research article on the long-range dependence of...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 842197 5 pageshttpdxdoiorg1011552013842197

Research ArticleOn the Long-Range Dependence of Fractional Brownian Motion

Ming Li

School of Information Science amp Technology East China Normal University No 500 Dong-Chuan Road Shanghai 200241 China

Correspondence should be addressed to Ming Li ming lihkyahoocom

Received 5 May 2013 Accepted 15 May 2013

Academic Editor Massimo Scalia

Copyright copy 2013 Ming Li This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper clarifies that the fractional Brownian motion 119861119867(119905) is of long-range dependence (LRD) for the Hurst parameter 0 lt

119867 lt 1 except 119867 = 12 In addition we note that the fractional Brownian motion is positively correlated for 0 lt 119867 lt 1 except119867 = 12 Moreover we present a theorem to state that the differential or integral of a random function 119883(119905) may substantiallychange the statistical dependence of 119883(119905) One example is that the differential of 119861

119867(119905) in the domain of generalized functions

changes the LRD of 119861119867(119905) to be of short-range dependence (SRD) when 0 lt 119867 lt 05

1 Introduction

Fractional Brownian motion (fBm) is widely used [1ndash10] Itstheory and applications attract the interests of researchers invarious fields ranging from telecommunications to biomedi-cal engineering see for example [11ndash44] simply citing a few

There is a set of statistical properties of fBm such asnonstationarity and being nondifferentiable in the domain ofordinary functions [45] Two properties namely nonstation-arity and nondifferentiable property are the basic propertiesof standard Brownian motion (Bm) [46ndash52] which is wellknown in the fields of time series as well as stochastic pro-cesses [53 54] As the substantial generalization of Bm fBmhas a property that Bm lacks that is its statistical dependence[1ndash4 45] The measure of the statistical dependence of fBm ischaracterized by the Hurst parameter119867 isin (0 1)

Note that the fBm for the Hurst parameter 119867 isin (0 1)

and 119867 = 12 is of LRD [11 12 45 55 56] In additionfBm is positively correlated for 119867 isin (0 1) but 119867 = 12

[57] However the LRD property of fBm may be sometimesconservatively expressed For example the LRD property offBm was restricted by 119867 isin (05 1) as can be seen from [58page 2341] and [59 page 708] For this reason it may bemeaningful to clarify which this paper aims at

The remaining paper is organized as follows In Section 2we describe that the range of 119867 for fBm to be of LRD is119867 isin (0 1) and119867 = 12 Discussions are in Section 3 which isfollowed by conclusions

2 FBm Is LRD for 0 lt 119867 lt 1 except 119867 = 05

In what follows a random function in general is denoted by119883(119905) for 119905 isin (0infin) We denote 119861

119867(119905) for 119905 isin (0infin) as fBm

with119867 isin (0 1)Without generality losing we assume that 119883(119905) is a ran-

dom function with mean zero The autocorrelation function(ACF) of119883(119905) is for 119905 119904 isin (0infin) denoted by

119862119883119883 (119905 119904) = 119864 [119883 (119905)119883 (119904)] (1)

By LRD [1 2] we mean that

int

infin

0

119862119883119883 (119905 119904) 119889119905 = infin (2)

If

int

infin

0

119862119883119883 (119905 119904) 119889119905 lt infin (3)

119883(119905) is of short-range dependence (SRD)Denote by 119878

119883119883(120596 119905) the power spectrumdensity function

(PSD) of 119883(119905) Denote by 119865 the operator of the Fouriertransform Then [60ndash64]

119878119883119883 (120596 119905) = 119865 [119862119883119883 (119905 119904)] (4)

The LRD condition described in the frequency domain isexpressed by

lim120596rarr0

119878119883119883 (120596 119904) = infin (5)

2 Mathematical Problems in Engineering

The above expression implies the property of 1119891 noiseregarding random functions with LRD [1ndash4 65ndash70] On theother hand119883(119905) is of SRD if

lim120596rarr0

119878119883119883 (120596 119904) lt infin (6)

