A law of iterated logarithm for the subfractional Brownian motion and an application

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S^{H}=\{S^{H}_{t},t\geq0\}$\end{document}SH={StH,t≥0} be a sub-fractional Brownian motion with Hurst index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$0< H<1$\end{document}0<H<1. In this paper, we give a local law of the iterated logarithm of the form \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\limsup_{s\downarrow0}\frac{ \vert S^{H}_{t+s}-S^{H}_{t} \vert }{ s^{H}\sqrt {2\log^{+}\log(1/s)}}=1, $$\end{document}lim sups↓0|St+sH−StH|sH2log+log(1/s)=1, almost surely, for all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t > 0$\end{document}t>0, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\log^{+}x=\max{\{1, \log x\}}$\end{document}log+x=max{1,logx} for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x\geq0$\end{document}x≥0. As an application, we introduce the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Phi_{H}$\end{document}ΦH-variation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$S^{H}$\end{document}SH driven by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Phi_{H}(x):= [x/\sqrt{2\log^{+}\log ^{+}(1/x)} ]^{1/H}$\end{document}ΦH(x):=[x/2log+log+(1/x)]1/H \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(x>0)$\end{document}(x>0) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Phi_{H}(0)=0$\end{document}ΦH(0)=0.


Introduction and main results
The quadratic variation and realized quadratic variation have been widely used in stochastic analysis and statistics of stochastic processes. The realized power variation of order p > 0 is a generalization of the quadratic variation, which is defined as where {X t , t > 0} is a stochastic process and κ = {0 = t 0 < t 1 < · · · < t n = t} is a partition of [0, t] with max 1≤i≤n {t it i-1 } → 0. It was introduced in Barndorff-Nielsen and Shephard [1,2] to estimate the integrated volatility in some stochastic volatility models used in quantitative finance and also, under an appropriate modification, to estimate the jumps of the processes under analysis. The main interest in these papers is the asymptotic behavior of the statistic (1.1), or some appropriate renormalized version of it, as n → ∞, when the process X t is a stochastic integral with respect to a Brownian motion. Refinements of their results have been obtained in Woerner [3]. A more general generalization to the realized quadratic variation is called -variation, and it is defined by where is a nonnegative, increasing continuous function on R + with (0) = 0. Let P ([0, t]) be a class of all partitions κ = {0 = t 0 < t 1 < · · · < t n = t} of [0, t] with |κ| := max 1≤i≤n {t it i-1 }. Then the -variation of a stochastic process {X t , t > 0} is defined as S (X, t) := lim sup δ→0 S (X, t, κ) : κ ∈ P [0, t] , |κ| < δ .

