Strong Local Non-Determinism of Sub-Fractional Brownian Motion


Let be a subfractional Brownian motion in . We prove that is strongly locally nondeterministic.

Share and Cite:

Luan, N. (2015) Strong Local Non-Determinism of Sub-Fractional Brownian Motion. Applied Mathematics, 6, 2211-2216. doi: 10.4236/am.2015.613194.

Received 30 August 2015; accepted 27 November 2015; published 30 November 2015

1. Introduction

The fractional Brownian motion (fBm for short) is the best known and most used process with long-dependence property for models in telecommunications, turbulence, image processing and finance. This process is first introduced by [1] and later studied by [2] . The self-similarity and stationarity of the increments are two main properties for which fBm enjoy success as a modeling tool. The fBm is the only continuous Gaussian process which is self-similar and has stationary increments; see [3] . Many authors have also proposed for using more general self-similar Gaussian processes and random fields as stochastic models; see e.g. [4] -[9] . Such applications have raised many interesting theoretical questions about self-similar Gaussian processes and fields in general. However, in contrast to the extensive studies on fractional Brownian motion, there has been little systematic investigation on other self-similar Gaussian processes until [10] fills the gap by developing systematic ways to study sample path properties of a class of self-similar Gaussian process, namely, the bifractional Brownian motion. Their main tools are the Lamperti transformation, which provides a powerful connection between self-similar processes and stationary processes; see [11] , and the strong local non-determinism of Gaussian processes; see [12] . In particular, for any self-similar Gaussian processes, the Lamperti transformation leads to a stochastic integal representation for X.

An extension of Bm which preserves many properties of the fBm, but not the stationarity of the increments, is so called sub-fractional Brownian motion (sub-fBm, in short) introduced by [13] . The sub-fBm is another class of self-similar Gaussian process which has properties analogous to those of fBm; see [13] -[15] . Given a constant, the sub-fractional Brownian motion in is a centered Gaussian process with covariance function



Let be independent copies of. We define the Gaussian process with values in by


By (1), one can verify easily that is a self-similar process with index H, that is, for every constant,


where means that the two processes have the same finite dimensional distributions. Note that does not have stationary increments.

The strong local non-determinism is an important tool to study the sample path properties of self-similar Gaussian process, such as the small ball probability and Chung’s law of the iterated logarithm. In this paper, we apply the Lamperti transformation to prove the strong local non-determinism of. Throughout this paper, a specified positive and finite constant is denoted by which may depend on H.

2. Strong Local Non-Determinism

Theorem 1. For all constants, is strongly locally -nondeterministic on with. That is, there exist positive constants and such that for all and all,


Proof. By Lamperti’s transformation (see [11] ), we consider the centered stationary Gaussian process defined by


The covariance function is given by


where is an even function. By (6) and Taylor expansion, we verify that, as, where. It follows that. Also, by using (6) and the Taylor expansion again, we also have


Using Bochner’s theorem, has the following stochastic integral representation


where W is a complex Gaussian measure with control measure whose Fourier transform is. The measure is called the spectral measure of.

Since, the spectral measure of has a continuous density function which can be represented as the inverse Fourier transform of:


We would like to prove that f has the following asymptotic property


where is an explicit constant depending only on H.

In the following we give a direct proof of (10) by using (9) and an Abelian argument similar to that in the proof of Theorem 1 of [16] . Without loss of generality, we assume that. Applying integration-by-parts to (9), we get




We need to distinguish three cases:, and. In the first case, it can be verified from (12) that, hence, and


We will also make use of the properties of higher order derivatives of. It is elementary to compute and verify that, when, we have


and as which implies.

The behavior of the derivatives of is simpler when. (12) becomes




Hence, we have, , and both and are in.

When, it can be shown that (14) still holds, and as.

Now, we proceed to prove (10). First, we consider the case when. By a change of variable, we can write




Let be a fixed constant. It follows from (13) and the dominated convergence theorem that


On the other hand, integration-by-parts yields


By Riemann-Lebesgue lemma,


Moreover, since by (13) and as, we have as. It follows that


Then for all large enough, we derive


Hence, we have


Combining (18), (19), and (24), we have


Then we see that, when, (10) holds with.

