In this paper, we deal with an Ornstein–Uhlenbeck process driven by sub-fractional Brownian motion of the second kind with Hurst index H∈(12,1). We provide a least squares estimator (LSE) of the drift parameter based on continuous-time observations. The strong consistency and the upper bound O(1/n) in Kolmogorov distance for central limit theorem of the LSE are obtained. We use a Malliavin–Stein approach for normal approximations.

Sub-fractional Ornstein–Uhlenbeck process of second kindleast squares estimatorBerry–Esséen boundMalliavin–Stein approach for normal approximations60G1560G2262F1262M0960H07SIMONS foundationM. F. Baldé acknowledges support from NLAGA project of SIMONS foundation. Introduction

Let SH:=StH,t≥0 be a sub-fractional Brownian motion (sub-fBm) with Hurst parameter H∈(0,1) that is a centered Gaussian process, defined on a complete probability space (Ω,F,P), with the covariance function
ESsHStH=t2H+s2H−12|t−s|2H+|t+s|2H,s,t≥0.
Note that, when H=12, S12 is a standard Brownian motion. We only refer to [21] for information about the sub-fBm and additional references.

Consider the sub-fractional Ornstein–Uhlenbeck process (sub-fOU) of the second kind, defined as the unique pathwise solution to
dXt=−θXtdt+dYt,t≥0,X0=0,
where Yt:=∫0te−sdSasH with at=HetH, and SH is a sub-fBm with Hurst parameter H∈(12,1), whereas θ>00$]]> is considered as unknown parameter. Equivalently, X is the process given explicitly by
Xt=e−θt∫0teθsdYs,
where the integral with respect to Y can be understood in the Skorohod sense. When H=12, the process Yt=∫0te−sdSas12 is a standard Brownian motion, by Lévy’s characterization theorem. Therefore, the process X given by (1) is a standard Ornstein–Uhlenbeck process. Notice also that the model (1) was originally introduced in [15], where the driving process is a fractional Brownian motion, and its definition is related to the Lamperti transform of the fractional Brownian motion.

Our aim is to estimate the parameter θ based the continuous observations of the process (Xt)t≥0 given by (1). We will restrict to the case when θ>00$]]> since the case when θ<0 has been treated in [1]. Throughout the paper we denote by ∫0nutdYt the Skorohod integral (or, say, a divergence-type integral) with respect to the Gaussian process Y (see Preliminaries for definition). Let us recall the idea to construct the least squares estimator (LSE) for the drift coefficient θ, introduced in [13]. The LSE is obtained by minimizing
θ⟼∫0n|X˙t+θXt|2dt.
In this way, we obtain the LSE proposed in (1.4) in the paper [13], which is defined by
θˆn=θ−∫0nXtdYt∫0nXt2dt.

In recent years, several researchers have been interested in studying statistical estimation problems for Gaussian Ornstein–Uhlenbeck processes. We aim to bring a new contribution to the statistical inference for fractional diffusions by estimating the drift parameter of a sub-fOU process of the second kind. Our paper is relevant to the literature on parameter estimation for processes with Gaussian long-memory processes, including [1–5, 8–10, 12, 13, 17, 20]. Estimation of the drift parameters for Ornstein–Uhlenbeck processes driven by fractional noise is a problem that is both well-motivated by practical needs and theoretically challenging. In the finance context, a practical motivation to study this estimation problem is to provide tools to understand volatility modeling in finance. Let us mention some important results in this field where the volatility exhibits long-memory, which means that the volatility today is correlated to past volatility values with a dependence that decays very slowly. Following the approach of [7], the authors of [6] considered the problem of option pricing under a stochastic volatility model that exhibits long-range dependence. More precisely they considered and analyzed the dynamics of the volatility that are described by the equation (1), where the driving process Y is replaced by a standard fractional Brownian motion (fBm) with Hurst parameter H greater than 1/2.

The study of the asymptotic distribution of an estimator is not very useful in general for practical purposes unless the rate of convergence is known. As far as we know, no result of the Berry–Esséen type is known for the distribution of the LSE θˆn of the drift parameter θ of the sub-fOU of the second kind (1).

