Local independence of fractional Brownian motion

Ilkka Norros (Corresponding Author), Eero Saksman

Research output: Contribution to journalArticleScientificpeer-review

2 Citations (Scopus)

Abstract

Let σ(t,t′) be the sigma-algebra generated by the differences Xs−Xs′ with s,s′∈(t,t′), where (Xt)−∞<t<∞ is the fractional Brownian motion with Hurst index H∈(0,1). We prove that for any two distinct timepoints t1 and t2 the sigma-algebras σ(t1−ε,t1+ε) and σ(t2−ε,t2+ε) are asymptotically independent as ε↘0. We show the independence in the strong sense that Shannon’s mutual information between the two σ-algebras tends to zero as ε↘0. Some generalizations and quantitative estimates are also provided.
Original languageEnglish
Pages (from-to)3155 - 3172
Number of pages18
JournalStochastic Processes and their Applications
Volume119
Issue number10
DOIs
Publication statusPublished - 2009
MoE publication typeA1 Journal article-refereed

Fingerprint

Sigma algebra
Local Independence
Brownian movement
Fractional Brownian Motion
Algebra
Hirsch Index
Mutual Information
Tend
Distinct
Zero
Estimate

Keywords

  • Fractional Brownian motion
  • Asymptotic
  • Independence
  • Local

Cite this

Norros, Ilkka ; Saksman, Eero. / Local independence of fractional Brownian motion. In: Stochastic Processes and their Applications. 2009 ; Vol. 119, No. 10. pp. 3155 - 3172.
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Local independence of fractional Brownian motion. / Norros, Ilkka (Corresponding Author); Saksman, Eero.

In: Stochastic Processes and their Applications, Vol. 119, No. 10, 2009, p. 3155 - 3172.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Local independence of fractional Brownian motion

AU - Norros, Ilkka

AU - Saksman, Eero

PY - 2009

Y1 - 2009

N2 - Let σ(t,t′) be the sigma-algebra generated by the differences Xs−Xs′ with s,s′∈(t,t′), where (Xt)−∞ is the fractional Brownian motion with Hurst index H∈(0,1). We prove that for any two distinct timepoints t1 and t2 the sigma-algebras σ(t1−ε,t1+ε) and σ(t2−ε,t2+ε) are asymptotically independent as ε↘0. We show the independence in the strong sense that Shannon’s mutual information between the two σ-algebras tends to zero as ε↘0. Some generalizations and quantitative estimates are also provided.

AB - Let σ(t,t′) be the sigma-algebra generated by the differences Xs−Xs′ with s,s′∈(t,t′), where (Xt)−∞ is the fractional Brownian motion with Hurst index H∈(0,1). We prove that for any two distinct timepoints t1 and t2 the sigma-algebras σ(t1−ε,t1+ε) and σ(t2−ε,t2+ε) are asymptotically independent as ε↘0. We show the independence in the strong sense that Shannon’s mutual information between the two σ-algebras tends to zero as ε↘0. Some generalizations and quantitative estimates are also provided.

KW - Fractional Brownian motion

KW - Asymptotic

KW - Independence

KW - Local

U2 - 10.1016/j.spa.2009.05.004

DO - 10.1016/j.spa.2009.05.004

M3 - Article

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JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

SN - 0304-4149

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