Remarks on Confidence Intervals for Self-Similarity Parameter of a Subfractional Brownian Motion

Abstract

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. [1] and Nualart [2] 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 𝑆^𝐻={𝑆^𝐻_𝑡,𝑡≥0} with 𝑆^𝐻₀=0 and the covariance 𝑅(𝑡,𝑠)≡𝐸𝑆^𝐻_𝑡𝑆^𝐻_𝑠=𝑠^2𝐻+𝑡^2𝐻−12(𝑠+𝑡)^2𝐻+|𝑡−𝑠|^2𝐻, (1.1) for all 𝑠,𝑡≥0. For 𝐻=1/2, 𝑆^𝐻 coincides with the standard Brownian motion. 𝑆^𝐻 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 𝑆^𝐻. The sub-fBm has properties analogous to those of fBm (self-similarity, long-range dependence, Hölder paths) and, for 𝑠≤𝑡, satisfies the following estimates: 2−2^2𝐻−1∧1(𝑡−𝑠)^2𝐻≤𝐸𝑆^𝐻_𝑡−𝑆^𝐻_𝑠²≤2−2^2𝐻−1∨1(𝑡−𝑠)^2𝐻. (1.2) 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 [7], Tudor [8–12], 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 [7], Tudor and Viens [17, 18], and references therein for more details. Recently, Breton et al. [19] 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. [3] 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 𝑉_𝑛(𝑆^𝐻) which is defined by 𝑉_𝑛𝑆^𝐻=^𝑛−1_𝑘=0𝑆^𝐻_{()𝑘+2/𝑛}+𝑆^𝐻_𝑘/𝑛−2𝑆^𝐻_{()𝑘+1/𝑛}²,𝑛≥1, (1.3) 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 [20] in order to derive an exact (i.e., nonasymptotic) confidence interval for the self-similar parameter of subfractional Brownian motion 𝑆^𝐻. Our formula hinges on the class of statistics 𝑉_𝑛(𝑆^𝐻) and 𝑍_𝑛=𝑛^2𝐻−1/2𝑉_𝑛−√𝑛(4−2^2𝐻).

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={𝑋_𝑘∶𝑘=0,…,𝑀} and write 𝑟(𝑘,𝑙)=𝐸(𝑋_𝑘𝑋_𝑙). 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)=𝑐^𝑀_𝑘=0𝑋²_𝑘−𝑟(𝑘,𝑘); (2.1) the second quadratic form is 𝑄₂(𝑐,𝑋)=2𝑐^2𝑀_𝑘,𝑙=0𝑋_𝑘𝑋_𝑙𝑟(𝑘,𝑙). (2.2)

The following result, whose proof relies on the Malliavin calculus techniques developed in Nourdin and Peccati [21], Nourdin and Viens [20],characterizes the tail behavior of 𝑄₁(𝑐,𝑋).

Theorem 2.1 (Theorem  2.1 in Breton et al. [19]). 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 𝑃𝑄₁()−𝑧𝑐,𝑋≥𝑧≤exp²𝑄2𝛼𝑧+2𝛽,𝑃₁()−𝑧𝑐,𝑋≤−𝑧≤exp²,2𝛽 (2.3) in particular, 𝑃𝑄₁(−𝑧𝑐,𝑋)≥𝑧≤2exp².2𝛼𝑧+2𝛽 (2.4)

On the other hand, to be sure that the second-order quadratic variation 𝑉_𝑛(𝑆^𝐻) converges almost surely to a deterministic limit, we need to normalize this quantity. A result of the form lim_𝑛→∞𝑛^1−𝜃𝑉_𝑛𝑆^𝐻=¹₀𝑓(𝑢)𝑑𝑢,a.s. (2.5) is expected, where 𝜃 is related to the regularity of the paths of the subfractional Brownian motion 𝑆^𝐻 and 𝑓 is related to the nondifferentiability of 𝑟 on the diagonal {𝑠=𝑡} and is called the singularity function of the process. Begyn [22] 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 [22] 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: 𝛿^ℎ₁𝑅()()()(),𝛿𝑠,𝑡=𝑅𝑠+ℎ,𝑡+𝑅𝑠−ℎ,𝑡−2𝑅𝑠,𝑡_ℎ2𝑅(𝑠,𝑡)=𝑅(𝑠,𝑡+ℎ)+𝑅(𝑠,𝑡−ℎ)−2𝑅(𝑠,𝑡). (2.6) First, we recall the result of asymptotic expansion of 𝑉_𝑛(𝑋) under some certain conditions on the covariance function.

