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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.

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

As a generalization of Brownian motion, recently, Bojdecki et al. [

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,

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. [

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

This note is organized as follows. In Section

Consider a finite centered Gaussian family

The following result, whose proof relies on the Malliavin calculus techniques developed in Nourdin and Peccati [

If the above assumptions are satisfied, suppose that

On the other hand, to be sure that the second-order quadratic variation

We define the second-order increments of the covariance function

Assume that the Gaussian process

The covariance function

If

If

Then, for all

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

Assume that the Gaussian process

The covariance function

Let

We assume that there exist

Then one has

In the following theorem the almost sure convergence of the second-order quadratic variations

For all

It is clear that the derivative

For the assumption 2(b) in Theorem

Therefore, the assumption 2(c) in Theorem

Next we study the weak convergence.

One has the following weak convergence

We apply Theorem

For assumption 2, the previous computation showed that, for all

For assumption 3 in Theorem

Let

For

The idea used here is essentially due to Breton et al. [

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).