In order to describe the asymptotic behavior of the LSE θˆn when n→∞, we first need the following proposition given in [16, Corollary 1]. This result is proved based on techniques relied on the combination of Malliavin calculus and Stein’s method (see, e.g., [18]). More precisely, the authors of [16] provided an upper bound of the Kolmogorov distance for central limit theorem of sequences of the form Fn/Gn, where Fn and Gn are functionals of Gaussian fields.

In the following proposition, H⊙2 denote the symmetric tensor product.

Letfn,gn∈H⊙2for alln≥1, and letbnbe a positive function of n such thatI2(gn)+bn>00$]]>almost surely for alln≥1. Define for all sufficiently large positive n,ψ1(n):=1bn2bn2−2‖fn‖H⊗222+8‖fn⊗1fn‖H⊗22,ψ2(n):=2bn22‖fn⊗1gn‖H⊗2+⟨fn,gn⟩H⊗22,ψ3(n):=2bn2‖gn‖H⊗24+2‖gn⊗1gn‖H⊗22.Suppose thatψi(n)→0fori=1,2,3, asn→∞. Then there exists a positive constant C such that for all sufficiently large positive n,supz∈RPI2(fn)I2(gn)+bn≤z−PZ≤z≤Cmaxi=1,2,3ψi(n).

Let us now describe the results we prove in the present paper. First, in (4) we show that the strong consistency of the LSE θˆn defined by (3), as n→∞. Then, in (5) we provide, when H∈(12,1), an upper bound of Kolmogorov distance for central limit theorem of the LSE θˆn.

AssumeH∈(12,1)and letθˆnbe given by (3). Then, asn→∞,θˆn⟶θalmost surely.Moreover, there exists a constant0<C<∞, depending only on θ and H, such that for all sufficiently large positive n,supz∈RPnσθ,Hθ−θˆn≤z−PZ≤z≤Cn,where Z denotes a standard normal random variable, and the positive constantσθ,His given byσθ,H:=θ2∫(0,∞)3F(y1,y2,y3)dy1dy2dy3Hβ(Hθ+1−H,2H−1)−k(θ,H)<∞,with β denoting the classical Beta function,σθ,H<∞(due to [3]and0≤|x−y|2H−2−|x+y|2H−2≤|x−y|2H−2for veryx,y≥0), whereas the function F is defined byF(y1,y2,y3):=e−θ|y1−y3|e−θy2e(1−1H)(y1+y2+y3)1−e−y1H2H−2−1+e−y1H2H−2×e−y2H−e−y3H2H−2−e−y2H+e−y3H2H−2.

The rest of the paper is structured as follows. Section 2 presents some basic elements of Malliavin calculus which are helpful for some of the arguments we use throughout the paper. Section 3 is devoted to the proof of Theorem 1.

Throughout the paper Z denotes a standard normal random variable, and C denotes a generic positive constant (perhaps depending on θ and H, but not on anything else), which may change from line to line.

Preliminaries

In this section, we briefly recall some basic elements of Gaussian analysis, and Malliavin calculus which are helpful for some of the arguments we use throughout the paper. For more details we refer to [18] and [19].

Consider the Gaussian process Yt=∫0te−sdSasH, t≥0, with at=HetH. Assume that 12<H<1. Setting au−1=Hlog(u/H), it follows from [5] that, for every f, g in C1, E∫stf(r)dYr∫uvg(r)dYr=H(2H−1)∫asat∫auavf(ax−1)g(ay−1)e−ax−1−ay−1[|x−y|2H−2−(x+y)2H−2]dxdy=∫st∫uvf(w)g(z)rH(w,z)dwdz, where rH(x,y) is a symmetric kernel given by
rH(w,z)=H2H−1(2H−1)(awaz)1−H[aw−az2H−2−aw+az2H−2]=H2H−1(2H−1)ew/Hez/H1−H×[ew/H−ez/H2H−2−ew/H+ez/H2H−2].
In particular, we obtain the following covariance:
E(Yt−Ys)(Yv−Yu)=∫st∫uvrH(w,z)dwdz.

Fix a time interval [0,T]. We denote by H the canonical Hilbert space associated to the Gaussian process Y. It is the closure of the linear span E generated by the indicator functions 1[0,t], t∈[0,T], with respect to the inner product
⟨1[s,t],1[u,v]⟩H=E(Yt−Ys)(Yv−Yu).
The mapping 1[0,t]↦Yt can be extended to a linear isometry between H and the Gaussian space H1 spanned by Y. We denote this isometry by φ∈H↦Y(φ).