Theorem 2.2 (Theorem  1 in Begyn [22]). Assume that the Gaussian process 𝑋 satisfies the following statements.(1)𝑡→𝑀_𝑡=𝔼𝑋_𝑡 has a bounded first derivative in [0,1].(2)The covariance function 𝑅 of 𝑋 has the following properties: (a) 𝑅 is continuous in [0,1]². (b) The derivative 𝜕⁴𝑅/𝜕𝑠²𝜕𝑡² 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∈0,1²{},𝜕𝑠=𝑡⁴𝑅𝜕𝑠²𝜕𝑡²(𝐿𝑠,𝑡)≤𝐶(|𝑠−𝑡|)|𝑠−𝑡|^2+𝛾. (2.7) (c) There exist 𝑞+1 functions (𝑞∈ℕ)𝑔₀,𝑔₁,…,𝑔_𝑞 from (0,1) to ℝ, 𝑞 real numbers 0<𝜈₁<⋯<𝜈_𝑞 and a function 𝜙∶(0,1)→(0,∞) such that (i) if 𝑞≥1, then ∀0≤𝑖≤𝑞−1, 𝑔_𝑖 is Lipschitz on (0,1); (ii) 𝑔_𝑞 is bounded on (0,1); (iii) one has sup_{ℎ≤𝑡≤1−ℎ}𝛿^ℎ₁∘𝛿^ℎ₂𝑅(𝑡,𝑡)ℎ^2−𝛾𝐿(ℎ)−𝑔₀(𝑡)−^𝑞_𝑖=1𝑔_𝑖(𝑡)𝜙(ℎ)^𝜈𝑖=𝑜𝜙(ℎ)^𝜈𝑞,𝑎𝑠ℎ⟶0+, (2.8) where the symbol “∘” denotes the composition of functions and if 𝑞=0, then ∑^𝑞_𝑖=1𝑔_𝑖(𝑡)𝜙(ℎ)^𝜈𝑖=0 and 𝜙(ℎ)^𝜈𝑞=1; else if 𝑞≠0, then lim_ℎ→0+𝜙(ℎ)=0.(3)If 𝑞≠0, we assume that lim_𝑛→+∞log𝑛𝑛𝜙(1/𝑛)^𝜈𝑞=0. (2.9)(4)If 𝑋 is not centered, we make the additional assumption lim_𝑛→+∞1𝑛^𝛾𝐿(1/𝑛)𝜙(1/𝑛)^𝜈𝑞=0, (2.10) where if 𝑞=0, then 𝜙(1/𝑛)^𝜈𝑞=1.
Then, for all 𝑡∈[0,1], one has almost surely lim_𝑛→+∞𝑛^1−𝛾𝐿()𝑉1/𝑛^𝑛(𝑋)=¹₀𝑔₀(𝑥)𝑑𝑥+^𝑞_𝑖=1¹₀𝑔_𝑖(𝑥)𝜙1𝑑𝑥𝑛^𝜈𝑖𝜙1+𝑜𝑛^𝜈𝑞. (2.11)

