# Asymptotic Normality of a Hurst Parameter Estimator Based on the Modified Allan Variance

Authors: Alessandra Bianchi; Massimo Campanino; Irene Crimaldi
Journal: International Journal of Stochastic Analysis
 (2012)

---

## Abstract

In order to estimate the memory parameter of Internet traffic data, it has been recently proposed a log-regression estimator based on the so-called modified Allan variance (MAVAR). Simulations have shown that this estimator achieves higher accuracy and better confidence when compared with other methods. In this paper we present a rigorous study of the MAVAR log-regression estimator. In particular, under the assumption that the signal process is a fractional Brownian motion, we prove that it is consistent and asymptotically normally distributed. Finally, we discuss its connection with the wavelets estimators.


---

## Body

## 1.Introduction

It is well known that different kinds of real data (hydrology, telecommunication networks, economics, and biology) display self-similarity and long-range dependence (LRD) on various time scales. By self-similarity we refer to the property that a dilated portion of a realization has the same statistical characterization as the original realization. This can be well represented by a self-similar random process with a given scaling exponentH (Hurst parameter). The long-range dependence, also called long memory, emphasizes the long-range time correlation between past and future observations and it is thus commonly equated to an asymptotic power law behaviour of the spectral density at low frequencies or, equivalently, to an asymptotic power-law decrease of the autocovariance function, of a given stationary random process. In this situation, the memory parameter of the process is given by the exponent d characterizing the power law of the spectral density. (For a review of historical and statistical aspects of the self-similarity and the long memory see [1].)Though a self-similar process cannot be stationary (and thus nor LRD), these two properties are often related in the following sense. Under the hypothesis that a self-similar process has stationary (or weakly stationary) increments, the scaling parameterH enters in the description of the spectral density and covariance function of the increments, providing an asymptotic power law with exponent d=2H-1. Under this assumption, we can say that the self-similarity of the process reflects on the long-range dependence of its increments. The most paradigmatic example of this connection is provided by the fractional Brownian motion and by its increment process, the fractional Gaussian noise [2].In this paper we will consider the problem of estimating the Hurst parameterH of a self-similar process with weakly stationary increments. Among the different techniques introduced in the literature, we will focus on a method based on the log-regression of the so-called modified Allan variance (MAVAR). The MAVAR is a well-known time-domain quantity generalizing the classic Allan variance [3, 4], which has been proposed for the first time as a traffic analysis tool in [5]. In a series of paper [5–7], its performance has been evaluated by simulation and comparison with the real IP traffic. These works have pointed out the high accuracy of the method in estimating the parameter H and have shown that it achieves better confidence if compared with the well-established wavelet log-diagram.The aim of the present work is to substantiate and enrich these results from the theoretical point of view, studying the rate of convergence of the estimator toward the memory parameter. In particular, our goal is to provide the limit properties and the precise asymptotic normality of the MAVAR log-regression estimator in order to compute the related asymptotic confidence intervals. This will be reached under the assumption that the signal process is a fractional Brownian motion. Although this hypothesis may look restrictive (indeed this estimator is designed for a larger class of stochastic processes), the obtained results are a first step toward the mathematical study of the MAVAR log-regression estimator. To our knowledge, there are no similar results in the literature. The present paper also provides the theoretical foundations (mathematical details and proofs) for [8]. Indeed the formulas analytically obtained here have been implemented and numerically tested in [8] for different choices of the regression weights and it has been shown that the numerical evidences are in good agreement with the theoretical results proven here.For a survey on Hurst parameter estimators of a fractional Brownian motion, we refer to [9, 10]. However we stress once again that the MAVAR-estimator is not specifically a target for the fractional Brownian motion, but it has been thought and successfully used for more general processes.Our theorems can be viewed as a counterpart of the already established results concerning the asymptotic normality of the wavelet log-regression estimators [11–13]. Indeed, although the MAVAR can be related in some sense to a suitable Haar-type wavelets family (see [14] for the classical Allan variance), the MAVAR and wavelets log-regression estimators do not match as the regression runs on different parameters (see Section 5). Hence, we adopt a different argument which in turns allows us to avoid some technical troubles due to the poor regularity of the Haar-type functions.The paper is organized as follows. In Section2 we recall the properties of self-similarity and long-range dependence for stochastic processes and the definition of fractional Brownian motion; in Section 3 we introduce the MAVAR and its estimator, with their main properties; in Section 4 we state and prove the main results concerning the asymptotic normality of the estimator; in Section 5 we make some comments on the link between the MAVAR and the wavelet estimators and on the modified Hadamard variance, which is a generalization of the MAVAR; in the Appendix we recall some results used along the proof.


