Packing-type Measures of the Sample Paths of Fractional Brownian Motion期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Packing-type Measures of the Sample Paths of Fractional Brownian Motion

Authors:

Affiliation:

(1) Department of Applied Mathematics, Xidian University, Xi’an 710071, China;(2) School of Information and Mathematics, Yangtze University, Hubei 434104, China;(3) School of Mathematics and Computer Science, Central University for Nationalities, Beijing 100081, China

Abstract:

Abstract Let Λ = {λ_k} be an infinite increasing sequence of positive integers with λ_k→∞. Let X = {X(t), t ∈? R ^N} be a multi-parameter fractional Brownian motion of index α(0 < α < 1) in R ^d. Subject to certain hypotheses, we prove that if N < αd, then there exist positive finite constants K ₁ and K ₂ such that, with unit probability,

$K_{1} leqslant varphi - p wedge {left( {X{left( {{left[ {0,1} right]}} right)}^{N} } right)} leqslant varphi - p wedge {left( {GrX{left( {{left[ {0,1} right]}^{N} } right)}} right)} leqslant K_{2}$

if and only if there exists γ > 0 such that

${sumlimits_{k = 1}^infty {frac{1} {{lambda ^{gamma }_{k} }}} } = infty ,$

where ϕ(s) = s ^N/α(log log 1/s)^N/(2α), ϕ-p _Λ(E) is the Packing-type measure of E,X([0, 1])^N is the image and GrX([0, 1]^N) = {(t,X(t)); ∈? [0, 1]^N} is the graph of X, respectively. We also establish liminf type laws of the iterated logarithm for the sojourn measure of X. Supported by the National Natural Science Foundation of China (No.10471148), Sci-tech Innovation Item for Excellent Young and Middle-Aged University Teachers and Major Item of Educational Department of Hubei (No.2003A005)

Keywords:

KeywordHeading" > Fractional Brownian motion packing-type measure image graph law of iterated logarithm sojourn measure

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