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

Theorem 2.3 (Theorem  2 in Begyn [22]). Assume that the Gaussian process 𝑋 is centered and satisfies the following statements.(1)The covariance function 𝑅 of 𝑋 is continuous in [0,1]².(2)Let 𝑇={0≤𝑡≤𝑠≤1}. We assume that the derivative 𝜕⁴𝑅/𝜕𝑠²𝜕𝑡² exists in (0,1]²/{𝑠=𝑡} and that there exists a continuous function 𝐶∶𝑇→ℝ, a real 𝛾∈(0,2) and a positive slowly varying function 𝐿∶(0,1)→ℝ such that ∀𝑡,𝑠∈()𝑇,𝑠−𝑡^2+𝛾𝐿()𝜕𝑠−𝑡⁴𝑅𝜕𝑠²𝜕𝑡²(𝑠,𝑡)=𝐶(𝑠,𝑡), (2.12) where 𝑇 denotes the interior of 𝑇 (i.e., 𝑇={0<𝑠<𝑡<1}).(3)We assume that there exist 𝑞+1 functions (𝑞∈ℕ) 𝑔₀,𝑔₁,…,𝑔_𝑞 from (0,1) to ℝ, 𝑞 real numbers 0<𝜈₁<⋯<𝜈_𝑞 and a function 𝜙∶(0,1)→(0,∞) such that (a) if 𝑞≥1, then for all 0≤𝑖≤𝑞−1, 𝑔_𝑖 is Lipschitz on (0,1); (b) 𝑔_𝑞 is (1/2+𝛼_𝑞)-Hölderian on (0,1) with 0<𝛼_𝑞≤1/2; (c) there exists 𝑡∈(0,1) such that 𝑔₀(𝑡)≠0; (d) one has lim_ℎ→0+1√ℎsup_{ℎ≤𝑡≤1−ℎ}𝛿^ℎ₁∘𝛿^ℎ₂𝑅(𝑡,𝑡)ℎ^2−𝛾𝐿(ℎ)−𝑔₀(𝑡)−^𝑞_𝑖=1𝑔_𝑖(𝑡)𝜙(ℎ)^𝜈𝑖=0, (2.13) where if 𝑞=0, then ∑^𝑞_𝑖=1𝑔_𝑖(𝑡)𝜙(ℎ)^𝜈𝑖=0 and where if 𝑞≠0, then lim_ℎ→0+𝜙(ℎ)=0. (e) there exists a bounded function 𝑔∶(0,1)→ℝ such that lim_ℎ→0+sup_{ℎ≤𝑡≤1−2ℎ}𝛿^ℎ₁∘𝛿^ℎ₂𝑅(𝑡+ℎ,𝑡)ℎ^2−𝛾𝐿(ℎ)−𝑔(𝑡)=0. (2.14)
Then one has √𝑛𝑛^1−𝛾𝐿()𝑉1/𝑛^𝑛(𝑋)−¹₀𝑔₀(𝑥)𝑑𝑥−^𝑞_𝑖=1¹₀𝑔_𝑖(𝑥)𝜙1𝑑𝑥𝑛^𝜈𝑖⟶𝒩0,𝜎² (2.15) in distribution as 𝑛 tends to infinity where 𝜎²=2¹₀𝑔₀(𝑥)²𝑑𝑥+4¹₀𝑔(𝑥)²𝑑𝑥+4𝜌_𝛾²¹₀𝐶(𝑥,𝑥)²𝑑𝑥 (2.16) and ‖𝜌_𝛾‖²=∑^+∞_𝑙=2𝜌_𝛾(𝑙)² with, if 𝛾≠1, 𝜌_𝛾(𝑙)=|𝑙−2|^2−𝛾−4|𝑙−1|^2−𝛾+6|𝑙|^2−𝛾−4|𝑙+1|^2−𝛾+|𝑙+2|^2−𝛾𝛾;𝛾−2𝛾−1𝛾+1 (2.17) 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|. (2.18)

3. Asymptotic Expansion and Central Limit Theorem

In the following theorem the almost sure convergence of the second-order quadratic variations 𝑉_𝑛(𝑆^𝐻) is proved.