## 2.Self-Similarity and Long-Range Dependence

In the sequel we consider a centered real-valued stochastic processX={X(t),t∈ℝ}, that can be interpreted as the signal process. Sometimes it is also useful to consider the τ-increment of the process X, which is defined, for every τ>0 and t∈ℝ, as
(2.1)Yτ(t)=X(t+τ)-X(t)τ.In order to reproduce the behavior of the real data, it is commonly assumed thatX satisfies one of the two following properties: (i) self-similarity; (ii) long range dependence.(i)
Theself-similarity of a centered real-valued stochastic process X, starting from zero, refers to the existence of a parameter H∈(0,1), called Hurst index or Hurst parameter of the process, such that, for all a>0, it holds
(2.2){X(t),t∈ℝ}=d{a-HX(at),t∈ℝ}.
In this case we say that X is an H-self-similar process.(ii)
We first recall that a centered real-valued stochastic processX is weakly stationary if it is square integrable and its autocovariance function, CX(t,s):=Cov(X(t),X(s)), is a translation invariant, namely, if
(2.3)CX(t,s)=CX(t+r,s+r),∀t,s,r∈ℝ.
In this case we also set RX(t):=CX(t,0). If X is a weakly stationary process, we say that it displays a long-range dependence, or long memory, if there exists d∈(0,1) such that the spectral density of the process, fX(λ), satisfies the condition
(2.4)fX(λ)~cf|λ|-dasλ→0,
for some finite constant cf≠0, where we write f(x)~g(x) as x→x0, if limx→x0(f(x)/g(x))=1. Due to the correspondence between the spectral density and the autocovariance function, given by
(2.5)RX(t)=12π∫ℝ‍eitλfX(λ)dλ,
the long-range condition (2.4) can be often stated in terms of the autocovariance of the process as
(2.6)RX(t)~cR|t|-βas|t|→+∞,
for β=(1-d)∈(0,1) and  some finite constant cR≠0.Notice that ifX is a self-similar process, then it obviously cannot be weakly stationary. On the other hand, assuming that X starts from zero and it is as H-self-similar process with weakly stationary increments, that is, the quantity
(2.7)𝔼[(X(τ2+s+t)-X(s+t))(X(τ1+s)-X(s))]
does not depend on s, it turns out that the autocovariance function is given by
(2.8)CX(s,t)=σH22(|t|2H-|t-s|2H+|s|2H),
with σH2:=𝔼[X2(1)], which is clearly not a translation invariant. Consequently, denoting by Yτ its τ-increment process (see (2.1)), the autocovariance function of Yτ is such that ([15])
(2.9)RYτ(t)~σH2H(2H-1)|t|2H-2as|t|→+∞.
In particular, if H∈(1/2,1), the process Yτ displays a long-range dependence in the sense of (2.6) with β=2-2H. Under this assumption, we thus embrace the two main empirical properties of a wide collection of real data.A basic example of the connection between self-similarity and long-range dependence is provided by thefractional Brownian motionBH={BH(t),t∈ℝ} [2]. This is a centered Gaussian process, starting from zero, with autocovariance function given by (2.8), where [16]
(2.10)σH2=1Γ(2H+1)sin(πH).
It can be proven that BH is a self-similar process with Hurst index H∈(0,1), which corresponds, for H=1/2, to the standard Brownian motion. Moreover, its increment process
(2.11)Gτ,H(t)=BH(t+τ)-BH(t)τ,
called fractional Gaussian noise, turns out to be a weakly stationary Gaussian process [2, 17].In the next sections we will perform the analysis of the modified Allan variance and of the related estimator of the memory parameter.


## 3.The Modified Allan Variance

In this section we introduce and recall the main properties of the modified Allan variance (MAVAR) and of the log-regression estimator of the memory parameter based on it.Suppose thatX is a centered real-valued stochastic process, starting from zero, with weakly stationary increments. Let τ0>0 be the “sampling period” and define the sequence of times {tk}k≥1 taking t1∈ℝ and setting ti-ti-1=τ0, that is, ti=t1+τ0(i-1).Definition 3.1.
For any fixed integerp≥1, the modified Allan variance (MAVAR) is defined [3] as
(3.1)σp2=σ2(τ0,p):=12τ02p2𝔼[(1p∑j=1p‍X(tj+2p)-2X(tj+p)+X(tj))2]=12τ2𝔼[(1p∑j=1p‍X(tj+2τ)-2X(tj+τ)+X(tj))2],
where we set τ:=τ0p. For p=1 we recover the well-known Allan variance.Let us assume that a finite sampleX1,…,Xn of the process X is given, and that the observations are taken at times t1,…,tn. In other words we set Xi=X(ti) for i=1,…,n.A standard estimator for the modified Allan variance (MAVAR estimator) is given by(3.2)σ^p2(n)=σ^2(τ0,p,n):=12τ02p4np∑r=1np‍(∑j′=rp+r-1‍(Xj′+2p-2Xj′+p+Xj′))2=1np∑k=0np-1‍(12τ0p2∑j=1p‍(Xk+j+2p-2Xk+j+p+Xk+j))2,
for p∈{1,…,⌊n/3⌋} and np:=n-3p+1.Fork∈ℤ, let us set
(3.3)dp,k=d(τ0,p,k):=12τ0p2∑j=1p‍(Xk+j+2p-2Xk+j+p+Xk+j),
so that the process {dp,k}k turns out to be weakly stationary for each fixed p, with 𝔼[dp,k]=0, and we can write
(3.4)σp2=𝔼[dp,02]=𝔼[dp,k2],σ^p2(n)=1np∑k=0np-1‍dp,k2.


