We first present two convergence results about the second-order quadratic variations of the subfractional Brownian motion: the first is a deterministic asymptotic expansion; the second is a central limit theorem. Next we combine these results and concentration inequalities to build confidence intervals for the self-similarity parameter associated with one-dimensional subfractional Brownian motion.

1. Introduction

A fundamental assumption in many statistical and stochastic models is that of independent observations. Moreover, many models that do not make the assumption have the convenient Markov property, according to which the future of the system is not affected by its previous states but only by the current one.

The long-range dependence property has become an important aspect of stochastic models in various scientific areas including hydrology, telecommunication, turbulence, image processing, and finance. The best known and most widely used process that exhibits the long-range dependence property is fractional Brownian motion (fBm in short). The fBm is a suitable generalization of the standard Brownian motion. The reader is referred, for example, to Alòs et al.  and Nualart  for a comprehensive introduction to fractional Brownian motion. On the other hand, many authors have proposed to use more general self-similar Gaussian processes and random fields as stochastic models. Such applications have raised many interesting theoretical questions about self-similar Gaussian processes.

As a generalization of Brownian motion, recently, Bojdecki et al. [3, 4] introduced and studied a rather special class of self-similar Gaussian processes which preserves many properties of the fractional Brownian motion. This process arises from occupation time fluctuations of branching particle systems with Poisson initial condition, which is called the subfractional Brownian motion. The so-called subfractional Brownian motion (sub-fBm in short) with index H(0,1) is a mean zero Gaussian process SH={StH,t0} with S0H=0 and the covariance R(t,s)E[StHSsH]=s2H+t2H-12[(s+t)2H+|t-s|2H], for all s,t0. For H=1/2, SH coincides with the standard Brownian motion. SH is neither a semimartingale nor a Markov process unless H=1/2, so many of the powerful techniques from stochastic analysis are not available when dealing with SH. The sub-fBm has properties analogous to those of fBm (self-similarity, long-range dependence, Hölder paths) and, for st, satisfies the following estimates: [(2-22H-1)1](t-s)2HE[(StH-SsH)2][(2-22H-1)1](t-s)2H. Thus, Kolmogorov continuity criterion implies that the subfractional Brownian motion is Hölder continuous of order ν for any ν<H. But its increments are not stationary. More works for sub-fBm can be found in Bojdecki et al. [3, 4], Liu and Yan [5, 6], Liu , Tudor , Yan and Shen [13, 14], and others.

The problem of the statistical estimation of the self-similarity parameter is of great importance. The self-similarity parameter characterizes all of the important properties of the self-similar processes and consequently describes the behavior of the underlying physical system. Therefore, properly estimating them is of the utmost importance. Several statistics have been introduced to this end, such as wavelets, k-variations, variograms, maximum likelihood estimators, and spectral methods. This issue has generated a vast literature. See Chronopoulou et al. [15, 16], Liu , Tudor and Viens [17, 18], and references therein for more details. Recently, Breton et al.  firstly obtained the nonasymptotic construction of confidence intervals for the Hurst parameter H of fractional Brownian motion. Observe that the knowledge of explicit nonasymptotic confidence intervals may be of great practical value, for instance in order to evaluate the accuracy of a given estimation of H when only a fixed number of observations are available.

Motivated by all these results, in the present note, we will construct the confidence intervals for the self-similarity parameter associated with the so-called subfractional Brownian motion. It is well known that, in contrast to the extensive studies on fractional Brownian motion, there has been little systematic investigation on other self-similar Gaussian processes. The main reasons are the complexity of dependence structures and the nonavailability of convenient stochastic integral representations for self-similar Gaussian processes which do not have stationary increments. As we know, in comparison with fractional Brownian motion, the subfractional Brownian motion has nonstationary increments, and the increments over nonoverlapping intervals are more weakly correlated and their covariance decays polynomially as a higher rate in comparison with fractional Brownian motion (for this reason in Bojdecki et al.  it is called subfractional Brownian motion). The above-mentioned properties make subfractional Brownian motion a possible candidate for models which involve long-range dependence, self-similarity, and nonstationary increments. Therefore, it seems interesting to construct the confidence intervals of self-similar parameter of subfractional Brownian motion. And we need more precise estimates to prove our results because of the nonstationary increments.