Theorem 3.1. For all 𝑡∈[0,1], one has almost surely lim_𝑛→∞𝑛^2𝐻−1𝑉_𝑛𝑆^𝐻=4−2^2𝐻. (3.1)

Proof. It is clear that the derivative (𝜕⁴/𝜕𝑠²𝜕𝑡²)𝑅(𝑠,𝑡) exists on (0,1]²/{𝑠=𝑡}. Moreover we can check that, for all 𝑠,𝑡∈(0,1]²/{𝑠=𝑡}, 𝜕⁴𝜕𝑠²𝜕𝑡²𝑅(𝑠,𝑡)=𝐻(2𝐻−1)(2𝐻−2)(2𝐻−3)−|𝑠−𝑡|^2𝐻−4−|𝑠+𝑡|^2𝐻−4. (3.2) 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 𝛿^ℎ₁∘𝛿^ℎ₂𝑅(𝑡,𝑡)ℎ^𝐻=4−2^2𝐻+𝜆_𝑡(ℎ)ℎ^2𝐻, (3.3) with 𝜆_𝑡(ℎ)=1ℎ^2𝐻−2^2𝐻−1(𝑡+ℎ)^2𝐻−3⋅2^2𝐻𝑡^2𝐻−2^2𝐻−1(𝑡−ℎ)^2𝐻+2(2𝑡+ℎ)^2𝐻+2(2𝑡−ℎ)^2𝐻, (3.4) and we can check 𝜆_𝑡(0)=𝜆^′_𝑡(0)=𝜆_𝑡^′′(0)=𝜆_𝑡⁽³⁾(0)=0. So that Taylor formula yields 𝜆_𝑡(ℎ)=^ℎ₀(ℎ−𝑥)³𝜆3!^𝑡(4)(𝑥)𝑑𝑥,∀ℎ≤𝑡≤1−ℎ. (3.5) Therefore, we have sup_{ℎ≤𝑡≤1−ℎ}sup_{0≤𝑥≤ℎ}𝜆_𝑡⁽⁴⁾(𝑥)=𝒪(1),asℎ⟶0+, (3.6) which yields sup_{ℎ≤𝑡≤1−ℎ}𝛿^ℎ₁∘𝛿^ℎ₂𝑅(𝑡,𝑡)ℎ^𝐻−4−2^2𝐻=𝒪ℎ^4−2𝐻,asℎ⟶0+. (3.7)
Therefore, the assumption 2(c) in Theorem 2.2 is fulfilled with 𝑔₀(𝑡)=4−2^2𝐻. (3.8) Consequently, we can apply Theorem 2.2 to 𝑉_𝑛(𝑆^𝐻) and obtain the desired result.

Next we study the weak convergence.

Theorem 3.2. One has the following weak convergence √𝑛𝑛^2𝐻−1𝑉_𝑛𝑆^𝐻−4−2^2𝐻⟶_(ℒ)𝑁0,𝜎²_𝐻,as𝑛⟶∞, (3.9) where 𝜎²_𝐻=24−2^2𝐻2+2^2𝐻+2−7−3^2𝐻2+[2𝐻(2𝐻−1)(2𝐻−2)(2𝐻−3)]²𝜌_2−2𝐻²,𝜌_2−2𝐻²=^∞_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. (3.10)

Proof. We apply Theorem 2.3 to 𝑉_𝑛(𝑆^𝐻). 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}, 𝜕⁴𝜕𝑠²𝜕𝑡²𝑅(𝑠,𝑡)=𝐻(2𝐻−1)(2𝐻−2)(2𝐻−3)−|𝑠−𝑡|^2𝐻−4−|𝑠+𝑡|^2𝐻−4. (3.11) Therefore (𝑠−𝑡)^4−2𝐻𝜕⁴𝜕𝑠²𝜕𝑡²𝑅(𝑠,𝑡)=−𝐻(2𝐻−1)(2𝐻−2)⋅(2𝐻−3)1+(𝑠−𝑡)^4−2𝐻(𝑠+𝑡)^2𝐻−4∶≡𝐶(𝑠,𝑡). (3.12) 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)=4−2^2𝐻 and 𝛼₀=1/2. Moreover, one can check that 𝛿^ℎ₁∘𝛿^ℎ₂𝑅(𝑡,𝑡+ℎ)ℎ^𝐻=122^2𝐻+2−3^2𝐻+𝜉−7^𝑡(ℎ)ℎ^2𝐻. (3.13) Using the same arguments as those used for 𝜆_𝑡(ℎ) in the previous proof, we obtain sup_{ℎ≤𝑡≤1−ℎ}𝜉_𝑡(ℎ)=𝒪ℎ⁴,asℎ⟶0+. (3.14) This shows that the assumption 3(e) in Theorem 2.3 is satisfied with (𝑡)=1𝑔22^2𝐻+2−3^2𝐻−7. (3.15) Consequently, we can apply Theorem 2.3 to 𝑉_𝑛(𝑆^𝐻) to obtain the desired result.

