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23
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 generaliza ..."
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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)
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.
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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Cited by 3 (0 self)
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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.
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 3 (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
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...
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...
Where Did The Brownian Particle Go?
, 2000
"... Consider the radial projection onto the unit sphere of the path a ddimensional Brownian motion W, started at the center of the sphere and run for unit time. Given the occupation measure µ of this projected path, what can be said about the terminal point W(1), or about the range of the original path ..."
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Consider the radial projection onto the unit sphere of the path a ddimensional Brownian motion W, started at the center of the sphere and run for unit time. Given the occupation measure µ of this projected path, what can be said about the terminal point W(1), or about the range of the original path? In any dimension, for each Borel set A ⊆ S d−1, the conditional probability that the projection of W(1) is in A given µ(A) is just µ(A). Nevertheless, in dimension d ≥ 3, both the range and the terminal point of W can be recovered with probability 1 from µ. In particular, for d ≥ 3 the conditional law of the projection of W(1) given µ is not µ. In dimension 2 we conjecture that the projection of W(1) cannot be recovered almost surely from µ, and show that the conditional law of the projection of W(1) given µ is not µ.
Remarks Regarding Queuing Model and Packet Loss Probability for the Traffic with Self Similar Characteristics
"... Abstract—Network management techniques have long been of interest to the networking research community. The queue size plays a critical role for the network performance. The adequate size of the queue maintains Quality of Service (QoS) requirements within limited network capacity for as many users a ..."
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Abstract—Network management techniques have long been of interest to the networking research community. The queue size plays a critical role for the network performance. The adequate size of the queue maintains Quality of Service (QoS) requirements within limited network capacity for as many users as possible. The appropriate estimation of the queuing model parameters is crucial for both initial size estimation and during the process of resource allocation. The accurate resource allocation model for the management system increases the network utilization. The present paper demonstrates the results of empirical observation of memory allocation for packetbased services.
Bayesian Analysis Of Long Memory Time Series
, 1997
"... (Statistics) BAYESIAN ANALYSIS OF LONG MEMORY TIME SERIES by Giovanni Petris Institute of Statistics and Decision Sciences Duke University Date: Approved: Dr. Mike West, Supervisor Dr. Michael Lavine Dr. Peter Muller Dr. David Rios Insua An abstract of a dissertation submitted in partial fulfillme ..."
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(Statistics) BAYESIAN ANALYSIS OF LONG MEMORY TIME SERIES by Giovanni Petris Institute of Statistics and Decision Sciences Duke University Date: Approved: Dr. Mike West, Supervisor Dr. Michael Lavine Dr. Peter Muller Dr. David Rios Insua An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Institute of Statistics and Decision Sciences in the Graduate School of Duke University Abstract In recent years there has been a growing interest for the statistical analysis of long memory processes, i.e., stationary processes with a spectral density presenting a pole at the origin. In this dissertation, I will present a Bayesian nonparametric approach to the problem. I propose a class of prior distributions on a family of spectral densities which properly includes a set of densities having a pole at the zero frequency. This allows to test for the presence of a long memory behaviour and to compare the prior and posteri...