Let 119882minusV be the Weyl integral of order V gt 0 Then forrandom function119883(119905) see for example [71ndash75] one has

119882minusV119883(119905) =

1

Γ (V)int

infin

119905

(119906 minus 119905)Vminus1119883 (119906) 119889119906 (7)

Thus the fBm of the Weyl type is in the form

119861119867 (119905) minus 119861119867 (0) =

1

Γ (119867 + 12)

times int

0

minusinfin

[(119905 minus 119906)119867minus05

minus (minus119906)119867minus05

] 119889119861 (119906)

+int

119905

0

(119905 minus 119906)119867minus05

119889119861 (119906)

(8)

Following [76] the PSD of the fBm of the Weyl type isexpressed by

119878119861119867119861119867

(120596 119905) =1

|120596|2119867+1

(1 minus 21minus2119867 cos 2120596119905) (9)

Therefore we have the following theorem

Theorem 1 FBm is of LRD for119867 isin (0 1) except119867 = 12

Proof Because lim120596rarr0

119878119861119867119861119867

(120596 119905) = infin for all 119905 gt 0 and for119867 isin (0 1) except119867 = 12 the theorem holds

As a matter of fact fBm reduces to the standard Bm if119867 = 12 The PSD of BM see [11] is given by

1198781198611211986112

(119905 120596) =1

1205962(1 minus cos 2120596119905) (10)

Thus

lim120596rarr0

1198781198611211986112

(119905 120596) = 21199052=infin (11)

From the theorem we have the following corollary

Corollary 2 FBm is not SRD for119867 isin (0 1)

In passing we mention that the ACF of 119861119867(119905) of theWeyl

type is in the form

119862119861119867119861119867

(119905 119904) =119881119867

(119867 + 12) Γ (119867 + 12)

times [|119905|2119867+ |119904|2119867minus |119905 minus 119904|

2119867]

(12)

where 119881119867is the strength of 119861

119867(119905) It is given by

119881119867= Var[119861

119867 (1)] = Γ (1 minus 2119867)cos120587119867120587119867

(13)

Following [57 page 4] we have the following remark

00

0

10

10

10

20

20

20

30

30

C

Figure 1 ACF of fBm for 119905 119904 = 0 1 30 and119867 = 02

00

10

1020

20

30

30

50

100

C


Remark 3 The ACF of fBm is positively correlated for 119867 isin

(0 1) except 119867 = 12 That is 119862119861119867119861119867

(119905 119904) ge 0 for 119905 119904 isin(0infin) Figures 1 and 2 indicate the plots of 119862

119861119867119861119867(119905 119904) for

119905 119904 = 0 1 30 with119867 = 02 and 04 respectively

3 Discussions

Let119866119867(119905) be the fractional Gaussian noise (fGn)Then in the

domain of generalized functions over the Schwartz space oftest functions [45] we write

119866119867 (119905) =

119889119861119867 (119905)

119889119905 (14)

Mathematical Problems in Engineering 3

Denote by 119862119866119867119866119867

(120591 119904) the ACF of 119866119867(119905) Then for 120576 gt 0 [19

45] one has

119862119866119867119866119867

(120591 120576) =1198811198671205762119867minus2

2

times [(|120591|

120576+ 1)

2119867

+

10038161003816100381610038161003816100381610038161003816

|120591|

120576minus 1

10038161003816100381610038161003816100381610038161003816

2119867

minus 2

1003816100381610038161003816100381610038161003816

120591

120576

1003816100381610038161003816100381610038161003816

2119867

]

(15)

From the contents in Section 2 we have the followingtheorem

Theorem 4 Let119883(119905) be a random functionThen the statisti-cal dependence of 119889119883(119905)119889119905 may substantially differ from thatof 119883(119905) where the differential is in the domain of generalizedfunctions

Proof To prove the theorem we only need an example toshow it Let 119883(119905) = 119861