### 3.1.Some Properties

Let us further assume thatX is an H-self-similar process (see (2.2)), with H∈(1/2,1). Applying the covariance formula (2.8), it holds
(3.5)σp2=𝔼[dp,02]=σH2τ2H-2K(H,p)=σH2K(H,p)τ2-2H,
with σH2=𝔼[X(1)2] and
(3.6)K(H,p):=2p(1-22H-2)+12p2∑ℓ=1p-1‍∑h=1ℓ‍P2H(hp),
where P2H is the polynomial of degree 2H given by
(3.7)P2H(x):=[-6|x|2H+4|1+x|2H+4|1-x|2H-|2+x|2H-|2-x|2H].Since we are interested in the limit forp→∞, we consider the approximation of the two finite sums in (3.6) by the corresponding double integral, namely,
(3.8)1p2∑ℓ=1p-1‍∑h=1ℓ‍P2H(hp)=∫01‍∫0y‍P2H(x)dxdy+OH(p-1).
Computing the integral and inserting the result in (3.6), we get
(3.9)K(H,p)=K(H)+OH(p-1),
where
(3.10)K(H):=22H+4+22H+3-32H+2-152(2H+1)(2H+2).
From (3.5) and (3.9), we get
(3.11)|σp2-σH2τ2H-2K(H)|≤σH2τ2H-2OH(p-1).Under the above hypotheses onX, one can also prove that the process {dp,k}p,k satisfies the stationary condition
(3.12)Cov(dp,k,dp-u,k′)=Cov(dp,k-k′,dp-u,0)for0≤u<p.
To verify this condition, we write explicitly the covariance as
(3.13)𝔼[dp,kdp-u,k′]=12τ2(p-u)2∑j=1p‍∑j'=1p-u‍𝔼[(Xk+j+2p-Xk+j+p)(Xk′+j′+2(p-u)-Xk′+j′+(p-u))]-𝔼[(Xk+j+2p-Xk+j+p)(Xk′+j′+(p-u)-Xk′+j′)]-𝔼[(Xk+j+p-Xk+j)(Xk′+j′+2(p-u)-Xk′+j′+(p-u))]+𝔼[(Xk+j+p-Xk+j)(Xk′+j′+(p-u)-Xk′+j′)].
Setting
(3.14)R(t;τ1,τ2):=𝔼[(X(τ2+s+t)-X(s+t))(X(τ1+s)-X(s))],r(q,h,p,u):=R(τ0(q+h+u);τ0(p-u),τ0p)-R(τ0(q+h+p);τ0(p-u),τ0p)-R(τ0(q+h-(p-u));τ0(p-u),τ0p)+R(τ0(q+h);τ0(p-u),τ0p),
we get
(3.15)𝔼[dp,kdp-u,k′]=12τ21(p-u)2∑j=1p‍‍∑j′=1p-u‍r(k-k′,j-j′,p,u).
This immediately provides the stationary condition (3.12).To better understand the behavior of the covariance𝔼[dp,kdp-u,k'] as k-k' varies, we apply the covariance formula (2.8) and get
(3.16)R(t;τ1,τ2)=σH22(|t+τ2|2H+|t-τ1|2H-|t+τ2-τ1|2H-|t|2H).
Thus, from (3.14),
(3.17)r(q,h,p,u)=σH2τ2H∑i=13‍gH(i)(qp,hp,up),
where
(3.18)gH(1)(qp,hp,up):=|q+hp+up+1|2H+|q+hp+up-1|2H-2|q+hp+up|2H,gH(2)(qp,hp,up):=-12(|q+hp+2|2H+|q+hp|2H-2|q+hp+1|2H),gH(3)(qp,hp,up):=-12(|q+hp+2up|2H+|q+hp+2up-2|2H-2|q+hp+2up-1|2H).
Inserting (3.17) in (3.15), we obtain
(3.19)𝔼[dp,kdp-u,k′]=σH2τ2H-2GH((k-k′)p,up,p),
where τ=τ0p as before, and
(3.20)GH(qp,up,p):=121(p-u)2∑j=1p‍‍∑j′=1p-u‍‍∑i=13‍gH(i)(qp,(j-j′)p,up).
Now we set q=mp+r with m∈ℤ and r an integer number in {0,…,p-1} and we study the asymptotic behavior of GH(m+(r/p),u/p,p) for |m|→+∞.The limit relation(3.21)|m+1|2H+|m-1|2H-2|m|2H~2H(2H-1)|m|2H-2as|m|→+∞,
that is,
(3.22)lim|m|→+∞|m+1|2H+|m-1|2H-2|m|2H|m|2H-2=2H(2H-1),
implies that
(3.23)gH(1)(m+(rp),hp,up)~2H(2H-1)|m+rp+hp+up|2H-2as|m|→+∞,gH(2)(m+(rp),hp,up)~-H(2H-1)|m+rp+hp+1|2H-2as|m|→+∞,gH(3)(m+(rp),x,up)~-H(2H-1)|m+rp+hp+2up-1|2H-2as|m|→+∞,
and so, for |m|→+∞,
(3.24)∑i=13‍gH(i)(m+(rp),hp,up)~-H(2H-1)(|m+r+hp+1|2H-2+|m+r+h+2up-1|2H-2-2|m+r+h+up|2H-2)=-H(2H-1)×(|m+r+u+hp+p-up|2H-2+|m+r+u+hp-p-up|2H-2-2|m+r+u+hp|2H-2)=-H(2H-1)(p-up)2H-2×(|mpp-u+r+u+hp-u+1|2H-2+|mpp-u+r+u+hp-u-1|2H-2-2|mpp-uq+u+hp-u|2H-2)~cH(p-up)2H-2|mpp-u+r+u+hp-u|2H-4=cH(p-up)2|m+r+u+hp|2H-4~cH(p-up)2|m|2H-4,
where cH:=(-4!/2)(2H4). We can conclude that
(3.25)GH(qp,up,p)=GH(m+(rp),up,p)~cH2(1-up)|m|2H-4as|m|→+∞.


### 3.2.The MAVAR Log-Regression Estimator

Letn be the sample size, that is, the number of the observations.Definition 3.2.
Letp-,ℓ-∈ℕ such that 1≤(1+ℓ-)p-≤pmax(n)=⌊n/3⌋, and let w_=(w0,…,wℓ-) be a vector of weights satisfying the conditions
(3.26)∑ℓ=0ℓ-‍wℓ=0,∑ℓ=0ℓ-‍wℓlog(1+ℓ)=1.
The MAVAR log-regression estimator associated to the weights w_ is defined as
(3.27)α^n(p-,w_)=α^n(τ0,p-,w_):=∑ℓ=0ℓ-‍wℓlog(σ^(1+ℓ)p-2(n)).
Roughly speaking, the idea behind this definition is to use the approximation
(3.28)(σ^p-2(n),…,σ^(1+ℓ-)p-2(n))=~(σp-2(n),…,σ(1+ℓ-)p-2(n)),
in order to get, by (3.11) and (3.26),
(3.29)α^n(p-,w_)=~∑ℓ=0ℓ-‍wℓlog(σ(1+ℓ)p-2)=~∑ℓ=0ℓ-‍wℓ[αlog(1+ℓ)+αlog(τ0p-)+log(σH2K(H))]=α,
where α:=2H-2. Thus, given the data X1,…,Xn, the following procedure is used to estimate H: (i)
compute the modified Allan variance by (3.2), for integer values (1+ℓ)p-, with 1≤(1+ℓ)p-≤pmax(n)=⌊n/3⌋;(ii)
compute the weighted MAVAR log-regression estimator by (3.27) in order to get an estimate α^ of α;(iii)
estimateH by H^=(α^+2)/2. In the sequel we will give, under suitable assumptions, two convergence results in order to justify these approximations and to get the rate of convergence of α^n(p-,w_) toward α=2H-2. Obviously, we need to take p-=p-(n)→+∞ as n→+∞ in order to reach jointly the above two approximations.


