, i.e. persistent correlation. It can be verified that such a process is statistically self similar with parameter H; i.e. at each single point x 0 in time, the pointwise Holder exponent is H. These processes are extremely useful ....

G. Samorodnitsky and M. Taqqu. Stable nonGaussian random processes: Stochastic models with infinite variance. Chapman and Hall, 1994.

....Cjj Gammaff as 0 for C a constant and 0 ff 1. One can check that for X a self similar process with 1=2 H 1, the increments of X have LRD or asymptotic self similarity with ff = 2H Gamma 1. However, there are LRD time series that do not arise from self similar processes (see [18] for an FARIMA(0; d; 0) example) 2.2. Fractals and multifractals Loosely speaking, we refer to self similarity as a global property of a process or (even more loosely) of a data set. The self similarity parameter measures how the entire process scales from one time scale to another. We might ....

G. Samorodnitsky and M. Taqqu. Stable non-Gaussian random processes: Stochastic models with infinite variance. Chapman and Hall, 1994.

.... based on other self similar # stable processes with stationary increments, e.g. # stable Levy motion, log fractional stable motion, have tried to address the problem of self similarity and heavy tails at the same time [2] 5] 11] For more information about these non Gaussian processes see [23]. The objective of our research in this paper is twofold: a) We provide a formal definition of the fractional Levy process, show some of its properties, and derive its probability density function following a novel approach, and b) we provide queuing results related to the asymptotic behavior of ....

....parameter of self similarity. The are many different self similar processes in the literature. We typically consider selfsimilar processes with stationary increments, and call them H sssi processes, since they are of great interest in applications. For details on self similar processes see [4] and [23] For example, from the above definition, it is not difficult to check that the Wiener process or (ordinary) Brownian motion (oBm) 4] is a self similar process with H =1=2and since it has stationary increments, it is a 1=2 sssi process. B. Definition of the fractional Levy motion The counterpart ....

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

....the estimated Hurst parameter because it is believed as a measure of burstiness. On the contrary, our example shows that if we are smoothing the traffic the Hurst parameter is increasing. If the process is a pure self similar process there is a good interpretation of the Hurst parameter (see e.g. [24]) but in practice where the traffic structure is modified by several mechanisms (shaping, queueing, multiplexing, etc. the process is not pure selfsimilar. The question is, what is the interpretation of the estimated Hurst parameter These results motivate our future research in that direction. ....

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

....distributions are Gaussian. The second is the assumption that trading may take place in continuous time. However, these assumptions fail to hold in practice, because empirical stock prices are not Gaussian and their marginal distribution usually possesses heavy tails (e.g. Mandelbrot (1997) and Samorodnitski (1994)) Moreover, trades can not be placed continuously in real markets. This discrepancy may results in differences between the Black Scholes price and the true option s price, and appears as the smile effect or implied volatility smile in real markets (see Chapter 17 of the book by Hull (1999) ....

Samorodnitski, G., and M.S. Taqqu. (1994). Stable non-Gaussian random processes: stochastic models with infinite variance, Chapman & Hall, New York.

....in the statistics, signal processing and econometrics community in # stable distributions. There is now a large literature. Nikias and Shao [27] and Kuruoglu [17] give review coverage of signal processing appliucations. Theoretical details are well covered in Feller [10] Samorodnitsky and Taqqu [31] and Zolotarev [39] while some recent applications are given in [1] There is physical justification for the occurence of heavy tailed stable law distributions in fields as diverse as wireless communications, sonar and econometrics (where pioneering work was done by Mandelbrot [26] Exact ....

....this section we summarize the basic results and definitions required for the inference procedures described in subsequent sections. 2.1 Characteristic function A random variable X is drawn from a Stable Law distribution f #,# (a, b) i# its characteristic function is given by [10, p.570, p. 581] [31]: #(#) # exp # a # # # 1 isign(#)# tan # ## 2 ## i b# # , # #= 1) exp ( a # [1 isign(#)# log ( # ) i b#) # = 1) 1) We will use the notation f #,# (a, b) to denote both the distribution and the density function. Where the name of the random variable is ....

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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.

....behavior as # varies, ranging from Gaussian to very heavy tailed. In fact, although the scale parameter # is analogous to a variance, the tails of many of these variables are so heavy that variances fail to exist. A classical example is the # stable family, which has been extensively studied (see [46]) The case # = 2 corresponds to the familiar Gaussian, whereas variables with smaller # 0 have increasingly heavy tails. A well known example with heavy tails is the Cauchy distribution, which corresponds to # = 1. The generalized Gaussian family, also known as the generalized Laplacian ....