119867(119905) Then 119889119883(119905)119889119905 = 119866

119867(119905) It is

well known that fGn is LRD when 119867 isin (05 1) as 119862119866119867119866119867

isnonintegrable if 119867 isin (05 1) On the other hand for 119867 isin

(0 05) the integral of119862119866119867119866119867

is zero Hence fGn is SRDwhen119867 isin (0 05) In passing we note that 119862

119866119867119866119867(120591 120576) changes

its sign and becomes negative for some 120591 proportional to 120576in this parameter domain [45 page 434] Since 119861

119867(119905) is LRD

for 119867 isin (0 1) except 119867 = 12 the statistical dependence of119866119867(119905) substantially differs from that of 119861

119867(119905) This completes

the proof

From Theorem 4 we immediately obtain the corollarybelow

Corollary 5 Let 119883(119905) be a random function Then thestatistical dependence of119863minus1119883(119905)may substantially differ fromthat of 119883(119905) where119863minus1 is the integral operator of order one

Proof Let 119883(119905) = 119866119867(119905) Then 119863minus1119883(119905) = 119861

119867(119905) Since

119861119867(119905) is LRD for119867 isin (0 05) while 119866

119867(119905) is SRD when119867 isin

(0 05) one sees that the statistical dependence of 119863minus1119883(119905)substantially differs from that of 119883(119905) Thus Corollary 5results

4 Conclusions

We have clarified that fBm is LRD and positively correlatedfor 119867 isin (0 1) except 119867 = 12 In addition we have provedthat the differential or integral of a random function mayconsiderably change its statistical dependence

Acknowledgments

This work was supported in part by the 973 plan under theproject Grant no 2011CB302800 and by the National NaturalScience Foundation of China under the project Grants nos61272402 61070214 and 60873264

References

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[2] J Beran Statistics for Long-Memory Processes Chapman andHall New York NY USA 1994

[3] J Levy-Vehel E Lutton and C Tricot Eds Fractals in Engi-neering Springer 1997

[4] P Doukhan G Oppenheim and M S Taqqu EdsTheory andApplications of Long-Range Dependence Birkhauser BostonMass USA 2002

[5] J B Bassingthwaighte L S Liebovitch and B J West FractalPhysiology Oxford University Press 1994

[6] J-P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990

[7] W Willinger and V Paxson ldquoWhere mathematics meets theinternetrdquo Notices of the American Mathematical Society vol 45no 8 pp 961ndash970 1998

[8] V Paxson and S Floyd ldquoWide area traffic the failure of Poissonmodelingrdquo IEEEACMTransactions on Networking vol 3 no 3pp 226ndash244 1995

[9] S C Lim and L P Teo ldquoModeling single-file diffusion withstep fractional Brownian motion and a generalized fractionalLangevin equationrdquo Journal of Statistical Mechanics vol 2009no 8 Article ID P08015 24 pages 2009

[10] S Kim S Y Nam and D K Sung ldquoEffective bandwidth for asingle server queueing system with fractional Brownian inputrdquoPerformance Evaluation vol 61 no 2-3 pp 203ndash223 2005

[11] V M Sithi and S C Lim ldquoOn the spectra of Riemann-Liouvillefractional Brownian motionrdquo Journal of Physics vol 28 no 11pp 2995ndash3003 1995

[12] S V Muniandy and S C Lim ldquoModeling of locally self-similarprocesses using multifractional Brownian motion of Riemann-Liouville typerdquo Physical Review E vol 63 no 4 part 2 ArticleID 461047 2001

[13] D Feyel and A de La Pradelle ldquoOn fractional Brownianprocessesrdquo Potential Analysis vol 10 no 3 pp 273ndash288 1999

[14] R F Peltier and J Levy-Vehel ldquoMultifractional Brownianmotion definition and preliminaries resultsrdquo INRIA TR 26451995

[15] T G Sinai ldquoDistribution of the maximum of a fractionalBrownian motionrdquo Russian Mathematical Surveys vol 52 no2 pp 119ndash138 1997

[16] S C Lim and S V Muniandy ldquoOn some possible generaliza-tions of fractional Brownianmotionrdquo Physics Letters A vol 266no 2-3 pp 140ndash145 2000

[17] M D Ortigueira and A G Batista ldquoOn the relation betweenthe fractional Brownian motion and the fractional derivativesrdquoPhysics Letters A vol 372 no 7 pp 958ndash968 2008