## 3.1.Some Properties

Let us further assume thatX is an H-self-similar process (see (2.2)), with H∈(1/2,1). Applying the covariance formula (2.8), it holds
(3.5)σp2=𝔼[dp,02]=σH2τ2H-2K(H,p)=σH2K(H,p)τ2-2H,
with σH2=𝔼[X(1)2] and
(3.6)K(H,p):=2p(1-22H-2)+12p2∑ℓ=1p-1‍∑h=1ℓ‍P2H(hp),
where P2H is the polynomial of degree 2H given by
(3.7)P2H(x):=[-6|x|2H+4|1+x|2H+4|1-x|2H-|2+x|2H-|2-x|2H].Since we are interested in the limit forp→∞, we consider the approximation of the two finite sums in (3.6) by the corresponding double integral, namely,
(3.8)1p2∑ℓ=1p-1‍∑h=1ℓ‍P2H(hp)=∫01‍∫0y‍P2H(x)dxdy+OH(p-1).
Computing the integral and inserting the result in (3.6), we get
(3.9)K(H,p)=K(H)+OH(p-1),
where
(3.10)K(H):=22H+4+22H+3-32H+2-152(2H+1)(2H+2).
From (3.5) and (3.9), we get
(3.11)|σp2-σH2τ2H-2K(H)|≤σH2τ2H-2OH(p-1).Under the above hypotheses onX, one can also prove that the process {dp,k}p,k satisfies the stationary condition
(3.12)Cov(dp,k,dp-u,k′)=Cov(dp,k-k′,dp-u,0)for0≤u<p.
To verify this condition, we write explicitly the covariance as
(3.13)𝔼[dp,kdp-u,k′]=12τ2(p-u)2∑j=1p‍∑j'=1p-u‍𝔼[(Xk+j+2p-Xk+j+p)(Xk′+j′+2(p-u)-Xk′+j′+(p-u))]-𝔼[(Xk+j+2p-Xk+j+p)(Xk′+j′+(p-u)-Xk′+j′)]-𝔼[(Xk+j+p-Xk+j)(Xk′+j′+2(p-u)-Xk′+j′+(p-u))]+𝔼[(Xk+j+p-Xk+j)(Xk′+j′+(p-u)-Xk′+j′)].
Setting
(3.14)R(t;τ1,τ2):=𝔼[(X(τ2+s+t)-X(s+t))(X(τ1+s)-X(s))],r(q,h,p,u):=R(τ0(q+h+u);τ0(p-u),τ0p)-R(τ0(q+h+p);τ0(p-u),τ0p)-R(τ0(q+h-(p-u));τ0(p-u),τ0p)+R(τ0(q+h);τ0(p-u),τ0p),
we get
(3.15)𝔼[dp,kdp-u,k′]=12τ21(p-u)2∑j=1p‍‍∑j′=1p-u‍r(k-k′,j-j′,p,u).
This immediately provides the stationary condition (3.12).To better understand the behavior of the covariance𝔼[dp,kdp-u,k'] as k-k' varies, we apply the covariance formula (2.8) and get
(3.16)R(t;τ1,τ2)=σH22(|t+τ2|2H+|t-τ1|2H-|t+τ2-τ1|2H-|t|2H).
Thus, from (3.14),
(3.17)r(q,h,p,u)=σH2τ2H∑i=13‍gH(i)(qp,hp,up),
where
(3.18)gH(1)(qp,hp,up):=|q+hp+up+1|2H+|q+hp+up-1|2H-2|q+hp+up|2H,gH(2)(qp,hp,up):=-12(|q+hp+2|2H+|q+hp|2H-2|q+hp+1|2H),gH(3)(qp,hp,up):=-12(|q+hp+2up|2H+|q+hp+2up-2|2H-2|q+hp+2up-1|2H).
Inserting (3.17) in (3.15), we obtain
(3.19)𝔼[dp,kdp-u,k′]=σH2τ2H-2GH((k-k′)p,up,p),
where τ=τ0p as before, and
(3.20)GH(qp,up,p):=121(p-u)2∑j=1p‍‍∑j′=1p-u‍‍∑i=13‍gH(i)(qp,(j-j′)p,up).
Now we set q=mp+r with m∈ℤ and r an integer number in {0,…,p-1} and we study the asymptotic behavior of GH(m+(r/p),u/p,p) for |m|→+∞.The limit relation(3.21)|m+1|2H+|m-1|2H-2|m|2H~2H(2H-1)|m|2H-2as|m|→+∞,
that is,
(3.22)lim|m|→+∞|m+1|2H+|m-1|2H-2|m|2H|m|2H-2=2H(2H-1),
implies that
(3.23)gH(1)(m+(rp),hp,up)~2H(2H-1)|m+rp+hp+up|2H-2as|m|→+∞,gH(2)(m+(rp),hp,up)~-H(2H-1)|m+rp+hp+1|2H-2as|m|→+∞,gH(3)(m+(rp),x,up)~-H(2H-1)|m+rp+hp+2up-1|2H-2as|m|→+∞,
and so, for |m|→+∞,
(3.24)∑i=13‍gH(i)(m+(rp),hp,up)~-H(2H-1)(|m+r+hp+1|2H-2+|m+r+h+2up-1|2H-2-2|m+r+h+up|2H-2)=-H(2H-1)×(|m+r+u+hp+p-up|2H-2+|m+r+u+hp-p-up|2H-2-2|m+r+u+hp|2H-2)=-H(2H-1)(p-up)2H-2×(|mpp-u+r+u+hp-u+1|2H-2+|mpp-u+r+u+hp-u-1|2H-2-2|mpp-uq+u+hp-u|2H-2)~cH(p-up)2H-2|mpp-u+r+u+hp-u|2H-4=cH(p-up)2|m+r+u+hp|2H-4~cH(p-up)2|m|2H-4,
where cH:=(-4!/2)(2H4). We can conclude that
(3.25)GH(qp,up,p)=GH(m+(rp),up,p)~cH2(1-up)|m|2H-4as|m|→+∞.


## 3.2.The MAVAR Log-Regression Estimator

Letn be the sample size, that is, the number of the observations.Definition 3.2.
Letp-,ℓ-∈ℕ such that 1≤(1+ℓ-)p-≤pmax(n)=⌊n/3⌋, and let w_=(w0,…,wℓ-) be a vector of weights satisfying the conditions
(3.26)∑ℓ=0ℓ-‍wℓ=0,∑ℓ=0ℓ-‍wℓlog(1+ℓ)=1.
The MAVAR log-regression estimator associated to the weights w_ is defined as
(3.27)α^n(p-,w_)=α^n(τ0,p-,w_):=∑ℓ=0ℓ-‍wℓlog(σ^(1+ℓ)p-2(n)).
Roughly speaking, the idea behind this definition is to use the approximation
(3.28)(σ^p-2(n),…,σ^(1+ℓ-)p-2(n))=~(σp-2(n),…,σ(1+ℓ-)p-2(n)),
in order to get, by (3.11) and (3.26),
(3.29)α^n(p-,w_)=~∑ℓ=0ℓ-‍wℓlog(σ(1+ℓ)p-2)=~∑ℓ=0ℓ-‍wℓ[αlog(1+ℓ)+αlog(τ0p-)+log(σH2K(H))]=α,
where α:=2H-2. Thus, given the data X1,…,Xn, the following procedure is used to estimate H: (i)
compute the modified Allan variance by (3.2), for integer values (1+ℓ)p-, with 1≤(1+ℓ)p-≤pmax(n)=⌊n/3⌋;(ii)
compute the weighted MAVAR log-regression estimator by (3.27) in order to get an estimate α^ of α;(iii)
estimateH by H^=(α^+2)/2. In the sequel we will give, under suitable assumptions, two convergence results in order to justify these approximations and to get the rate of convergence of α^n(p-,w_) toward α=2H-2. Obviously, we need to take p-=p-(n)→+∞ as n→+∞ in order to reach jointly the above two approximations.