For 12<H<1, we introduce |H| as the set of measurable functions φ on [0,T] such that
‖φ‖|H|2:=∫0T∫0T|φ(u)||φ(v)|rH(u,v)dudv<∞.
Note that, if φ,ψ∈|H|,
E(Y(φ)Y(ψ))=∫0T∫0Tφ(u)ψ(v)rH(u,v)dudv.
The space |H| is a Banach space with the norm ‖.‖|H| and it is included in H. Let Cb∞(Rn,R) be the class of infinitely differentiable functions f:Rn⟶R such that f and all its partial derivatives are bounded. We denote by S the class of smooth cylindrical random variables G of the form
F=f(Y(φ1),…,Y(φn)),
where n≥1, f∈Cb∞(Rn,R) and φ1,…,φn∈H.

The derivative operator D of a smooth cylindrical random variable G of the form (10) is defined as the H-valued random variable
DG=∑i=1n∂f∂xi(Y(φ1),…,Y(φn))φi.
In this way the derivative DG is an element of L2(Ω;H). We denote by D1,2 the closure of S with respect to the norm defined by
‖G‖1,22=E(G2)+E(‖DG‖H2).
The divergence operator δ is the adjoint of the derivative operator D. Concretely, a random variable u∈L2(Ω;H) belongs to the domain of the divergence operator Domδ if
E⟨DG,u⟩H≤cu‖G‖L2(Ω)
for every G∈S, where cu is a constant which depends only on u. In this case δ(u) is given by the duality relation
E(Gδ(u))=EDG,uH_{\mathfrak{H}}}\]]]>
for any F∈D1,2. We will make use of the notation
δ(u)=∫0TusdYs,u∈Domδ.
In particular, for h∈H, Y(h)=δ(h)=∫0ThsdYs.

For every n≥1, let Hn be the nth Wiener chaos of B, that is, the closed linear subspace of L2(Ω) generated by the random variables {Hn(Y(h)),h∈H,‖h‖H=1} where Hn is the nth Hermite polynomial. The mapping In(h⊗n)=n!Hn(Y(h)) provides a linear isometry between the symmetric tensor product H⊙n (equipped with the modified norm ‖.‖H⊙n=n!‖.‖H⊗n) and Hn. For every f,g∈H⊙n the following product formula holds
EIn(f)In(g)=n!⟨f,g⟩H⊗n.
Notice that for every nonrandom Hölder continuous function φ of order α∈(1−H,1), we have
∫0tφsdYs=∫0tφsdYs=Y(φ).

For a smooth and cylindrical random variable F=Y(φ1),…,Y(φn), with φi∈H, i=1,…,n, and f∈Cb∞(Rn) (f and all of its partial derivatives are bounded), we define its Malliavin derivative as the H-valued random variable given by
DF=∑i=1n∂f∂xiY(φ1),…,Y(φn)φi.

For every q≥1, Hq denotes the qth Wiener chaos of Y, defined as the closed linear subspace of L2(Ω) generated by the random variables {Hq(Y(h)),h∈H,‖h‖H=1} where Hq is the qth Hermite polynomial. Wiener chaoses of different orders are orthogonal in L2Ω.

The mapping Iq(h⊗q):=q!Hq(Y(h)) is a linear isometry between the symmetric tensor product H⊙q (equipped with the modified norm ‖.‖H⊙q=q!‖.‖H⊗q) and Hq. For every f,g∈H⊙q the following extended isometry property holds
EIq(f)Iq(g)=q!⟨f,g⟩H⊗q.
We will only need to know the product formula for q=1 (see [18, Section 2.7.3]), which is
I1(f)I1(g)=I2f⊗g+⟨f,g⟩H.

