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35
Laws of the iterated logarithm for αtime Brownian motion
, 2008
"... We introduce a class of iterated processes called αtime Brownian motion for 0 < α ≤ 2. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric αstable process. We prove a Chungtype law of the iterated logarithm (LIL) for these processes which is a general ..."
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Cited by 5 (2 self)
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We introduce a class of iterated processes called αtime Brownian motion for 0 < α ≤ 2. These are obtained by taking Brownian motion and replacing the time parameter with a symmetric αstable process. We prove a Chungtype law of the iterated logarithm (LIL) for these processes which is a generalization of LIL proved in [14] for iterated Brownian motion. When α = 1 it takes the following form liminf T → ∞ T −1/2 (log log T) sup Zt  = π 0≤t≤T 2 √ λ1 a.s. where λ1 is the first eigenvalue for the Cauchy process in the interval [−1, 1]. We also define the local time L ∗ (x, t) and range R ∗ (t) = {x: Z(s) = x for some s ≤ t}  for these processes for 1 < α < 2. We prove that there are universal constants cR, cL ∈ (0, ∞) such that limsup t→∞ R ∗ (t) (t / log log t) 1/2α = cR a.s. log log t liminf t→∞ supx∈R L ∗ (x, t) = cL a.s. (t / log log t)
Multifractality in foreign currency markets
 Multinational Finance Journal
, 2002
"... The standard hypothesis concerning the behavior of asset returns states that they follow a random walk in discrete time or a Brownian motion in continuous time. The Brownian motion process is characterized by a quantity, called the Hurst exponent, which is related to some fractal aspects of the proc ..."
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Cited by 4 (1 self)
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The standard hypothesis concerning the behavior of asset returns states that they follow a random walk in discrete time or a Brownian motion in continuous time. The Brownian motion process is characterized by a quantity, called the Hurst exponent, which is related to some fractal aspects of the process itself. For a standard Brownian motion (sBm) this exponent is equal to 0.5. Several empirical studies have shown the inadequacy of the sBm. To correct for this evidence some authors have conjectured that asset returns may be independently and identically ParetoLévy stable (PLs) distributed, whereas others have asserted that asset returns may be identically but not independently fractional Brownian motion (fBm) distributed with Hurst exponents, in both cases, that differ from 0.5. In this paper we empirically explore such nonstandard assumptions for both spot and (nearby) futures returns for five foreign currencies: the British Pound, the Canadian Dollar, the German Mark, the Swiss Franc, and the Japanese Yen. We assume that the Hurst exponent belongs to a suitable neighborhood of 0.5 that allows us to verify if the socalled Fractal Market Hypothesis (FMH) can be a “reasonable ” generalization of the Efficient Market hypothesis. Furthermore, we also allow the Hurst exponent to vary over time which permits the
Limit Theorems and Estimation for Structural and Aggregate Teletraffic Models
, 2003
"... The thesis proposes models for aggregate data network traffic which incorporate the additional randomness arising from the randomness in the number of data sources. A conditionallyGaussian scale mixture process is shown to be a limit for the cumulative work from a random superposition of alternatin ..."
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Cited by 4 (1 self)
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The thesis proposes models for aggregate data network traffic which incorporate the additional randomness arising from the randomness in the number of data sources. A conditionallyGaussian scale mixture process is shown to be a limit for the cumulative work from a random superposition of alternating onoff processes. SubFractional Brownian Motion is shown to be the limit in a particular case. Queueing and estimation results for processes which are conditionally Fractional Gaussian Noise are included. A model with a superposition of alternating onoff processes with independent lifetimes is also considered.
A wavelet based estimator for the parameter of selfsimilarity of fractional Brownian motion
, 1996
"... Let X = fX(t); t 2 IRg be a continuous fractional Brownian motion with parameter of selfsimilarity H (0 ! H ! 1). Let / be a wavelet function with compact support and let / j;k be rescaled versions of the function / at position k. We investigate the basic properties of the estimator for H H k n ..."