4. Confidence Intervals

Let 𝑆^𝐻 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 𝑉_𝑛(𝑆^𝐻) defined in (1.3), fix 𝑛≥0 and a real number 𝑎 such that 0<𝑎<(4−2^2𝐻∗)√𝑛. For 𝑥∈(0,1), set 𝑔_𝑛(𝑥)=𝑥−log(4−2^2𝑥)/2log𝑛. Then, with probability at least 𝜑(𝑎)=−𝑎1−2exp²4𝐶_𝐻∗𝑎/√𝑛+3+𝐶_𝐻∗/𝑛₊, (4.1) where 𝐶_𝐻∗ is a positive constant depending only on 𝐻_∗ and [⋅]₊ stands for the positive part function; the unknown quantity 𝑔_𝑛(𝐻) belongs to the following confidence intervals 𝐼_𝑛=𝐼_𝑙(𝑛),𝐼_𝑟(𝑛=1)2−log𝑉_𝑛+√2log𝑛log1−𝑎/𝑛4−2^2𝐻∗,12log𝑛2−log𝑉_𝑛+√2log𝑛log1+𝑎/𝑛4−2^2𝐻∗.2log𝑛 (4.2)

Proof. The idea used here is essentially due to Breton et al. [19]. Define 𝑋_𝑛={𝑋_𝑛,𝑘;𝑘=0,1,…,𝑛−1}, where 𝑋_𝑛,𝑘=𝑆^𝐻_{()𝑘+2/𝑛}+𝑆^𝐻_𝑘/𝑛−2𝑆^𝐻_{()𝑘+1/𝑛}. (4.3) One can prove by standard computations that the covariance structure of Gaussian family 𝑋_𝑛,𝑘 is described by the relation 𝐸𝑋_𝑛,𝑘𝑋_𝑛,𝑙=1𝑛^2𝐻𝜌_𝐻(𝑘,𝑙), (4.4) where 𝜌_𝐻()1𝑘,𝑙=−2(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. (4.5) Now let 𝑍_𝑛=𝑛^2𝐻−1/2𝑉_𝑛−√𝑛(4−2^2𝐻), where 𝑉_𝑛 is defined in (1.3). It is easy to see that 𝑍_𝑛=𝑄₁𝑛^2𝐻−1/2,𝑋_𝑛+1√𝑛^𝑛−1_𝑘=0𝑎_𝑘,𝐻, (4.6) where 𝑎_𝑘,𝐻1=−2(2𝑘+4)^2𝐻+2(2𝑘+3)^2𝐻−3(2𝑘+2)^2𝐻+2(2𝑘+1)^2𝐻−12(2𝑘)^2𝐻. (4.7) On the other hand 