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

....two estimations, but actually it tells that the contributions of the different timescales at which variability occurs has changed. The crucial information given by the spectrum is the behavior near the origin, which should tend to 0 in the case of 0 H 0:5 and to 1 in the case of 0:5 H 1 [ST94] TTW95] TT98] What we see here is that heavy tails change the importance of the low frequency components relative to higher frequency ones. Heavy tails make the contribution of all frequencies more even, as illustrated by the slope of the spectrum that tends to 0. This section has shown that ....

G. Samorodnitsky and M. Taqqu. Stable non-Gaussian Random Processes: stochastic models with infinite variance. Chapman & Hall, 1994.

....distributions. Formally, a random variable X is distributed with a tail distribution if P (X u) u ;ff h(u)# where a u b u means a u =b u 1asu 1, ff 0 is the tail index of the distribution, and h(u) is a slowvarying function at infinity# i.e. lim u 1 h(tu) h(u) 1 for any t 0 [see Samorodnitsky and Taqqu (1994)] When 0 ff 2, X is referred to as having a stable distribution with index ff [DuMuouchel (1983) In this article we will refer to those distributions having a tail index between 1 and 3 as fat tailed distributions. If ff =2,X will be Gaussian with moments of all orders. When ff 2, X is a ....

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

....one hand, methods usually used when the second order moments exist cannot be applied but, on the other hand, it it possible to obtain some interesting results about unconditional distributions of R GARCH processes. All relevant properties of stable random variables and processes can be found in [6]. Let us recall only a few facts that are important for the rest of the paper: ffl The characteristic function of a stable random variable Z S ff (oe; fi; is given by E expfiZg = 8 : expf Gammaoe ff jj ff (1 Gamma ifi(sign)tg ff 2 ) ig if ff 6= 1, expf Gammaoejj(1 ifi 2 ....

....ff=2 and h n S ff=2 0 B 2 0 1 X j=1 ffi ff=2 j 1 A 2=ff cos ff 4 2=ff ; 1; 0 1 C A : 3.3) The ffi j s are the coefficients in the series expansion of Theta(z) 1 Gamma Phi(z) jzj 1. P r o o f. This proposition may be proved in the analogous way as Theorem 7. 12.2 in [6] but here the coefficients ffi j are determined by the following system of equations 8 : ffi 1 = 1 ffi 2 Gamma ffi 1 OE 1 = 2 ffi 3 Gamma ffi 2 OE 1 Gamma ffi 1 OE 2 = 3 . ffi r Gamma ffi r Gamma1 OE 1 Gamma ffi r Gamma2 OE 2 Gamma : ....

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

....main theorem below. Before giving the general result, we first need to focus on the class of symmetric stable distributions. 1 Let the distribution of # t be equal to S # (c) where S # denotes a symmetric stable law with index of stability # # (0, 2] and where c 0 is the scale parameter, cf. Samorodnitsky and Taqqu (1994, p. 20) In the following we assume that c = 1. It follows from Definition 1.1.4 of Samorodnitsky and Taqqu (1994, p. 3) and Theorem 1 in Feller (1971, ch. VI.1) that r n t = n 1 X i=0 # t i d = n 1 # # t . Consequently, E r n t # = E 0 n 1 X i=0 # ....

....1 Let the distribution of # t be equal to S # (c) where S # denotes a symmetric stable law with index of stability # # (0, 2] and where c 0 is the scale parameter, cf. Samorodnitsky and Taqqu (1994, p. 20) In the following we assume that c = 1. It follows from Definition 1.1. 4 of Samorodnitsky and Taqqu (1994, p. 3) and Theorem 1 in Feller (1971, ch. VI.1) that r n t = n 1 X i=0 # t i d = n 1 # # t . Consequently, E r n t # = E 0 n 1 X i=0 # t i # 1 A = n # # E # t # , and hence log E r n t # log E ....

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Samorodnitsky, G., and M.S. Taqqu (1994): Stable non-Gaussian random processes: Stochastic models with infinite variance. New York: Chapman & Hall.

....have been used to model phenomena in areas as diverse as economics, statistical physics, and geophysics. More concretely, they have been applied in stock market analysis, Brownian motion, weather forecasts, earthquake prediction, and recently, for modeling time delays on the World Wide Web (e.g. [1, 44, 56]) Various researchers studying the computational nature of search methods on combinatorial problems have informally observed the erratic behavior of the mean and the variance of the search cost. In fact, this phenomenon has led researchers studying the nature of computationally hard problems to ....

....is a substantial literature on heavy tailed distributions. Mandelbrot in [44] provides a good introduction to these distributions with a discussion of their inherently self similar or fractal nature. For a complete treatment of stable distributions see either [66] or the more modern approach of [56]. 3.2. Empirical Results In order to check for the existence of heavy tails in our runtime distributions for backtrack search, we proceed in two steps. First, we graphically analyze the tail behavior of the sample distributions. Second, we formally estimate the index of stability. From (1) we ....