The first aim of this note is to prove a deterministic asymptotic expansion and a central limit theorem of the so-called second-order quadratic variation Vn(SH) which is defined by Vn(SH)=k=0n-1(S(k+2)/nH+Sk/nH-2S(k+1)/nH)2,n1, because the standard quadratic variation does not satisfy a central limit theorem in general. The second aim is to exploit the concentration inequality proved by Nourdin and Viens  in order to derive an exact (i.e., nonasymptotic) confidence interval for the self-similar parameter of subfractional Brownian motion SH. Our formula hinges on the class of statistics Vn(SH) and Zn=n2H-1/2Vn-n(4-22H).

This note is organized as follows. In Section 2 we present some preliminaries for concentration inequality and two convergence results about the quadratic variations of some Gaussian processes. In Section 3 we prove the asymptotic expansion and central limit theorem for the second-order quadratic variations of subfractional Brownian motion with H(0,1). In Section 4 we state and prove the main result of this note.

Notation. Most of the estimates of this paper contain unspecified constants. An unspecified positive and finite constant will be denoted by C or c, which may not be the same in each occurrence. Sometimes we will emphasize the dependence of these constants upon parameters.

2. Preliminaries

Consider a finite centered Gaussian family X={Xk:k=0,,M} and write r(k,l)=E(XkXl). In what follows, we will consider two quadratic forms associated with X and with some real coefficient c. The first is obtained by summing up the squares of the elements of X and by subtracting the corresponding variances Q1(c,X)=ck=0M(Xk2-r(k,k)); the second quadratic form is Q2(c,X)=2c2k,l=0MXkXlr(k,l).

The following result, whose proof relies on the Malliavin calculus techniques developed in Nourdin and Peccati , Nourdin and Viens ,characterizes the tail behavior of Q1(c,X).

Theorem 2.1 (Theorem  2.1 in Breton et al. [<xref ref-type="bibr" rid="B5">19</xref>]).

If the above assumptions are satisfied, suppose that Q1(c,X) is not a.s. zero and fix α>0 and β>0. Assume that Q2(c,X)αQ1(c,X)+β, a.s.-P. Then, for all z>0, one has P(Q1(c,X)z)exp(-z22αz+2β),P(Q1(c,X)-z)exp(-z22β), in particular, P(|Q1(c,X)|z)2exp(-z22αz+2β).

On the other hand, to be sure that the second-order quadratic variation Vn(SH) converges almost surely to a deterministic limit, we need to normalize this quantity. A result of the form limnn1-θVn(SH)=01f(u)du,  a.s. is expected, where θ is related to the regularity of the paths of the subfractional Brownian motion SH and f is related to the nondifferentiability of r on the diagonal {s=t} and is called the singularity function of the process. Begyn  considered a class of processes for which a more general normalization is needed. Moreover, he presented a better result about the asymptotic expansion of the left hand of (2.5) and proved a central limit theorem. Because Theorems  1 and 2 in Begyn  are crucial in the proofs of Theorems 3.1 and 3.2, it is useful to recall the results.

We define the second-order increments of the covariance function R of a Gaussian process X as follows: δ1hR(s,t)=R(s+h,t)+R(s-h,t)-2R(s,t),δ2hR(s,t)=R(s,t+h)+R(s,t-h)-2R(s,t). First, we recall the result of asymptotic expansion of Vn(X) under some certain conditions on the covariance function.

Theorem 2.2 (Theorem  1 in Begyn [<xref ref-type="bibr" rid="B2">22</xref>]).

Assume that the Gaussian process X satisfies the following statements.

tMt=𝔼Xt has a bounded first derivative in [0,1].

The covariance function R of X has the following properties: (a)   R is continuous in [0,1]2. (b) The derivative 4R/s2t2 exists and is continuous in (0,1]2/{s=t}. There exists a constant C>0, a real γ(0,2) and a positive slowly varying function L:(0,1)(0,+) such that t,s(0,1]2{s=t},|4Rs2t2(s,t)|CL(|s-t|)|s-t|2+γ. (c) There exist q+1 functions (q)g0,g1,,gq from (0,1) to , q real numbers 0<ν1<<νq and a function ϕ:(0,1)(0,) such that (i) if q1, then 0iq-1, gi is Lipschitz on (0,1); (ii)   gq is bounded on (0,1); (iii) one has supht1-h|(δ1hδ2hR)(t,t)h2-γL(h)-g0(t)-i=1qgi(t)ϕ(h)νi|=o(ϕ(h)νq),as  h0+, where the symbol “” denotes the composition of functions and if q=0, then i=1qgi(t)ϕ(h)νi=0 and ϕ(h)νq=1; else if q0, then limh0+ϕ(h)=0.