[18] S C Lim and L P Teo ldquoWeyl and Riemann-Liouvillemultifrac-tional Ornstein-Uhlenbeck processesrdquo Journal of Physics A vol40 no 23 pp 6035ndash6060 2007

[19] M Li and S C Lim ldquoA rigorous derivation of power spectrumof fractional Gaussian noiserdquo Fluctuation and Noise Letters vol6 no 4 pp C33ndashC36 2006

[20] M Li andW Zhao ldquoOn bandlimitedness and lag-limitedness offractional Gaussian noiserdquo Physica A vol 392 no 9 pp 1955ndash1961 2013

[21] M Li and W Zhao ldquoQuantitatively investigating locally weakstationarity of modified multifractional Gaussian noiserdquo Phys-ica A vol 391 no 24 pp 6268ndash6278 2012


[22] V Paxson ldquoFast approximate synthesis of fractional gaussiannoise for generating self-similar network trafficrdquo ACM SIG-COMM Computer Communication Review vol 27 no 5 pp 5ndash18 1997

[23] C Cattani M Scalia E Laserra I Bochicchio and K K NandildquoCorrect light deflection in Weyl conformal gravityrdquo PhysicalReview D vol 87 no 4 Article ID 47503 4 pages 2013

[24] C Cattani ldquoHarmonic wavelet approximation of randomfractal and high frequency signalsrdquoTelecommunication Systemsvol 43 no 3-4 pp 207ndash217 2010

[25] C Cattani G Pierro and G Altieri ldquoEntropy and multifrac-tality for the myeloma multiple TET 2 generdquo MathematicalProblems in Engineering vol 2012 Article ID 193761 14 pages2012

[26] C Cattani ldquoFractals and hidden symmetries in DNArdquo Mathe-matical Problems in Engineering vol 2010 Article ID 507056 31pages 2010

[27] S Hu Z Liao and W Chen ldquoSinogram restoration for low-dosed X-ray computed tomography using fractional-orderPerona-Malik diffusionrdquo Mathematical Problems in Engineer-ing vol 2012 Article ID 391050 13 pages 2012

[28] Z W Liao S X Hu D Sun and W F Chen ldquoEnclosed Lapla-cian operator of nonlinear anisotropic diffusion to preservesingularities and delete isolated points in image smoothingrdquoMathematical Problems in Engineering vol 2011 Article ID749456 15 pages 2011

[29] J W Yang Y J Chen and M Scalia ldquoConstruction of affineinvariant functions in spatial domainrdquo Mathematical Problemsin Engineering vol 2012 Article ID 690262 11 pages 2012

[30] H E Stanley S V Buldyrev A L Goldberger S Havlin C-KPeng and M Simons ldquoLong-range power-law correlations incondensed matter physics and biophysicsrdquo Physica A vol 200no 1ndash4 pp 4ndash24 1993

[31] B Podobnik P C Ivanov K Biljakovic D Horvatic H E Stan-ley and I Grosse ldquoFractionally integrated process with power-law correlations in variables and magnitudesrdquo Physical ReviewE vol 72 no 2 Article ID 026121 7 pages 2005

[32] A Carbone G Castelli and H E Stanley ldquoTime-dependentHurst exponent in financial time seriesrdquo Physica A vol 344 no1-2 pp 267ndash271 2004

[33] Z Chen P C Ivanov KHu andH E Stanley ldquoEffect of nonsta-tionarities on detrended fluctuation analysisrdquoPhysical ReviewEvol 65 no 4 Article ID 041107 15 pages 2002

[34] C Toma ldquoAdvanced signal processing and command synthesisfor memory-limited complex systemsrdquoMathematical Problemsin Engineering vol 2012 Article ID 927821 13 pages 2012

[35] E G Bakhoum and C Toma ldquoSpecific mathematical aspectsof dynamics generated by coherence functionsrdquo MathematicalProblems in Engineering vol 2011 Article ID 436198 10 pages2011