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

....form of a stable distribution is known in analytic form. These cases include the Gaussian distribution, which corresponds to a stable distribution with # = 2. For # 2, this parameter is the maximal moment exponent of the stable law. A modern treatment of stable distributions can be found in Samorodnitsky and Taqqu (1994). Pareto laws A random variable X is said to be of the Pareto type if its probability density function is of the form f(x) #C # x # 1 , for 0 C # x #, 0 #. 3) This is a Pareto distribution of the first of three kinds. Although the estimators we are going to discuss could be ....

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

....the form of a stable distribution is known in analytic form. These cases include the Gaussian distribution, which corresponds to a stable distribution with # = 2. For # 2, this parameter is the maximal moment exponent of the stable law. A modern treatment of stable distributions can be found in Samorodnitsky and Taqqu (1994). Pareto laws We will consider Pareto random variables X with probability density functions of the form f(x) #C # x # 1 , for 0 C#x #,0 #. 3) This is a Pareto distribution of the first of three kinds. Although the estimators we are going to discuss could be applied to Pareto laws without ....

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

.... that a Poisson model is appropriate [72,91] However, recent studies on buffered networks of various kinds argue that the Poisson assumption fails [77, 128] Instead, the traffic has a self similar nature [101,151,156,159,160] This is attributed to high variability and long range dependence [42, 131]. We conjecture that the traffic in wormhole switched networks has similar characteristics. Study if the GSMP view facilitates a parallel implementation [48] of GMSim. We believe that the consistent structure and object oriented view is well suited for a distributed implementation. 32 32 ....

SAMORODNITSKY, G., AND TAQQU, M. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, 1994.

....Example 5.3 (Nonsingularity of discrete spectral measures) A spectral measure of an stable random vector is concentrated on a nite number of points on the unit sphere s i can be expressed as a linear transformation of independent stable real random variables (Proposition 2.3. 7 of [43]) Now, let (k) be d independent stable real random variables and A = a jk ) be a real (s; d) matrix with rank equal to s. Then the random vector with components j = d P k=1 a jk (k) j = 1; s) is stable with (discrete) spectral measure = d X k=1 f 1 2 (1 ....

.... (k) j = 1; s) is stable with (discrete) spectral measure = d X k=1 f 1 2 (1 k ) k s k 1 2 (1 k ) k s k g ; 25 where k 2 IR, k 0 depend on the parameters of (k) and on A, and the vectors s k 2 s are normalized columns of A (see Example 2.3. 6 of [43]) Hence, it holds Z s jhx; ij 2 (d ) d X k=1 k jhx; s k ij 2 ; and is nonsingular since spanfs 1 ; s d g = IR s . Let us nally consider empirical estimates n and n of some unknown pair ( 2 (1; 2) P( s ) of parameters of a stable random vector . If the ....

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

....the stability property, which states that if , and are stable independent random variables of the same distribution, then there exist and satisfying (1) where , and are constants and denotes equality in distribution. Second, it satisfies the generalized central limit theorem [2] 9] [10] stating: is stable if and only if is the limit in distribution of the sum (2) where , are i.i.d. r.v. s and . Parameter is real and is real and positive. GEORGIOU et al. ALPHA STABLE MODELING 293 Fig. 2. The tails of the probability density function of a symmetric ff stable distribution ....

....pens clicking, or objects falling can give rise to the impulsiveness in the noise. The class of stable distributions does not possess finite second (or higher) moments. In fact, stable distributions with have finite moments only for order lower than Gaussian: 8) References [1] 3] and [10] treat the stable theory further. For the purposes of this paper, we will deal with the class of symmetric stable (S S) distributions ( with finite mean, i.e. III. PARAMETER ESTIMATION FOR S SDISTRIBUTIONS The possibility that heavy tailed noise behavior may adequately be described by the ....

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

.... infinitely divisible, they are particularly attractive for continuous time modeling, as emphasized by Samuelson (1965) and McCulloch (1978) The stable generalization of the familiar standard Brownian motion is often called the Levy # stable motion and is the subject of two recent monographs by Samorodnitsky and Taqqu (1994) and Janicki and Weron (1994) The relevance of stable motions for option pricing has been recognized previously. For example, Janicki, Popova, Ritchken, and Woyczynski (1997) and Popova and Ritchken (1998) derive option pricing bounds in an # stable security market. However, in the nondegenerate ....

Samorodnitsky, Gennady, and Murad S. Taqqu, 1994, Stable non-Gaussian random processes : stochastic models with infinite variance. (Chapman & Hall New York).

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

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

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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. 16

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