If q0, we assume that limn+lognnϕ(1/n)νq=0.

If X is not centered, we make the additional assumption limn+1nγL(1/n)ϕ(1/n)νq=0, where if q=0, then ϕ(1/n)νq=1.

Then, for all t[0,1], one has almost surely limn+n1-γL(1/n)Vn(X)=01g0(x)dx+i=1q(01gi(x)dx)ϕ(1n)νi+o(ϕ(1n)νq).

Second, let us recall the result of central limit theorem.

Theorem 2.3 (Theorem  2 in Begyn [<xref ref-type="bibr" rid="B2">22</xref>]).

Assume that the Gaussian process X is centered and satisfies the following statements.

The covariance function R of X is continuous in [0,1]2.

Let T={0ts1}. We assume that the derivative 4R/s2t2 exists in (0,1]2/{s=t} and that there exists a continuous function C:T, a real γ(0,2) and a positive slowly varying function L:(0,1) such that t,sT¯,(s-t)2+γL(s-t)4Rs2t2(s,t)=C(s,t), where T¯ denotes the interior of T (i.e., T¯={0<s<t<1}).

We assume that there exist q+1 functions (q) g0,g1,,gq from (0,1) to , q real numbers 0<ν1<<νq and a function ϕ:(0,1)(0,) such that (a) if q1, then for all 0iq-1, gi is Lipschitz on (0,1); (b)   gq is (1/2+αq)-Hölderian on (0,1) with 0<αq1/2; (c) there exists t(0,1) such that g0(t)0; (d) one has limh0+1h(supht1-h|(δ1hδ2hR)(t,t)h2-γL(h)-g0(t)-i=1qgi(t)ϕ(h)νi|)=0, where if q=0, then i=1qgi(t)ϕ(h)νi=0 and where if q0, then limh0+ϕ(h)=0. (e) there exists a bounded function g̃:(0,1) such that limh0+supht1-2h|(δ1hδ2hR)(t+h,t)h2-γL(h)-g̃(t)|=0.

Then one has n(n1-γL(1/n)Vn(X)-01g0(x)dx-i=1q(01gi(x)dx)ϕ(1n)νi)N(0,σ2) in distribution as n tends to infinity where σ2=201g0(x)2dx+401g̃(x)2dx+4ργ201C(x,x)2dx and ργ2=l=2+ργ(l)2 with, if γ1, ργ(l)=|l-2|2-γ-4|l-1|2-γ+6|l|2-γ-4|l+1|2-γ+|l+2|2-γ(γ-2)(γ-1)γ(γ+1); if γ=1, ρ1(l)=12(|l-2|log|l-2|-4|l-1|log|l-1|+6|l|log|l|-4|l+1|log|l+1|+2|l+2|log|l+2|).

3. Asymptotic Expansion and Central Limit Theorem

In the following theorem the almost sure convergence of the second-order quadratic variations Vn(SH) is proved.

Theorem 3.1.

For all t[0,1], one has almost surely limnn2H-1Vn(SH)=4-22H.

Proof.

It is clear that the derivative (4/s2t2)R(s,t) exists on (0,1]2/{s=t}. Moreover we can check that, for all s,t(0,1]2/{s=t}, 4s2t2R(s,t)=H(2H-1)(2H-2)(2H-3)[-|s-t|2H-4-|s+t|2H-4]. Therefore the assumption 2(b) in Theorem 2.2 is satisfied with L(H)=1 and γ=2-2H.

For the assumption 2(b) in Theorem 2.2, standard computations yield (δ1hδ2hR)(t,t)hH=4-22H+λt(h)h2H, with λt(h)=1h2H[-22H-1(t+h)2H-322Ht2H-22H-1(t-h)2H+2(2t+h)2H+2(2t-h)2H], and we can check λt(0)=λt(0)=λt′′(0)=λt(3)(0)=0. So that Taylor formula yields λt(h)=0h(h-x)33!λt(4)(x)dx,ht1-h. Therefore, we have supht1-hsup0xh|λt(4)(x)|=O(1),as  h0+, which yields supht1-h|(δ1hδ2hR)(t,t)hH-(4-22H)|=O(h4-2H),as  h0+.

Therefore, the assumption 2(c) in Theorem 2.2 is fulfilled with g0(t)=4-22H. Consequently, we can apply Theorem 2.2 to Vn(SH) and obtain the desired result.

Next we study the weak convergence.

Theorem 3.2.