𝑄₂𝑛^2𝐻−1/2,𝑋_𝑛=2𝑛^{4𝐻−1𝑛−1}_𝑘,𝑙=0𝑋_𝑛,𝑘𝑋_𝑛,𝑙𝜌_𝐻(𝑘,𝑙)≤2𝑛^{2𝐻−1𝑛−1}_𝑘,𝑙=0𝑋_𝑛,𝑘𝑋_𝑛,𝑙𝜌_𝐻(𝑘,𝑙)≤𝑛^{2𝐻−1𝑛−1}_𝑘,𝑙=0𝑋_𝑛,𝑘²+𝑋_𝑛,𝑙²𝜌_𝐻(𝑘,𝑙)=2𝑛^{2𝐻−1𝑛−1}_𝑘,𝑙=0𝑋_𝑛,𝑘²𝜌_𝐻(𝑘,𝑙)≤2𝑛^{2𝐻−1𝑛−1}_𝑘=0𝑋_𝑛,𝑘²_{𝑖,𝑗∈ℤ}𝜌_𝐻≤2𝑖,𝑗√𝑛_{𝑖,𝑗∈ℤ}𝜌_𝐻𝑖,𝑗𝑍_𝑛+4−2^2𝐻√𝑛≤2√𝑛_{𝑖,𝑗∈ℤ}𝜌_𝐻𝑖,𝑗𝑍_𝑛+3√𝑛=2√𝑛_{𝑖,𝑗∈ℤ}𝜌_𝐻𝑖,𝑗𝑄₁𝑛^2𝐻−1/2,𝑋_𝑛+1√𝑛^𝑛−1_𝑘=0𝑎_𝑘,𝐻+3√𝑛≤2√𝑛_{𝑖,𝑗∈ℤ}𝜌_𝐻𝑖,𝑗𝑄₁𝑛^2𝐻−1/2,𝑋_𝑛+1√𝑛^𝑛−1_𝑘=0𝑎_𝑘,𝐻+3√𝑛=𝛼_𝑛𝑄₁𝑛^2𝐻−1/2,𝑋_𝑛+𝛽_𝑛, (4.8) with 𝛼_𝑛=2√𝑛_{𝑖,𝑗∈ℤ}𝜌_𝐻𝑖,𝑗,𝛽=2_{𝑖,𝑗∈ℤ}𝜌_𝐻1𝑖,𝑗3+𝑛^𝑛−1_𝑘=0𝑎_𝑘,𝐻. (4.9) Since 𝑍_𝑛≠0, Theorem 2.1 yields 𝑃𝑍_𝑛−𝑎≥𝑎≤2exp²4∑_{𝑖,𝑗∈ℤ}𝜌_𝐻𝑖,𝑗𝑎/√𝑛+3+(1/𝑛)∑^𝑛−1_𝑘=0𝑎_𝑘,𝐻. (4.10) Now let us find bounds on ∑_{𝑖,𝑗∈ℤ}|𝜌_𝐻(𝑖,𝑗)|. Using (1+𝑥)^𝛼=1+^∞_𝑘=1𝛼(𝛼−1)⋯(𝛼−𝑘+1)𝑥𝑘!^𝑘,for−1<𝑥<1. (4.11) We denote by 𝜌_𝐻𝑖,𝑗=𝜌_𝐻,1𝑖,𝑗+𝜌_𝐻,2𝑖,𝑗, (4.12) where 𝜌_𝐻,11𝑖,𝑗=−2(𝑘+𝑙+4)^2𝐻+2(𝑘+𝑙+3)^2𝐻−3(𝑘+𝑙+2)^2𝐻+2(𝑘+𝑙+1)^2𝐻−12(𝑘+𝑙)^2𝐻,𝜌_𝐻,2𝑖,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. (4.13) The second term 𝜌_𝐻,2(𝑖,𝑗) has been bounded by Breton et al. [19]. They proved that _{𝑖,𝑗∈ℤ}𝜌_𝐻,2𝑖,𝑗≤714. (4.14) Now let us bound the first term 𝜌_𝐻,1(𝑖,𝑗). We denote by 𝜌_𝐻,1𝑖,𝑗=𝜌_𝐻,1(𝑟),𝑟=𝑖+𝑗. (4.15) We can write for any 𝑟≥5, 𝜌_𝐻,1(𝑟)=𝑟^2𝐻2−41+𝑟^2𝐻3+41+𝑟^2𝐻2−61+𝑟^2𝐻1+41+𝑟^2𝐻=𝑟−1_2𝐻2^∞_𝑘=12𝐻(2𝐻−1)⋯(2𝐻−𝑘+1)𝑘!−4^𝑘+4⋅3^𝑘−6⋅2^𝑘+4𝑟^−𝑘. (4.16) 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|<2⋅1⋅2⋯(𝑘−1)=2(𝑘−1)!. (4.17) Hence we can write, for any 𝑟≥5, 𝜌_𝐻,1(𝑟≤𝑟)^2𝐻2^∞_𝑘=11𝑘−4^𝑘+4⋅3^𝑘−6⋅2^𝑘+4𝑟^−𝑘=𝑟^2𝐻2^∞_𝑘=11𝑘−4𝑟^𝑘3+4⋅𝑟^𝑘2−6⋅𝑟^𝑘1+4𝑟^𝑘=𝑟^2𝐻24log1−𝑟3−4log1−𝑟2+6log1−𝑟1−4log1−𝑟≤𝑟^2𝐻24log1−𝑟1−4log1−𝑟+3−4log1−𝑟2+6log1−𝑟. (4.18) One