scholarly journals Numerical aspects of shot noise representation of infinitely divisible laws and related processes

2021 ◽  
Vol 18 (none) ◽  
Author(s):  
Sida Yuan ◽  
Reiichiro Kawai
Keyword(s):  
Shot Noise ◽  
Infinitely Divisible ◽  
Infinitely Divisible Laws

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.

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