One has the following weak convergence n(n2H-1Vn(SH)-(4-22H))(L)N(0,σH2),as  n, where σH2=2(4-22H)2+(22H+2-7-32H)2+[2H(2H-1)(2H-2)(2H-3)]2ρ2-2H2,ρ2-2H2=l=2(-|l-2|2H+4|l-1|2H-6|l|2H+4|l+1|2H-|l+2|2H2H(2H-1)(2H-2)(2H-3))2.

Proof.

We apply Theorem 2.3 to Vn(SH). As in the proof of Theorem 3.1, we need only to show that the assumptions  2 and 3 in Theorem 2.3 are satisfied.

For assumption  2, the previous computation showed that, for all s,t(0,1]2/{s=t}, 4s2t2R(s,t)=H(2H-1)(2H-2)(2H-3)[-|s-t|2H-4-|s+t|2H-4]. Therefore (s-t)4-2H4s2t2R(s,t)=-H(2H-1)(2H-2)(2H-3)[1+(s-t)4-2H(s+t)2H-4]:C(s,t). This means that the assumption 2 in Theorem 2.3 is satisfied with L(H)=1, γ=2-2H and C(s,t) defined by (3.12).

For assumption 3 in Theorem 2.3, the expression (3.7) shows that the assumption 3 in Theorem 2.3 is fulfilled with q=0, g0(t)=4-22H and α0=1/2. Moreover, one can check that (δ1hδ2hR)(t,t+h)hH=12(22H+2-32H-7)+ξt(h)h2H. Using the same arguments as those used for λt(h) in the previous proof, we obtain supht1-h|ξt(h)|=O(h4),as  h0+. This shows that the assumption 3(e) in Theorem 2.3 is satisfied with g̃(t)=12(22H+2-32H-7). Consequently, we can apply Theorem 2.3 to Vn(SH) to obtain the desired result.

4. Confidence Intervals

Let SH is a subfractional Brownian motion with unknown Hurst parameter H(0,H*], with H*<1/2 known. The following result is the main finding of the present note.

Theorem 4.1.

For Vn(SH) defined in (1.3), fix n0 and a real number a such that 0<a<(4-22H*)n. For x(0,1), set gn(x)=x-log(4-22x)/2logn. Then, with probability at least φ(a)=[1-2exp(-a24CH*(a/n+3+CH*/n))]+, where CH* is a positive constant depending only on H* and [·]+ stands for the positive part function; the unknown quantity gn(H) belongs to the following confidence intervals In=[Il(n),Ir(n)]=[12-logVn2logn+log(1-a/n(4-22H*))2logn,12-logVn2logn+log(1+a/n(4-22H*))2logn].

Proof.

