Series representation and simulation of multifractional Lévy motions

2004 ◽  
Vol 36 (1) ◽  
pp. 171-197 ◽  
Author(s):  
Céline Lacaux
Keyword(s):  
Brownian Motion ◽  
Asymptotic Behaviour ◽  
Shot Noise ◽  
Error Rate ◽  
Control Measure ◽  
Series Representation ◽  
Sufficient Conditions ◽  
Infinitely Divisible ◽  
Multifractional Brownian Motion ◽  
Infinitely Divisible Laws

This paper introduces a method of generating real harmonizable multifractional Lévy motions (RHMLMs). The simulation of these fields is closely related to that of infinitely divisible laws or Lévy processes. In the case where the control measure of the RHMLM is finite, generalized shot-noise series are used. An estimation of the error is also given. Otherwise, the RHMLM Xh is split into two independent RHMLMs, Xε,1 and Xε,2. More precisely, Xε,2 is an RHMLM whose control measure is finite. It can then be rewritten as a generalized shot-noise series. The asymptotic behaviour of Xε,1 as ε → 0+ is further elaborated. Sufficient conditions to approximate Xε,1 by a multifractional Brownian motion are given. The error rate in terms of Berry-Esseen bounds is then discussed. Finally, some examples of simulation are given.

Series representation and simulation of multifractional Lévy motions

2004 ◽  
Vol 36 (01) ◽  
pp. 171-197
Author(s):  
Céline Lacaux
Keyword(s):  
Brownian Motion ◽  
Asymptotic Behaviour ◽  
Shot Noise ◽  
Error Rate ◽  
Control Measure ◽  
Series Representation ◽  
Sufficient Conditions ◽  
Infinitely Divisible ◽  
Multifractional Brownian Motion ◽  
Infinitely Divisible Laws

This paper introduces a method of generating real harmonizable multifractional Lévy motions (RHMLMs). The simulation of these fields is closely related to that of infinitely divisible laws or Lévy processes. In the case where the control measure of the RHMLM is finite, generalized shot-noise series are used. An estimation of the error is also given. Otherwise, the RHMLMXhis split into two independent RHMLMs,Xε,1andXε,2. More precisely,Xε,2is an RHMLM whose control measure is finite. It can then be rewritten as a generalized shot-noise series. The asymptotic behaviour ofXε,1as ε → 0+is further elaborated. Sufficient conditions to approximateXε,1by a multifractional Brownian motion are given. The error rate in terms of Berry-Esseen bounds is then discussed. Finally, some examples of simulation are given.

scholarly journals A martingale central limit theorem without negligibility conditions

1978 ◽  
Vol 18 (1) ◽  
pp. 13-19 ◽  
Author(s):  
Robert J. Adler
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Sufficient Conditions ◽  
Infinitely Divisible ◽  
Martingale Central Limit Theorem ◽  
Triangular Arrays ◽  
Infinitely Divisible Laws ◽  
Martingale Case

We obtain sufficient conditions for the convergence of martingale triangular arrays to infinitely divisible laws with finite variances, without making the usual assumptions of uniform asymptotic negligibility. Our results generalise known results for both the martingale case under a negligibility assumption and the classical (independence) case without such assumptions.

The central limit theorem for the Poisson shot-noise process

1984 ◽  
Vol 21 (02) ◽  
pp. 287-301 ◽  
Author(s):  
John A. Lane
Keyword(s):  
Shot Noise ◽  
Sufficient Conditions ◽  
Infinitely Divisible ◽  
Noise Process ◽  
Shot Noise Process ◽  
The Central Limit Theorem ◽  
Cluster Point Process ◽  
Necessary And Sufficient ◽  
Poisson Cluster ◽  
Poisson Shot Noise

The Poisson shot-noise process discussed here takes the form f:oo H(t, s)N(ds), where N(· is the counting measure of a Poisson process and the H(·, s) are independent stochastic processes. Necessary and sufficient conditions are obtained for convergence in distribution, as t ∼ OC, to any infinitely divisible distribution. The main interest is in the explosive transient one-sided shot-noise, Y(t) = f:1 H(t, s)N(ds) where Var Y(t)∼ oc, Here conditions for asymptotic normality are discussed in detail. Important examples include the Poisson cluster point process and the integrated stationary shotnoise.

The central limit theorem for the Poisson shot-noise process

1984 ◽  
Vol 21 (2) ◽  
pp. 287-301 ◽  
Author(s):  
John A. Lane
Keyword(s):  
Shot Noise ◽  
Sufficient Conditions ◽  
Infinitely Divisible ◽  
Noise Process ◽  
Shot Noise Process ◽  
The Central Limit Theorem ◽  
Cluster Point Process ◽  
Necessary And Sufficient ◽  
Poisson Cluster ◽  
Poisson Shot Noise

The Poisson shot-noise process discussed here takes the form f:oo H(t, s)N(ds), where N(· is the counting measure of a Poisson process and the H(·, s) are independent stochastic processes. Necessary and sufficient conditions are obtained for convergence in distribution, as t ∼ OC, to any infinitely divisible distribution. The main interest is in the explosive transient one-sided shot-noise, Y(t) = f:1 H(t, s)N(ds) where Var Y(t)∼ oc, Here conditions for asymptotic normality are discussed in detail. Important examples include the Poisson cluster point process and the integrated stationary shotnoise.

Infinitely Divisible Laws Associated with Hyperbolic Functions

2003 ◽  
Vol 55 (2) ◽  
pp. 292-330 ◽  
Author(s):  
Jim Pitman ◽  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Analytic Number Theory ◽  
Laplace Transforms ◽  
Bessel Processes ◽  
Infinitely Divisible ◽  
Infinitely Divisible Laws ◽  
Weak Limits ◽  
The Riemann Zeta Function ◽  
Riemann Zeta ◽  
Lévy Measures

AbstractThe infinitely divisible distributions on of random variables Ct, St and Tt with Laplace transformsrespectively are characterized for various t > 0 in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their Lévy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for t = 1 or 2 in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of C1 and S2 are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet L-function associated with the quadratic character modulo 4. Related families of infinitely divisible laws, including the gamma, logistic and generalized hyperbolic secant distributions, are derived from St and Ct by operations such as Brownian subordination, exponential tilting, and weak limits, and characterized in various ways.

Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

2007 ◽  
Vol 44 (02) ◽  
pp. 393-408 ◽  
Author(s):  
Allan Sly
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
White Noise ◽  
Gaussian Process ◽  
Discrete Data ◽  
Hölder Exponent ◽  
Scaling Properties ◽  
Multifractional Brownian Motion ◽  
Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

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