## 4.The Asymptotic Normality of the Estimator

Since now on we will always assume thatX is a fractional Brownian motion (with H∈(1/2,1)) so that the process {dp,k}p,k is also Gaussian. Under this assumption, and with the notation introduced before, we can state the following results.Theorem 4.1.
Letp-=p-(n) be a sequence of integers such that
(4.1)p-(n)→+∞,np-(n)-1→+∞.
Let ℓ- be a given integer, σ_^n2(p-,ℓ-) the vector (σ^p-2(n),σ^2p-2(n),…,σ^(1+ℓ-)p-2(n)) and, analogously, set σ_2(p-,ℓ-)=(σp-2,σ2p-2,…,σ(1+ℓ-)p-2). Then, it holds
(4.2)np-(τ0p-)2-2H(σ_^n2(p-,ℓ-)-σ_2(p-,ℓ-))→dn→∞𝒩(0,W(H)),
where W(H) is a suitable symmetric matrix.From this theorem, as an application of theδ-method, we can state the following result.Theorem 4.2.
Letα^n(p-,w_) be defined as in (3.27), for some finite integer ℓ- and a weight-vector w_ satisfying (3.26). If p-=p-(n) is a sequence of integers such that
(4.3)p-(n)→+∞,np-(n)-1→+∞,np-(n)-3→0,
then
(4.4)np-(α^n(p-,w_)-α)→dn→∞𝒩(0,w_*TV(H)w_*),
where α=2H-2, the column-vector w_* is such that [w_*]ℓ:=wℓ(1+ℓ)2-2H, and V(H)=(σH2K(H))-2W(H).Let us stress that from the above result, and due to the conditionnp-(n)-1→+∞, it follows that the estimator α^n(p-,w_) is consistent.Before starting the proof of the above theorems, we need the following lemma.Lemma 4.3.
Letp-=p-(n) be a sequence of integers such that
(4.5)p-(n)→+∞,np-(n)-1→+∞.
For two given integers ℓ',ℓ, with 0≤ℓ'≤ℓ, set pℓ′=(ℓ'+1)p- and pℓ=(ℓ+1)p-. Then
(4.6)np-(τ0p-)4-4HCov(σ^pℓ2(n),σ^pℓ′2(n))→n→+∞Wℓ′,ℓ(H),
where Wℓ′,ℓ(H) is a finite quantity.Proof.
Sincen/p-→+∞, without loss of generality we can assume that (1+ℓ-)p-≤pmax(n)=⌊n/3⌋ for each n. Recall the notation np=n-3p+1, and set nℓ=npℓ~n and nℓ′=npℓ′~n. From the definition of the MAVAR estimator and applying the Wick’s rule for jointly Gaussian random variables (see the Appendix), we get
(4.7)Cov(σ^pℓ2(n),σ^pℓ′2(n))=1nℓnℓ′∑k=0nℓ-1‍∑k′=0nℓ′-1‍Cov(dpℓ,k2,dpℓ′,k′2)=2nℓnℓ′∑k=0nℓ-1‍∑k′=0nℓ′-1‍Cov(dpℓ,k,dpℓ′,k′)2=2σH4(1+ℓ)4H-4(τ0p-)4H-4nℓnℓ′∑k=0nℓ-1‍∑k′=0nℓ′-1‍GH2(k-k′pℓ,uℓ′ℓpℓ,pℓ),
where uℓ′ℓ=pℓ-pℓ′. Since, by (3.20), the function GH(q/pℓ,uℓ′ℓ/pℓ,pℓ) only depends on q/pℓ and uℓ′ℓ/pℓ=(ℓ-ℓ')/(1+ℓ) as n→+∞, we rewrite the last line as
(4.8)2σH4(1+ℓ)4H-4(τ0p-)4H-4nℓnℓ′∑k=0nℓ-1‍∑k′=0nℓ′-1‍GH2(k-k′pℓ,(ℓ-ℓ′)1+ℓ).
This term, multiplied by (n/p-)(τ0p-)4-4H, is equal to
(4.9)2σH4(1+ℓ)4H-41p-nnℓnℓ′∑k=0nℓ-1‍∑k′=0nℓ′-1‍GH2(k-k′pℓ,(ℓ-ℓ′)(1+ℓ)).
It is easy to see that this quantity converges (to a finite strictly positive limit) if and only if
(4.10)1pℓ∑m=0[n/pℓ]‍∑r=0pℓ-1‍GH2(m+(rpℓ),(ℓ-ℓ′)(1+ℓ))
converges. From (3.25) it holds
(4.11)GH2(m+(rpℓ),(ℓ-ℓ′)(1+ℓ))~cH24(1-ℓ-ℓ'1+ℓ)2m4H-8asm→+∞.
Thus, the quantity (4.10) is controlled in the limit n→∞ by the sum ∑m=1∞‍m4H-8 that is convergent.Proof of Theorem4.1.
As before, without loss of generality, we can assume that(1+ℓ-)p-≤pmax(n) for each n. Moreover, set again nℓ=npℓ and nℓ′=npℓ′. For a given real vector v_T=(v0,…,vℓ-), let us consider the random variable Tn=T(p-(n),ℓ-,v_) defined as a linear combination of the empirical variances σ^p-2(n),…,σ^(1+ℓ-)p-2(n) as follows:
