Abstract
In this article, some properties of multifractional Brownian motion (MFBM) are discussed. It is shown that it has persistence of signs long range dependence (LRD) and persistence of magnitudes LRD properties. A generalization called here nth order multifractional Brownian motion (n-MFBM) that allows to take the functional parameter H(t) values in the range (n−1,n) is discussed. Two representations of the n-MFBM are given and their relationship with each other is obtained.
Highlights
We show that multifractional Brownian motion (MFBM)
To overcome these limitations MFBM was introduced in literature where pointwise irregularities represented by a Hölder continuous function
It is shown that nth order multifractional Brownian motion (n-MFBM) has long range dependence (LRD) property
Summary
(i) A second order process X (t) is said to have persistence of signs LRD property if for fixed s and large t, there exist functions c1 (s) and d1 (s) ∈ (0, 1) : Corr(signX (s), signX (t)) ≈ c1 (s)t−d1 (s) as t → ∞. If for all s, H (t) + H (s) > 1 for all sufficiently large t, by (19) the MFBM with functional parameter H has persistence of signs LRD in the sense of Definition 2, with functional. In case H (t) + H (s) < 1, MFBM may not have persistence of signs and persistence of magnitudes LRD properties when there exist a sequence {tn } tends to infinity s.t. H (tn ) +.
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