can easily check that |log(1−4𝑥)−4log(1−𝑥)|≤(243/20)𝑥², if 0≤𝑥≤1/5. And moreover, −4log(1−3𝑥)+6log(1−2𝑥)≤log(1−4𝑥)−4log(1−𝑥). (4.19) Then we have, for any 𝑟≥5, 𝜌_𝐻,1(𝑟)≤243𝑟20^{−3/2+𝐻∗}. (4.20) Consequently, we get _𝑟∈ℤ𝜌_𝐻,1(𝑟)≤𝜌_𝐻,1(0)+𝜌_𝐻,1(1)+𝜌_𝐻,1(2)+𝜌_𝐻,1(3)+𝜌_𝐻,1(4)+_𝑟≥5𝜌_𝐻,1(𝑟=1)2−4^2𝐻+4⋅3^2𝐻−6⋅2^2𝐻+1+42−5^2𝐻+4⋅4^2𝐻−6⋅3^2𝐻+4⋅2^2𝐻+1−12−6^2𝐻+4⋅5^2𝐻−6⋅4^2𝐻+4⋅3^2𝐻−2^2𝐻+12−7^2𝐻+4⋅6^2𝐻−6⋅5^2𝐻+4⋅4^2𝐻−3^2𝐻+12−8^2𝐻+4⋅7^2𝐻−6⋅6^2𝐻+4⋅5^2𝐻−4^2𝐻+_𝑟≥5𝜌_𝐻,1(𝑟)≤𝐶₁+𝐶_𝐻∗∶≡𝐶_𝐻∗<∞, (4.21) and the positive constant 𝐶_𝐻∗ does not depend on the unknown parameter 𝐻. Putting this bound in (4.10) yields 𝑃𝑍_𝑛−𝑎≥𝑎≤2exp²4𝐶_𝐻∗𝑎/√𝑛+3+𝐶_𝐻∗./𝑛 (4.22) Now we can construct the confidence interval for 𝑔_𝑛(𝐻)=𝐻−log(4−2^2𝐻)/2log𝑛. First observe that 𝑍_𝑛=𝑛^2𝐻−1/2𝑉_𝑛−√𝑛(4−2^2𝐻). Using the assumption 𝐻≤𝐻_∗<1/2 on the one hand and (4.22) on the other hand, we get 𝑃12−log𝑉_𝑛+2log𝑛log1−𝑎/4−2^2𝐻∗√𝑛2log𝑛≤𝑔_𝑛(𝐻)≤12−log𝑉_𝑛+2log𝑛log1+𝑎/4−2^2𝐻∗√𝑛12log𝑛≥𝑃2−log𝑉_𝑛+2log𝑛log1−𝑎/4−2^2𝐻√𝑛2log𝑛≤𝐻−log4−2^2𝐻≤12log𝑛2−log𝑉_𝑛+2log𝑛log1+𝑎/4−2^2𝐻√𝑛12log𝑛=𝑃4−log𝑉_𝑛+2log𝑛log4−2^2𝐻√𝑛−𝑎12log𝑛≤𝐻≤4−log𝑉_𝑛+2log𝑛log4−2^2𝐻√𝑛+𝑎𝑍2log𝑛=𝑃_𝑛−𝑎≤𝑎≥1−2exp²4𝐶_𝐻∗𝑎/√𝑛+3+𝐶_𝐻∗,/𝑛 (4.23) where ∑_{𝑖,𝑗∈ℤ}|𝜌(𝑖,𝑗)|≤𝐶_𝐻∗ and the positive constant 𝐶_𝐻∗ 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).