[36] E G Bakhoum and C Toma ldquoMathematical transform oftraveling-wave equations and phase aspects of quantum interac-tionrdquoMathematical Problems in Engineering Article ID 69520815 pages 2010

[37] G Korvin Fractal Models in the Earth Science Elsevier 1992[38] E E Peters Fractal Market AnalysismdashApplying ChaosTheory to

Investment and Economics John Wiley amp Sons 1994[39] S-C Liu and S Chang ldquoDimension estimation of discrete-time

fractional Brownian motion with applications to image textureclassificationrdquo IEEE Transactions on Image Processing vol 6 no8 pp 1176ndash1184 1997

[40] S Chang S-T Mao S-J Hu W-C Lin and C-L ChengldquoStudies of detrusor-sphincter synergia and dyssynergia duringmicturition in rats via fractional brownian motionrdquo IEEETransactions on Biomedical Engineering vol 47 no 8 pp 1066ndash1073 2000

[41] C Fortin R Kumaresan W Ohley and S Hoefer ldquoFractaldimension in the analysis of medical imagesrdquo IEEE Engineeringin Medicine and Biology Magazine vol 11 no 2 pp 65ndash71 1992

[42] A Eke E PHerman J B Bassingthwaighte et al ldquoPhysiologicaltime series distinguishing fractal analysis noises frommotionsrdquoEuropean Journal of Physiology vol 439 no 4 pp 403ndash4152000

[43] GM Raymond D B Percival and J B Bassingthwaighte ldquoThespectra and periodograms of anti-correlated discrete fractionalGaussian noiserdquo Physica A vol 322 pp 169ndash179 2003

[44] H E Schepers J H GM van Beek and J B BassingthwaighteldquoFour methods to estimate the fractal dimension from self-affine signalsrdquo IEEE Engineering in Medicine and Biology Mag-azine vol 11 no 2 pp 57ndash64 1992

[45] B B Mandelbrot and J W Van Ness ldquoFractional Brownianmotions fractional noises and applicationsrdquo SIAM Review vol10 pp 422ndash437 1968

[46] G E Uhlenbeck and L S Ornstein ldquoOn the theory of theBrownian motionrdquo Physical Review vol 36 no 5 pp 823ndash8411930

[47] T Hida Brownian Motion Springer 1980[48] J Dunkel andPHanggi ldquoRelativistic BrownianmotionrdquoPhysics

Reports vol 471 no 1 pp 1ndash73 2009[49] M Peligrad and S Utev ldquoAnother approach to Brownian

motionrdquo Stochastic Processes and their Applications vol 116 no2 pp 279ndash292 2006

[50] E Frey and K Kroy ldquoBrownian motion a paradigm of softmatter and biological physicsrdquo Annalen der Physik vol 14 no1ndash3 pp 20ndash50 2005

[51] P Hanggi F Marchesoni and F Nori ldquoBrownian motorsrdquoAnnalen der Physik vol 14 no 1ndash3 pp 51ndash70 2005

[52] F CecconiMCenciniM Falcioni andAVulpiani ldquoBrownianmotion and diffusion from stochastic processes to chaos andbeyondrdquo Chaos vol 15 no 2 Article ID 026102 p 9 2005

[53] A Papoulis Probability Random Variables and Stochastic Pro-cesses McGraw-Hill Book 2nd edition 1984

[54] W A Fuller Introduction to Statistical Time Series John Wileyamp Sons New York NY USA 2nd edition 1996

[55] S C Lim and S V Muniandy ldquoGeneralized Ornstein-Uhlen-beck processes and associated self-similar processesrdquo Journal ofPhysics A vol 36 no 14 pp 3961ndash3982 2003

[56] H Qian ldquoFractional brownian motion and fractional gaussiannoiserdquo Lecture Notes in Physics vol 621 pp 22ndash33 2003

[57] F Aurzada and C Baumgarten ldquoPersistence of fractionalBrownian motion with moving boundaries and applicationsrdquoJournal of Physics A vol 46 no 12 Article ID 125007 2013

[58] VMaroulas and J Xiong ldquoLarge deviations for optimal filteringwith fractional Brownianmotionrdquo Stochastic Processes and theirApplications vol 123 no 6 pp 2340ndash2352 2013