(4.12)Tn:=∑ℓ=0ℓ-‍vℓσ^(1+ℓ)p-2(n)=∑ℓ=0ℓ-‍vℓnℓ∑k=0nℓ-1‍d(1+ℓ)p-,k2.
In order to prove the convergence stated in Theorem 4.1, we have to show that the random variable n/p-(τ0p-)2-2H(Tn-∑ℓ=0ℓ-‍vℓσ(1+ℓ)p-2) converges to the normal distribution with zero mean and variance v_TW(H)v_. To this purpose, we note that
(4.13)np-(τ0p-)2-2H(Tn-∑ℓ=0ℓ-‍vℓσ(1+ℓ)p-2)=VnTAnVn-𝔼[VnTAnVn],
where Vn is the random vector with entries d(1+ℓ)p-,k, for 0≤ℓ≤ℓ- and 0≤k≤nℓ-1, and An is the diagonal matrix with entries
(4.14)[An]((1+ℓ)p-,k),((1+ℓ)p-,k)=vℓnℓnp-(τ0p-)2-2H=O((p-)3/2-2Hn-1/2).
By Lemma 4.3, if W(H) is the symmetric matrix with [W(H)]ℓ′,ℓ=Wℓ′,ℓ when 0≤ℓ'≤ℓ≤ℓ-, it holds
(4.15)Var[VnTAnVn]=np-(τ0p-)4-4H∑ℓ=0ℓ-‍∑ℓ′=0ℓ-‍vℓvℓ′Cov(σ^(1+ℓ)p-2(n),σ^p-(1+ℓ′)2(n))→n→+∞∑ℓ=0ℓ-‍∑ℓ′=0ℓ-‍vℓvℓ′Wℓ′,ℓ(H)=v_TW(H)v_,
therefore, condition (1) of Lemma A.2 is satisfied.
In order to verify condition(2) of Lemma A.2, let Cn denote the covariance matrix of the random vector Vn, and let ρ[Cn] denote its spectral radius. By Lemma A.3, we have
(4.16)ρ[Cn]≤∑ℓ=0ℓ-‍ρ[Cn((1+ℓ)p-)],
where Cn((1+ℓ)p-) is the covariance matrix of the subvector (d(1+ℓ)p-,k)k=0,…,nℓ-1. Applying the spectral radius estimate (A.3), and from equality (3.19), we then have
(4.17)ρ[Cn((1+ℓ)p-)]≤σH2[τ0(1+ℓ)p-]2H-2[GH(0,0,pℓ)+2∑q=1nℓ-1‍GH(qpℓ,0,pℓ)]=O((p-)2H-1).
In order to conclude, it is enough to note that
(4.18)ρ[An]ρ[Cn]=O((p-)3/2-2Hn-1/2)O((p-)2H-1)=O((p-)1/2n-1/2)→n→∞0.Proof of Theorem4.2.
By assumptions (4.3) on the sequence p-=p-(n), and in particular from the condition n/(p-)3→0, and inequality (3.11), it holds
(4.19)np-(τ0p-)2-2H|σ(1+ℓ)p-2-σH2[τ0(1+ℓ)p-]2H-2K(H)|≤σH2(1+ℓ)2H-2O(n1/2(p-)-3/2)→n→+∞0.
Thus, from Theorem 4.1, we get
(4.20)np-[(τ0p-)2-2Hσ^_n2(p-,ℓ-)-σ_*2]→dn→∞𝒩(0,W(H)),
where σ_*2 is the vector with elements
(4.21)[σ_*2]ℓ:=σH2(1+ℓ)2H-2K(H)for0≤ℓ≤ℓ-.
Now we observe that if f(x_):=∑ℓ=0ℓ-‍wℓlog(xℓ), then, by (3.26) and (3.27), we have
(4.22)α^n(p-,w_)=f(σ_^2(p-,ℓ-))=f((τ0p-)2-2Hσ_^2(p-,ℓ-)).
Moreover, α=f(σ_*2) and ∇f(σ_*2)=(σH2K(H))-1w_*. Therefore, by the application of the δ-method, the convergence (4.20) entails
(4.23)np-(α^n(p-,w_)-α)→dn→∞𝒩(0,∇f(σ_*2)TW(H)∇f(σ_*2)),
where
(4.24)∇f(σ_*2)TW(H)∇f(σ_*2)=w_*TV(H)w_*,
and thus concludes the proof.Remark 4.4.
By Lemma4.3, and since by definition Vℓ′,ℓ(H)=(σH2K(H))-2Wℓ′,ℓ(H), an estimate of Vℓ′,ℓ(H), with 0≤ℓ'≤ℓ≤ℓ-, is given by
(4.25)np-(τ0p-)4-4HσH4K(H)2Cov(σ^pℓ2(n),σ^pℓ′2(n)).
Therefore, setting ρn2(w_,H) equal to the corresponding estimate of w_*TV(H)w_* divided by n/p-, that is
(4.26)ρn2(w_,H)=2K(H)2∑ℓ=0ℓ-‍∑ℓ′=0ℓ-‍(1+ℓ∧ℓ'1+ℓ∨ℓ')2-2Hwℓwℓ′nℓnℓ′×∑k=0nℓ∨ℓ′-1‍∑k′=0nℓ∧ℓ′-1‍GH2(k-k′p-(1+ℓ∨ℓ'),|ℓ-ℓ′|1+ℓ∨ℓ',p-(1+ℓ∨ℓ′)),
and from (4.6), we obtain the convergence
(4.27)ρn-1(w_,H)(α^n(p-,w_)-α)→dn→∞𝒩(0,1),
which can be used to obtain an asymptotic confidence interval for the parameter H.