Secondly, we consider the case. Since is continuous and, (19) becomes


Using (20) and integration-by-parts again we derive


It follows from the (27), (16) and Riemann-Lebesgue lemma that


We see from the above and (17) that


This verifies that (10) holds when.

Finally we consider the case. Note that (19) and (24) are not useful anymore and we need to modify the above argument. By using integration-by-parts to (11) we obtain


Note that we have. Hence is integrable in the neighborhood of. Consequently, the proof for this case is very similar to the case of. From (30) and (14), we can verify that (10) holds as well and the constant is explicitly determined by H. Hence we have proved (10) in general.

It follows from (10) and Lemma 1 of [17] (see also [12] for more general results) that is strongly locally -nondeterministic on any interval with in the following sense: There exist positive constants and such that for all and all,


Now we prove the strong local nondeterminism of on I. To this end, note that for all. We choose. Then for all with we have


Hence, it follows from (31) and (32) that for all and,


where. This proves Theorem 1.


Supported by NSFC (No. 11201068) and “The Fundamental Research Funds for the Central Universities” in UIBE (No. 14YQ07).

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Kolmogorov, A.N. (1940) Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. C.R. (Doklady) Acad. Sci. URSS (N.S.), 26, 115-118.
[2] Mandelbrot, B. and van Ness, J.W. (1968) Fractional Brownian Motions, Fractional Noises and Applications. SIAM Review, 10, 422-437.
[3] Samorodnitsky, G. and Taqqu, M.S. (1994) Stable Non-Gaussian Random Processes. Stochastic Models with Infinite Variance. Stochastic Modeling. Chapman & Hall, New York.
[4] Anh, V.V., Angulo, J.M. and Ruiz-Medina, M.D. (1999) Possible Long-Range Dependence in Fractional Random Fields. Journal of Statistical Planning and Inference, 80, 95-110.
[5] Benassi, A., Bertrand, P., Cohen, S. and Istas, J. (2000) Identification of the Hurst Index of a Step Fractional Brownian Motion. Statistical Inference for Stochastic Processes, 3, 101-111.
[6] Benson, D.A., Meerschaert, M.M. and Baeumer, B. (2006) Aquifer Operator-Scaling and the Effect on Solute Mixing and Dispersion. Water Resources Research, 42, W01415.
[7] Michta, M. (2011) On Set-Valued Stochastic Integrals and Fuzzy Stochastic Equations. Fuzzy Sets Systems, 177, 1-19.
[8] Cheridito, P., Kawaguchi, H. and Maejima, M. (2003) Fractional Ornstein-Uhlenbeck Processes. Electronic Journal of Probability, 8, 14 p.
[9] Mannersalo, P. and Norros, I. (2002) A Most Probable Path Approach to Queueing Systems with General Gaussian Input. Computer Network, 40, 399-412.
[10] Tudor, C.A. and Xiao, Y. (2007) Sample Path Properties of Bifractional Brownian Motion. Bernoulli, 13, 1023-1052.
[11] Lamperti, J. (1962) Semi-Stable Stochastic Processes. Transactions of the American Mathematical Society, 104, 62-78.
[12] Xiao, Y. (2007) Strong Local Non-Determinism of Gaussian Random Fields and Its Applications. In: Lai, T.-L., Shao, Q.-M. and Qian, L., Eds., Asymptotic Theory in Probability and Statistics with Applications, Higher Education Press, Beijing, 136-176.
[13] Bojdecki, T., Gorostiza, L.G. and Talarczyk, A. (2004) Sub-Fractional Brownian Motion and Its Relation to Occupation Times. Statistics and Probability Letters, 69, 405-419.
[14] Dzhaparidze, K. and Van Zanten, H. (2004) A Series Expansion of Fractional Brownian Motion. Probability Theory and Related Fields, 103, 39-55.
[15] Tudor, C. (2007) Some Properties of the Sub-Fractional Brownian Motion. Stochastics, 79, 431-448.
[16] Pitman, E.J.G. (1968) On the Behavior of the Characteristic Function of a Probability Distribution in the Neighborhood of the Origin. Journal of the Australian Mathematical Society, 8 423-443.
[17] Cuzick, J. and Du Preez, J.P. (1982) Joint Continuity of Gaussian Local Times. Annals of Probability, 10, 810-817.

Copyright © 2024 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.