[59] R Scheffer and R Maciel Filho ldquoThe fractional Brownianmotion as a model for an industrial airlift reactorrdquo ChemicalEngineering Science vol 56 no 2 pp 707ndash711 2001

[60] M Sun C C Li L N Sekhar and R J Sclabassi ldquoWignerspectral analyzer for nonstationary signalsrdquo IEEE Transactionson Instrumentation and Measurement vol 38 no 5 pp 961ndash966 1989


[61] M B Priestley ldquoEvolutionary spectra and non-stationary pro-cessesrdquo Journal of the Royal Statistical Society B vol 27 pp 204ndash237 1965

[62] J S Bendat and A G Piersol Random Data Analysis andMeasurement Procedure John Wiley amp Sons 3rd edition 2010

[63] J Xiao and P Flandrin ldquoMultitaper time-frequency reas-signment for nonstationary spectrum estimation and chirpenhancementrdquo IEEE Transactions on Signal Processing vol 55no 6 pp 2851ndash2860 2007

[64] P Borgnat P Flandrin P Honeine C Richard and J XiaoldquoTesting stationarity with surrogates a time-frequency ap-proachrdquo IEEE Transactions on Signal Processing vol 58 no 7pp 3459ndash3470 2010

[65] M Li and W Zhao ldquoOn 1119891 noiserdquo Mathematical Problems inEngineering vol 2012 Article ID 673648 22 pages 2012

[66] K Fraedrich U Luksch and R Blender ldquo1119891 model forlong-time memory of the ocean surface temperaturerdquo PhysicalReview E vol 70 no 3 Article ID 037301 pp 1ndash4 2004

[67] W T Li and D Holste ldquoUniversal 1119891 noise crossovers of scal-ing exponents and chromosome-specific patterns of guanine-cytosine content in DNA sequences of the human genomerdquoPhysical Review E vol 71 no 4 Article ID 041910 9 pages 2005

[68] F NHooge ldquo1119891 noiserdquoPhysica B vol 83 no 1 pp 14ndash23 1976[69] M S Keshner ldquo1119891 noiserdquo Proceedings of the IEEE vol 70 no

3 pp 212ndash218 1982[70] G AquinoM Bologna P Grigolini and B JWest ldquoBeyond the

death of linear response 1119891 optimal information transportrdquoPhysical Review Letters vol 105 no 6 Article ID 069901 1 page2010

[71] J Klafter S C Lim and RMetzler Fractional Dynamics RecentAdvances World Scientific 2012

[72] A A Kilbas H M Srivastava and J J Trujillo Theory andApplications of Fractional Differential Equations Elsevier Sci-ence Amsterdam The Netherlands 2006

[73] S SamkoAAKilbas andD IMaritchev Integrals andDeriva-tives of the Fractional Order and Some of Their ApplicationsGordon and Breach Amsterdam The Netherlands 1993

[74] G M Zaslavsky Hamiltonian Chaos and Fractional DynamicsOxford University Press Oxford UK 2008

[75] R C Blei Analysis in Integer and Fractional Dimensions Uni-versity Press Cambridge UK 2003

[76] P Flandrin ldquoOn the spectrum of fractional BrownianmotionsrdquoIEEE Transactions on InformationTheory vol 35 no 1 pp 197ndash199 1989

Submit your manuscripts athttpwwwhindawicom

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Stochastic AnalysisInternational Journal of


The above expression implies the property of 1119891 noiseregarding random functions with LRD [1ndash4 65ndash70] On theother hand119883(119905) is of SRD if

lim120596rarr0

119878119883119883 (120596 119904) lt infin (6)

Let 119882minusV be the Weyl integral of order V gt 0 Then forrandom function119883(119905) see for example [71ndash75] one has

119882minusV119883(119905) =

1

Γ (V)int

infin

119905

(119906 minus 119905)Vminus1119883 (119906) 119889119906 (7)