### 4.1.An Alternative Representation of the Covariance

It is well known that an FBM (withX(0)=0 and 𝔼[X2(1)]=σH2) has the following stochastic integral representation (see [2, 16]):
(4.28)X(t)=1Γ(H+1/2)∫ℝ‍{[(t-s)+]H-1/2-(s-)H-1/2}dWs.
Using this representation and Ito’s formula, we can obtain another representation of the covariance Cov(σ^p2(n),σ^p-u2(n)) and, consequently, another formula for the coefficient ρn(w_,H).Indeed, assuming without loss of generalityt1=τ0 so that Xi=X(ti)=X(iτ0), by (3.3) and (4.28), we can write for 0≤u<p and k∈ℤ(4.29)dp-u,k=σH2(τ0p)H-3/2p2(p-u)2∫ℝ‍γH(kp,p,u,s(τ0p))dWs,
where
(4.30)γH(kp,p,u,s(τ0p)):=[Γ(H+12)σH]-1×1p∑j=1p-u‍{[(k+jp+2(p-u)p-sτ0p)+]H-1/2-2[(k+jp+p-up-sτ0p)+]H-1/2+[(k+jp-sτ0p)+]H-1/2}.
By Ito’s formula we get
(4.31)dp-u,k2=𝔼[dp-u,k2]+σH2(τ0p)2H-3p4(p-u)4∫ℝ‍Z(kp,p,u,ν)γH(kp,p,u,ν(τ0p))dWν,
where
(4.32)Z(kp,p,u,ν):=∫-∞ν‍γH(kp,p,u,s(τ0p))dWs.
Inserting this expression in (3.4), we obtain
(4.33)σ^p-u2(n)-σp-u2=σH2(τ0p)2H-3p4(p-u)4∫ℝ‍[1np-u∑k=0np-u-1‍Z(kp,p,u,ν)γH(kp,p,u,ντ0p)]dWν.
It follows, using Ito’s formula again and after the change of variables s=τ0(pv+k') and ν=τ0(pr+k'),
(4.34)Cov(σ^p2(n),σ^p-u2(n))=σH4(τ0p)4H-4p4(p-u)41npnp-u×∑k=0np-1‍∑k′=0np-u-1‍∫ℝ‍[γH((k-k′)p,p,0,r)γH(0,p,u,r)×∫-∞r‍γH((k-k′)p,p,0,v)γH(0,p,u,v)dv]dr.


## 4.1.An Alternative Representation of the Covariance

It is well known that an FBM (withX(0)=0 and 𝔼[X2(1)]=σH2) has the following stochastic integral representation (see [2, 16]):
(4.28)X(t)=1Γ(H+1/2)∫ℝ‍{[(t-s)+]H-1/2-(s-)H-1/2}dWs.
Using this representation and Ito’s formula, we can obtain another representation of the covariance Cov(σ^p2(n),σ^p-u2(n)) and, consequently, another formula for the coefficient ρn(w_,H).Indeed, assuming without loss of generalityt1=τ0 so that Xi=X(ti)=X(iτ0), by (3.3) and (4.28), we can write for 0≤u<p and k∈ℤ(4.29)dp-u,k=σH2(τ0p)H-3/2p2(p-u)2∫ℝ‍γH(kp,p,u,s(τ0p))dWs,
where
(4.30)γH(kp,p,u,s(τ0p)):=[Γ(H+12)σH]-1×1p∑j=1p-u‍{[(k+jp+2(p-u)p-sτ0p)+]H-1/2-2[(k+jp+p-up-sτ0p)+]H-1/2+[(k+jp-sτ0p)+]H-1/2}.
By Ito’s formula we get
(4.31)dp-u,k2=𝔼[dp-u,k2]+σH2(τ0p)2H-3p4(p-u)4∫ℝ‍Z(kp,p,u,ν)γH(kp,p,u,ν(τ0p))dWν,
where
(4.32)Z(kp,p,u,ν):=∫-∞ν‍γH(kp,p,u,s(τ0p))dWs.
Inserting this expression in (3.4), we obtain
(4.33)σ^p-u2(n)-σp-u2=σH2(τ0p)2H-3p4(p-u)4∫ℝ‍[1np-u∑k=0np-u-1‍Z(kp,p,u,ν)γH(kp,p,u,ντ0p)]dWν.
It follows, using Ito’s formula again and after the change of variables s=τ0(pv+k') and ν=τ0(pr+k'),
(4.34)Cov(σ^p2(n),σ^p-u2(n))=σH4(τ0p)4H-4p4(p-u)41npnp-u×∑k=0np-1‍∑k′=0np-u-1‍∫ℝ‍[γH((k-k′)p,p,0,r)γH(0,p,u,r)×∫-∞r‍γH((k-k′)p,p,0,v)γH(0,p,u,v)dv]dr.


## 5.Some Comments

### 5.1.The Modified Allan Variance and the Wavelet Estimators

Suppose thatX is a self-similar process with weakly stationary increments and consider the generalized process Y={Y(t),t∈ℝ} defined through the set of identities
(5.1)〈Y;I[t1,t2]〉=∫t1t2‍Y(t)dt=X(t2)-X(t1),∀t1,t2∈ℝ.
In short, we write Y=X˙. From this definition and with the notation introduced in Section 3, we can rewrite the MAVAR as
(5.2)σ2(τ0,p)=12τ02p2𝔼[(1p∑i=1p‍(∫ti+pti+2p‍Y(t)dt-∫titi+p‍Y(t)dt))2]
and its related estimator as
(5.3)σ^2(τ0,p,n)=12τ02p2np∑k=0np-1‍[1p∑i=1p‍(∫ti+k+pti+k+2p‍Y(t)dt-∫ti+kti+k+p‍Y(t)dt)]2.
Now we claim that, for p fixed, the quantity
(5.4)d(τ0,p,k):=12p2τ0∑i=1p‍(∫ti+k+pti+k+2p‍Y(t)dt-∫ti+kti+k+p‍Y(t)dt)
recalls a family of discrete wavelet transforms of the process Y, indexed by τ0 and k. To see that, let us fix j∈ℕ and k∈ℤ and set τ0=2j and t1=2j, so that ti+k=2j(i+k), for all i∈ℕ. With this choice on the sequence of times, it is not difficult to construct a function ψ(s) such that
(5.5)dk,j:=d(2j,p,k)=〈Y;ψk,j〉withψk,j(s):=2-jψ(2-js-k).
An easy check shows that the function
(5.6)ψ(s):=1pp∑i=1p‍ψi(s),ψi(s):=12p(I[i+p,i+2p](s)-I[i,i+p](s)),
satisfies (5.5). Notice also that the components ψi, i=1,…p, are suitably translated and renormalized Haar functions.In the casep=1, corresponding to the classical Allan variance, the function ψ is exactly given by the Haar mother wavelet, as was already pointed out in [14].Though the wavelet representation could be convenient in many respects, the Haar mother wavelet does not satisfy one of the conditions which are usually required in order to study the convergence of the estimator (see condition (W2) in [18]). Moreover, there is a fundamental difference between the two methods: in the wavelet setting the log-regression is done over the scale parameter τ0 for p fixed, while the MAVAR log-regression given in (3.27) is performed over p and for τ0 fixed.