The idea used here is essentially due to Breton et al. . Define Xn={Xn,k;  k=0,1,,n-1}, where Xn,k=S(k+2)/nH+Sk/nH-2S(k+1)/nH. One can prove by standard computations that the covariance structure of Gaussian family Xn,k is described by the relation E(Xn,kXn,l)=1n2HρH(k,l), where ρH(k,l)=-12(k+l+4)2H+2(k+l+3)2H-3(k+l+2)2H+2(k+l+1)2H-12(k+l)2H-3(k-l)2H+2(k-l+1)2H-12(k-l+2)2H+2(k-l-1)2H-12(k-l-2)2H. Now let Zn=n2H-1/2Vn-n(4-22H), where Vn is defined in (1.3). It is easy to see that Zn=Q1(n2H-1/2,Xn)+1nk=0n-1ak,H, where ak,H=-12(2k+4)2H+2(2k+3)2H-3(2k+2)2H+2(2k+1)2H-12(2k)2H. On the other hand Q2(n2H-1/2,Xn)=2n4H-1k,l=0n-1Xn,kXn,lρH(k,l)2n2H-1k,l=0n-1|Xn,kXn,lρH(k,l)|n2H-1k,l=0n-1(|Xn,k|2+|Xn,l|2)|ρH(k,l)|=2n2H-1k,l=0n-1|Xn,k|2|ρH(k,l)|2n2H-1k=0n-1|Xn,k|2(i,jZ|ρH(i,j)|)2n(i,jZ|ρH(i,j)|)(Zn+(4-22H)n)2n(i,jZ|ρH(i,j)|)(Zn+3n)=2n(i,jZ|ρH(i,j)|)(Q1(n2H-1/2,Xn)+1nk=0n-1ak,H+3n)2n(i,jZ|ρH(i,j)|)(Q1(n2H-1/2,Xn)+1nk=0n-1|ak,H|+3n)=αnQ1(n2H-1/2,Xn)+βn, with αn=2ni,jZ|ρH(i,j)|,β=2i,jZ|ρH(i,j)|(3+1nk=0n-1|ak,H|). Since Zn0, Theorem 2.1 yields P(|Zn|a)2exp(-a24i,jZ|ρH(i,j)|(a/n+3+(1/n)k=0n-1|ak,H|)). Now let us find bounds on i,j|ρH(i,j)|. Using (1+x)α=1+k=1α(α-1)(α-k+1)k!xk,for  -1<x<1. We denote by ρH(i,j)=ρH,1(i,j)+ρH,2(i,j), where ρH,1(i,j)=-12(k+l+4)2H+2(k+l+3)2H-3(k+l+2)2H+2(k+l+1)2H-12(k+l)2H,ρH,2(i,j)=-3(k-l)2H+2(k-l+1)2H-12(k-l+2)2H+2(k-l-1)2H-12(k-l-2)2H. The second term ρH,2(i,j) has been bounded by Breton et al. . They proved that i,jZρH,2(i,j)714. Now let us bound the first term ρH,1(i,j). We denote by ρH,1(i,j)=ρH,1(r),r=i+j. We can write for any r5, ρH,1(r)=r2H2[-(1+4r)2H+4(1+3r)2H-6(1+2r)2H+4(1+1r)2H-1]=r2H2[k=12H(2H-1)(2H-k+1)k!(-4k+43k-62k+4)r-k]. Note that the sign of 2H(2H-1)(2H-k+1) is the same as that of (-1)k-1, and |2H(2H-1)(2H-k+1)|=2H|(2H-1)||2H-k+1|<212(k-1)=2(k-1)!. Hence we can write, for any r5, |ρH,1(r)|r2H2k=1|1k(-4k+43k-62k+4)r-k|=r2H2k=11k|-(4r)k+4(3r)k-6(2r)k+4(1r)k|=r2H2|log(1-4r)-4log(1-3r)+6log(1-2r)-4log(1-1r)|r2H2[|log(1-4r)-4log(1-1r)|+|-4log(1-3r)+6log(1-2r)|]. One can easily check that |log(1-4x)-4log(1-x)|(243/20)x2, if 0x1/5. And moreover, |-4log(1-3x)+6log(1-2x)||log(1-4x)-4log(1-x)|. Then we have, for any r5, |ρH,1(r)|24320r-3/2+H*. Consequently, we get rZ|ρH,1(r)||ρH,1(0)|+|ρH,1(1)|+|ρH,1(2)|+|ρH,1(3)|+|ρH,1(4)|+r5|ρH,1(r)|=12|-42H+432H-622H+4|+12|-52H+442H-632H+422H-1|+12|-62H+452H-642H+432H-22H|+12|-72H+462H-652H+442H-32H|+12|-82H+472H-662H+452H-42H|+r5|ρH,1(r)|C1+CH*:CH*<, and the positive constant CH* does not depend on the unknown parameter H. Putting this bound in (4.10) yields P(|Zn|a)2exp(-a24CH*(a/n+3+CH*/n)). Now we can construct the confidence interval for gn(H)=H-log(4-22H)/2logn. First observe that Zn=n2H-1/2Vn-n(4-22H). Using the assumption HH*<1/2 on the one hand and (4.22) on the other hand, we get P(12-logVn2logn+log(1-a/(4-22H*)n)2logngn(H)12-logVn2logn+log(1+a/(4-22H*)n)2logn)P(12-logVn2logn+log(1-a/(4-22H)n)2lognH-log(4-22H)2logn12-logVn2logn+log(1+a/(4-22H)n)2logn)=P(14-logVn2logn+log((4-22H)n-a)2lognH14-logVn2logn+log((4-22H)n+a)2logn)=P(|Zn|a)1-2exp(-a24CH*(a/n+3+CH*/n)), where i,j|ρ(i,j)|CH* and the positive constant CH* does not depend on the unknown parameter H. This is the desired result.

Acknowledgments

The authors want to thank the academic editor and anonymous referee whose remarks and suggestions greatly improved the presentation of this paper. The project is sponsored by NSFC (10871041), NSFC (81001288), NSRC (10023). Innovation Program of Shanghai Municipal Education Commission (12ZZ063) and NSF of Jiangsu Educational Committee (11KJD11002).

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