Thus the fBm of the Weyl type is in the form

119861119867 (119905) minus 119861119867 (0) =

1

Γ (119867 + 12)

times int

0

minusinfin

[(119905 minus 119906)119867minus05

minus (minus119906)119867minus05

] 119889119861 (119906)

+int

119905

0

(119905 minus 119906)119867minus05

119889119861 (119906)

(8)

Following [76] the PSD of the fBm of the Weyl type isexpressed by

119878119861119867119861119867

(120596 119905) =1

|120596|2119867+1

(1 minus 21minus2119867 cos 2120596119905) (9)

Therefore we have the following theorem

Theorem 1 FBm is of LRD for119867 isin (0 1) except119867 = 12

Proof Because lim120596rarr0

119878119861119867119861119867

(120596 119905) = infin for all 119905 gt 0 and for119867 isin (0 1) except119867 = 12 the theorem holds

As a matter of fact fBm reduces to the standard Bm if119867 = 12 The PSD of BM see [11] is given by

1198781198611211986112

(119905 120596) =1

1205962(1 minus cos 2120596119905) (10)

Thus

lim120596rarr0

1198781198611211986112

(119905 120596) = 21199052=infin (11)

From the theorem we have the following corollary

Corollary 2 FBm is not SRD for119867 isin (0 1)

In passing we mention that the ACF of 119861119867(119905) of theWeyl

type is in the form

119862119861119867119861119867

(119905 119904) =119881119867

(119867 + 12) Γ (119867 + 12)

times [|119905|2119867+ |119904|2119867minus |119905 minus 119904|

2119867]

(12)

where 119881119867is the strength of 119861

119867(119905) It is given by

119881119867= Var[119861

119867 (1)] = Γ (1 minus 2119867)cos120587119867120587119867

(13)

Following [57 page 4] we have the following remark

00

0

10

10

10

20

20

20

30

30

C


00

10

1020

20

30

30

50

100

C


Remark 3 The ACF of fBm is positively correlated for 119867 isin

(0 1) except 119867 = 12 That is 119862119861119867119861119867

(119905 119904) ge 0 for 119905 119904 isin(0infin) Figures 1 and 2 indicate the plots of 119862

119861119867119861119867(119905 119904) for

119905 119904 = 0 1 30 with119867 = 02 and 04 respectively

3 Discussions

Let119866119867(119905) be the fractional Gaussian noise (fGn)Then in the

domain of generalized functions over the Schwartz space oftest functions [45] we write

119866119867 (119905) =

119889119861119867 (119905)

119889119905 (14)


Denote by 119862119866119867119866119867


45] one has

119862119866119867119866119867

(120591 120576) =1198811198671205762119867minus2

2

times [(|120591|

120576+ 1)

2119867

+

10038161003816100381610038161003816100381610038161003816

|120591|

120576minus 1

10038161003816100381610038161003816100381610038161003816

2119867

minus 2

1003816100381610038161003816100381610038161003816

120591

120576

1003816100381610038161003816100381610038161003816

2119867

]

(15)




119867(119905) Then 119889119883(119905)119889119905 = 119866

119867(119905) It is



(0 05) the integral of119862119866119867119866119867


119866119867119866119867(120591 120576) changes


119867(119905) is LRD



the proof




119867(119905) Since


119867(119905) is SRD when119867 isin


4 Conclusions


Acknowledgments


References






















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Denote by 119862119866119867119866119867


45] one has

119862119866119867119866119867

(120591 120576) =1198811198671205762119867minus2

2

times [(|120591|

120576+ 1)

2119867

+

10038161003816100381610038161003816100381610038161003816

|120591|

120576minus 1

10038161003816100381610038161003816100381610038161003816

2119867

minus 2

1003816100381610038161003816100381610038161003816

120591

120576

1003816100381610038161003816100381610038161003816

2119867

]

(15)




119867(119905) Then 119889119883(119905)119889119905 = 119866

119867(119905) It is



(0 05) the integral of119862119866119867119866119867


119866119867119866119867(120591 120576) changes


119867(119905) is LRD



the proof




119867(119905) Since


119867(119905) is SRD when119867 isin


4 Conclusions


Acknowledgments


References






















































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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