### 5.2.The Modified Hadamard Variance

Further generalizing the notion of the MAVAR, one can define the modified Hadamard variance (MHVAR). For fixed integersp,Q, and τ0∈ℝ, set
(5.7)σ2(τ0,p,Q):=1Q!τ02p2𝔼[(1p∑i=1p‍∑q=0Q‍c(Q,q)X(ti+qp))2],
where c(Q,q):=(-1)q(Q!/q!(Q-q)!). Notice that for Q=2 we recover the modified Allan variance. The MHVAR is again a time-domain quantity which has been introduced in [7] for the analysis of the network traffic. A standard estimator for this variance is given by
(5.8)σ^n2(τ0,p,Q):=1Q!τ02p4nQ,p∑h=1nQ,p‍(∑i′=hp+h-1‍∑q=0Q‍c(Q,q)Xi′+qp)2=1nQ,p∑k=0nQ,p-1‍(1Q!τ0p2∑i=1p‍∑q=0Q‍c(Q,q)Xk+i+qp)2,
for p=1,…,[n/(Q+1)] and nQ,p:=n-(Q+1)p+1.Similarly to the analysis performed for the MAVAR, let us set(5.9)dp,k=d(τ0,p,Q,k):=1Q!τ0p2∑i=1p‍∑q=0Q‍c(Q,q)Xk+i+qp,
so that we can write
(5.10)σ2(τ0,p,Q)=𝔼[dp,02],σ^n2(τ0,p,Q)=1nQ,p∑k=0nQ,p-1‍dp,k2.
This suggests that convergence results, similar to Theorems 4.1 and 4.2, can be achieved also for the MHVAR and its related log-regression estimator.


### 5.3.The Case of Stationary Processes

In applications, MAVAR and MHVAR are also used in order to estimate the memory parameter of long-range dependent processes. This general case is not included in our analysis (which is restricted to the fractional Brownian motion) and it requires a more involved investigation. To our knowledge, there are no theoretical results along this direction.


## 5.1.The Modified Allan Variance and the Wavelet Estimators

Suppose thatX is a self-similar process with weakly stationary increments and consider the generalized process Y={Y(t),t∈ℝ} defined through the set of identities
(5.1)〈Y;I[t1,t2]〉=∫t1t2‍Y(t)dt=X(t2)-X(t1),∀t1,t2∈ℝ.
In short, we write Y=X˙. From this definition and with the notation introduced in Section 3, we can rewrite the MAVAR as
(5.2)σ2(τ0,p)=12τ02p2𝔼[(1p∑i=1p‍(∫ti+pti+2p‍Y(t)dt-∫titi+p‍Y(t)dt))2]
and its related estimator as
(5.3)σ^2(τ0,p,n)=12τ02p2np∑k=0np-1‍[1p∑i=1p‍(∫ti+k+pti+k+2p‍Y(t)dt-∫ti+kti+k+p‍Y(t)dt)]2.
Now we claim that, for p fixed, the quantity
(5.4)d(τ0,p,k):=12p2τ0∑i=1p‍(∫ti+k+pti+k+2p‍Y(t)dt-∫ti+kti+k+p‍Y(t)dt)
recalls a family of discrete wavelet transforms of the process Y, indexed by τ0 and k. To see that, let us fix j∈ℕ and k∈ℤ and set τ0=2j and t1=2j, so that ti+k=2j(i+k), for all i∈ℕ. With this choice on the sequence of times, it is not difficult to construct a function ψ(s) such that
(5.5)dk,j:=d(2j,p,k)=〈Y;ψk,j〉withψk,j(s):=2-jψ(2-js-k).
An easy check shows that the function
(5.6)ψ(s):=1pp∑i=1p‍ψi(s),ψi(s):=12p(I[i+p,i+2p](s)-I[i,i+p](s)),
satisfies (5.5). Notice also that the components ψi, i=1,…p, are suitably translated and renormalized Haar functions.In the casep=1, corresponding to the classical Allan variance, the function ψ is exactly given by the Haar mother wavelet, as was already pointed out in [14].Though the wavelet representation could be convenient in many respects, the Haar mother wavelet does not satisfy one of the conditions which are usually required in order to study the convergence of the estimator (see condition (W2) in [18]). Moreover, there is a fundamental difference between the two methods: in the wavelet setting the log-regression is done over the scale parameter τ0 for p fixed, while the MAVAR log-regression given in (3.27) is performed over p and for τ0 fixed.


## 5.2.The Modified Hadamard Variance

Further generalizing the notion of the MAVAR, one can define the modified Hadamard variance (MHVAR). For fixed integersp,Q, and τ0∈ℝ, set
(5.7)σ2(τ0,p,Q):=1Q!τ02p2𝔼[(1p∑i=1p‍∑q=0Q‍c(Q,q)X(ti+qp))2],
where c(Q,q):=(-1)q(Q!/q!(Q-q)!). Notice that for Q=2 we recover the modified Allan variance. The MHVAR is again a time-domain quantity which has been introduced in [7] for the analysis of the network traffic. A standard estimator for this variance is given by
(5.8)σ^n2(τ0,p,Q):=1Q!τ02p4nQ,p∑h=1nQ,p‍(∑i′=hp+h-1‍∑q=0Q‍c(Q,q)Xi′+qp)2=1nQ,p∑k=0nQ,p-1‍(1Q!τ0p2∑i=1p‍∑q=0Q‍c(Q,q)Xk+i+qp)2,
for p=1,…,[n/(Q+1)] and nQ,p:=n-(Q+1)p+1.Similarly to the analysis performed for the MAVAR, let us set(5.9)dp,k=d(τ0,p,Q,k):=1Q!τ0p2∑i=1p‍∑q=0Q‍c(Q,q)Xk+i+qp,
so that we can write
(5.10)σ2(τ0,p,Q)=𝔼[dp,02],σ^n2(τ0,p,Q)=1nQ,p∑k=0nQ,p-1‍dp,k2.
This suggests that convergence results, similar to Theorems 4.1 and 4.2, can be achieved also for the MHVAR and its related log-regression estimator.


## 5.3.The Case of Stationary Processes

In applications, MAVAR and MHVAR are also used in order to estimate the memory parameter of long-range dependent processes. This general case is not included in our analysis (which is restricted to the fractional Brownian motion) and it requires a more involved investigation. To our knowledge, there are no theoretical results along this direction.



---

*Source: 905082-2012-11-27.xml*