Consider the function
with H (0) = 0 and 0 < H < 1, where log + x = max{1, log x} for x > 0. When X is a standard Brownian motion B, Taylor [4] first considered the 1/2 -variation and proved S 1/2 (B, t) = t for all t > 0. Kawada and Kôno [5] extended this to some stationary Gaussian processes W and proved S 1/2 (W , t) = t for all t > 0 by using an estimate given by Kôno [6]. Recently, Dudley and Norvaiša [7] extended this to the fractional Brownian motion B H with Hurst index H ∈ (0, 1) and proved S H (B H , t) = t for all t > 0. More generally, for a bi-fractional Brownian motion B H,K , Norvaiša [8] showed that S H, On the other hand, since Chung's law and Strassen's functional law of the iterated logarithm appeared, the functional law of the iterated logarithm and its rates for some classes of Gaussian processes have been discussed by many authors (see, for example, Csörgö and Révész [9], Lin et al. [10], Dudley and Norvaiša [7], Malyarenko [11]). However, almost all results considered only some Gaussian processes with stationary increments, and there has been little systematic investigation on other self-similar Gaussian processes (see, for example, Norvaiša [8], Tudor and Xiao [12], and Yan et al. [13]). The main reason for this is the complexity of dependence structures for self-similar Gaussian processes which do not have stationary increments.
Motivated by these results, in this paper, we consider the law of the iterated logarithm and -variation of a sub-fractional Brownian motion. Recall that a mean-zero Gaussian process S H = {S H t , t ≥ 0} is said to be a sub-fractional Brownian motion (in short, sub-fBm) with Hurst index H ∈ (0, 1), if S H 0 = 0 and for all s, t > 0. When H = 1 2 , this process coincides with the standard Brownian motion B. Sub-fBm was first introduced by Bojdecki et al. [14] as an extension of Brownian motion, and it arises from occupation time fluctuations of branching particle systems with Poisson initial condition. A sub-fBm with Hurst index H is H-self-similar, Hölder continuous, and it is long/short-range dependent. A process X is long-range dependent if n≥α ρ n (α) = ∞ for any α > 0, and it is short-range dependent if n≥α ρ n (α) < ∞, where ρ n (α) = E[(X α+1 -X α )(X n+1 -X n )], α > 0. However, when H = 1 2 , it has no stationary increments. Moreover, it admits the following (quasi-helix) estimates: for all t, s ≥ 0. More works on sub-fractional Brownian motion can be found in Bojdecki et al. [15,16], Shen and Yan [17], Sun and Yan [18], Tudor [19,20], Yan et al. [21,22], and the references therein. For the above discussions, we find that the complexity of sub-fractional Brownian motion is very different from that of fractional Brownian motion or bi-fractional Brownian motion. Therefore, it seems interesting to study the iterated logarithm andvariation of sub-fractional Brownian motion. In the present paper, our main objectives are to expound and to prove the following theorems. for all T > 0.
As an immediate question driven by Theorem 1.2, one can consider the following asymptotic behavior: as δ tends to zero, where L denotes a distribution, φ(δ) ↑ ∞ (δ → 0), and S H (S H , T, δ) is defined as follows: We have known that when H = 1 2 , the sub-fBm S H coincides with a standard Brownian motion B. So, the two results above are some natural extensions to Brownian motion (see, for example, Csörgö and Révész [9], Dudley and Norvaivsa [7], Lin et al. [10]). This paper is organized as follows. In Sect. 2, we prove Theorem 1.1. In Sect. 3, we give the proof of Theorem 1.2.

Proof of Theorem 1.1
In this section and the next section, we prove our main results. When H = 1 2 , the sub-fBm S H is a standard Brownian motion, and Theorem 1.1 and Theorem 1.2 are given in Taylor [4]. In this section and the next section, we assume throughout that H = 1 2 .
Lemma 2.1 Let μ be a centered Gaussian measure in a linear space E, and let A ⊂ E be a symmetric convex set. Then we have for any h ∈ E.
Inequality (2.1) is called Anderson's inequality (see, for example, [23]). It admits the following version: Let X 1 , . . . , X n and Y 1 , . . . , Y n both be jointly Gaussian with mean zero and such that for any x > 0. We also will need the next tail probability estimate which is introduced (Lemma 12.18) in Dudley and Norvaiša [7]. [7]) Let B be a Banach space, and let S ⊂ B be a compact set such that cS ⊂ S for each c ∈ (0, 1]. Assume that S(δ 0 ) ⊂ S is closed for some 0 < δ 0 ≤ 1 and that

Lemma 2.3 (Dudley and Norvaiša
The above result is Lemma 12.20 in Dudley and Norvaiša [7].
By Kolmogorov's consistency theorem, we find that there is a mean-zero Gaussian pro- for all t, s ≥ 0.
as s ↓ 0, where the notation ∼ denotes the equivalence as s ↓ 0 for every fixed t > 0, and for all u, v ≥ 0, and t > 0.
Proof Clearly, we have for all s, t ≥ 0, where x = s s+t . An elementary calculus may show that as x → 0, which implies that estimate (2.5) holds.
Given t > 0. Consider the Gaussian process ζ H with the covariance ρ H defined by (2.4). Then we have To see that the inequality holds, we define the function on R 2 admit a unique solution (x, y) = (0, 0). Thus, we get for all u, v ≥ 0 and t > 0. Combining this with (1.3), we give estimate (2.6) and the lemma follows.