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Let X = fX(t); t 2 IRg be a continuous fractional Brownian motion with parameter of selfsimilarity H (0 ! H ! 1). Let / be a wavelet function with compact support and let / j;k be rescaled versions of the function / at position k. We investigate the basic properties of the estimator for H H k n = n\Gamma1 X j=0 wj;n log 2 jD j;k j \Gamma 1 2 ; where D j;k = R R X(t)/ j;k (t) dt are wavelet coefficients, wj;n are certain weights, and n is the number of scales used. This estimator is a weighted sum of random variables Y k j = log 2 jD j;k j which form a stationary, strongly mixing sequence. We show that the estimator is unbiased, consistent and has asymptotically a normal distribution. Keywords and phrases: selfsimilar processes, fractional Brownian motion, wavelets. Lieve.Delbekewis.kuleuven.ac.be and walterwis.kuleuven.ac.be, the second author is a Research Director of the Belgian National Fund for Scientific Research 1 Introduction Selfsimilar processes are stocha...
Performance of EnergyConserving Access Protocols Under SelfSimilar Traffic
 IN PROC. IEEE WIRELESS COMMUNICATIONS NETWORKING CONF. (WCNC
, 1999
"... Since widespread commercial use of wireless technology is still many years off, there is very little data to determine what constitutes an accurate model for wireless traffic. Many recent studies have shown that landline network traffic does not follow a traditional Poisson model but instead exhibit ..."
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Since widespread commercial use of wireless technology is still many years off, there is very little data to determine what constitutes an accurate model for wireless traffic. Many recent studies have shown that landline network traffic does not follow a traditional Poisson model but instead exhibits selfsimilar behavior. Previous papers have introduced and analyzed the performance of fundamental classes of energy conserving protocols under traditional Poisson models. In order to evaluate these protocols under a distribution which might more accurately depict future wireless network traffic, we use simulation and proven models to consider the tradeoff between energy and delay under a selfsimilar arrival distribution.
THE LAMPERTI TRANSFORMATION FOR SELFSIMILAR PROCESSES
, 1997
"... In this paper we establish the uniqueness of the Lamperti transformation leading from selfsimilar to stationary processes, and conversely. We discuss #stable processes, which allow to understand better the di#erence between the Gaussian and nonGaussian cases. As a byproduct we get a natural cons ..."
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In this paper we establish the uniqueness of the Lamperti transformation leading from selfsimilar to stationary processes, and conversely. We discuss #stable processes, which allow to understand better the di#erence between the Gaussian and nonGaussian cases. As a byproduct we get a natural construction of two distinct #stable OrnsteinUhlenbeck processes via the Lamperti transformation for 0 < # < 2. Also a new class of mixed linear fractional #stable motions is introduced.
Performance Evaluation of the ServerNet R SAN under SelfSimilar Traffic
"... Selfsimilar traffic distributions have been observed in a wide range of networking applications and models such as LANs, WANs, telnet, FTP, WWW, ISDN, SS7 and VBR traffic over ATM. Therefore, it has been suggested that many other theoretical protocols and systems need to be reevaluated under this d ..."
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Cited by 2 (0 self)
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Selfsimilar traffic distributions have been observed in a wide range of networking applications and models such as LANs, WANs, telnet, FTP, WWW, ISDN, SS7 and VBR traffic over ATM. Therefore, it has been suggested that many other theoretical protocols and systems need to be reevaluated under this different type of traffic before practical implementations potentially show their faults. The ServerNet SAN is a new core technology for server architectures that focuses on moving data. It is a wormholerouted, packetswitched, pointtopoint network with special attention paid to reducing latency and assuring reliability. In this paper we investigate the implications of selfsimilar traffic distributions in the ServerNet SAN, and compare the results with those obtained on the basis of the Poisson assumption. 1.
Some New Statistical Approaches to the Analysis of Long Memory Processes
, 1994
"... This thesis describes methods of analysis and synthesis of long memory processes. Long memory processes are those which exhibit correlations between events separated by a long period of time. This phenomenon is characterized in the frequency domain by a sharp peak in the spectral density function as ..."
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This thesis describes methods of analysis and synthesis of long memory processes. Long memory processes are those which exhibit correlations between events separated by a long period of time. This phenomenon is characterized in the frequency domain by a sharp peak in the spectral density function as the frequency approaches zero. This characteristic is observed in many physical time series, including those in the fields of geophysics, astronomy and finance. A class of models that captures such long memory behaviour are fractionally differenced processes, the simplest of these processes is obtained by differencing white noise a fractional number of times. We employ two methods of analyzing such processes: Multitaper spectral estimation and Wavelet analysis. Multitaper spectral analysis uses the average of several direct spectral estimators evaluated using orthogonal tapers. We look at two sets of tapers: the discrete prolate spheroidal sequences and sinusoidal tapers. This method of s...