Lemma 2.6
For 0 < H < 1, we then have almost surely, for all t > 0.
Proof Let ε ∈ (0, 1) and t > 0. We see that = lim sup n→∞ X t r n 2 r n 2H log(-n log r) for every r ∈ (0, 1), by the fact log(-n log r) ∼ log n (n → ∞). Now, we verify that lim sup almost surely, for r ∈ (0, 1) small enough. In fact, by Lemma 2.3 we only need to prove for any ε ∈ (0, 1), where D n = {(k, m)|k, m ≥ n, k = m}. Some elementary calculations may show that the following inequalities hold: for any x ∈ (0, 1). It follows from Lemma 2.5 that there is a real r ∈ (0, 1) small enough such that for each k = m, which implies that (2.11) holds and (2.10) follows with probability one.
To prove Theorem 1.1, we now need to introduce the reverse inequality of (2.8), i.e., almost surely, for all t > 0. The used method is due to the decomposition (2.7), i.e., for all u, v ≥ 0 and t > 0. Recall that a mean-zero Gaussian process B H = {B H t , t ≥ 0} is said to be a fractional Brownian motion with Hurst index H ∈ (0, 1), if B H 0 = 0 and for all s, t > 0. When H = 1 2 , this process coincides with the standard Brownian motion B. Moreover, for all t > 0, the process {B H t+s -B H t , s ≥ 0} also is a fractional Brownian motion with Hurst index H ∈ (0, 1). It follows that for all u, v ≥ 0 and t > 0. More works on fractional Brownian motion can be found in Biagini et al. [24], Hu [25] and Mishura [26], Nourdin [27], and the references therein.
Therefore, by the Borel-Cantelli lemma, we have that almost surely. Noting that ϕ H is increasing, we see that for all t ≥ 0, where ξ t is a suitable Gaussian process. We can show that a similar limit theorem holds. However, for a different self-similar Gaussian process (weighted-fractional Brownian motion, bi-fractional Brownian motion, etc.) one needs to consider some concrete estimates.

Proof of Theorem 1.2
In order to prove Theorem 1.2, we first give a lemma which extends the related result for Brownian motion.
almost surely.
Finally, at the end of this paper, we give the proof of Theorem 1.2. We will use the local law of the iterated logarithm (Theorem 1.1) for S H and the Vitali covering lemma to introduce the next inequality and its reverse: for all T > 0, where H is defined in Sect. 1.
Proof of Theorem 1.2 Let H = 1 2 . We first show that inequality (3.3) holds. Given δ > 0. Let 0 < ε < 1 and Clearly, we have that there exists ξ > 0 such that for each 0 < v < ξ since H is regularly varying of order H -1 and is asymptotic to ϕ -1 H near zero. Therefore, by Theorem 1.1, for all t ∈ (0, T] and δ ∈ (0, ξ ), we have P({ω : (t, ω) ∈ E δ }) = 1. It follows from the Fubini theorem that P({m(E δ ) = T}) = 1 for each 0 < v < ξ , where m(·) denotes the Lebesgue measure on [0, T]. Clearly, the set of all intervals [t, t + s] with t ∈ [0, T] and arbitrarily small s > 0 is a Vitali covering of the set and P(E) = 1. According to the Vitali lemma, we can choose a finite sub-collection E δ of intervals of length less than δ which are disjoint and have total length at least Tε. Then almost surely, where κ = {t i , i = 0, 1, 2, . . . , n} ∈ P ([0, T]) with mesh |κ| ≤ δ such that for each of the disjoint intervals [t j , t j + s j ] from E δ with total length at least Tε, there is some i with t i-1 = t j and t i = t j + s j . Therefore, for each δ > 0 small enough, we obtain that with probability one. Let 1 be a subset of such that P( 1 ) = 1, and for every ω ∈ 1 , there exists δ 2 (ω) > 0 such that for all δ ≤ δ 2 (ω). We choose δ 2 (ω) ≤ η (A, ε). If a partition κ ∈ P ([0, T]) with |κ| ≤ δ 2 (ω) such that there exists an interval [t i-1 , t i ] contains a point of U δ 2 (ω) , then i ∈ I 1 . So, the total length of such intervals is at least Tε, and in particular, This shows that by (3.5) with c = A, provided |κ| ≤ δ 2 (ω).

Results, discussion, and conclusions
In this paper, we give an iterated logarithm and -variation for a sub-fBm by using some precise estimations and inequalities. It is important to note that the method used here is also applicative to many similar Gaussian processes.