Let {ek,k≥1} be a complete orthonormal system in the Hilbert space H. Given f∈H⊙n, g∈H⊙m, and p=1,…,n∧m, the pth contraction between f and g is the element of H⊗(m+n−2p) defined by
f⊗pg=∑i1,…,ip=1∞⟨f,ei1⊗⋯⊗eip⟩H⊗p⊗⟨g,ei1⊗⋯⊗eip⟩H⊗p.
Let us also recall the hypercontractivity property in Wiener chaos. For h∈H⊗q, the multiple Wiener integrals Iq(h), which exhaust the set Hq, satisfy a hypercontractivity property (equivalence in Hq of all Lp norms for all p≥2), which implies that for any G∈⊕l=1qHl (i.e. in a fixed sum of Wiener chaoses), we have
E[|G|p]1/p⩽cp,qE[|G|2]1/2for anyp≥2.
It should be noted that the constants cp,q above are known with some precision when G is a single chaos term: indeed, by Corollary 2.8.14 in [18], cp,q=p−1q/2.

The following result is a direct consequence of the Borel–Cantelli Lemma (the proof is elementary; see, e.g., [14, Lemma 2.1]). It is convenient for establishing almost sure convergences from Lp convergences.

Letγ>00$]]>. Let(Zn)n≥1be a sequence of random variables. If for everyp≥1there exists a constantcp>00$]]>such that for alln≥1,‖Zn‖Lp(Ω)⩽cp·n−γ,then for allε>00$]]>there exists a random variableαεwhich is almost surely finite such that|Zn|⩽αε·n−γ+εalmost surelyfor alln≥1. Moreover,E|αε|p<∞for allp≥1.

Proof of Theorem <xref rid="j_vmsta179_stat_002">1</xref>

From (3) we can write
θ−θˆn=∫0nXtdYt∫0nXt2dt.
It follows from (2) that
1n∫0nXtdYt=I2hn,
with
hn(s,t):=12ne−θ|t−s|1[0,n]2(s,t).
On the other hand, using the product formula (12),
Xt2=I1e−θ(t−.)1[0,t](.)2=I2e−2θteθueθv1[0,t]2(u,v)+e−θ(t−.)1[0,t](.)H2.
Let us introduce the positive constant
ρθ,H:=H2H(2H−1)θ[β(Hθ+1−H,2H−1)−k(θ,H)],
with
k(θ,H):=∫01uHθ−H(1+u)2H−2du.
Thus
1nρθ,H∫0nXt2dt=I21nρθ,H∫0ne−2θteθueθv1[0,t]2(u,v)dt+1nρθ,H∫0ne−2θteθu1[0,t](u)H2dt=:I2(gn)+bn,
where
bn:=1nρθ,H∫0ne−2θteθu1[0,t](u)H2dt,
and
gn(u,v):=1nρθ,Heθueθve−2θ(u∨v)−e−2θn2θ1[0,n]2(u,v)=12θρθ,Hne−θ|u−v|−e−2θneθueθv1[0,n]2(u,v)=1θρθ,Hnhn(u,v)−ln(u,v),
with hn given by (15), and
ln(u,v):=12nθρθ,He−2θneθueθv1[0,n]2(u,v).
Therefore, combining (14), (15) and (17), we get
nσθ,Hθ−θˆn=I2(fn)I2(gn)+bn,
where σθ,H is given by (6), and
fn:=1ρθ,Hσθ,Hhn.

In order to prove our main result we make use of the following technical lemmas.

LetH∈(12,1), and letbnandfnbe the functions given by (18) and (21), respectively. Then, for alln≥1,|bn−1|≤Cn,1−2‖fn‖H⊗22≤Cn.Consequently, for alln≥1,bn2−2‖fn‖H⊗22≤Cn.

Using (8) and making the change of variables u=x/y, we get
eθu1[0,t](u)H2=H(2H−1)∫a0at∫a0at(x/H)Hθ−H(y/H)Hθ−H[|x−y|2H−2−(x+y)2H−2]dxdy=2H2H(1−θ)+1(2H−1)∫a0aty2Hθ−1∫a0/y1uHθ−H[|1−u|2H−2−(1+u)2H−2]dudy=2H2H(1−θ)+1(2H−1)∫a0aty2Hθ−1∫01uHθ−H[|1−u|2H−2−(1+u)2H−2]dudy−2H2H(1−θ)+1(2H−1)∫a0aty2Hθ−1∫0a0/yuHθ−H[|1−u|2H−2−(1+u)2H−2]dudy:=At−Bt,
where
At=2H2H(1−θ)+1(2H−1)∫a0aty2Hθ−1dy×(∫01uHθ−H|1−u|2H−2−(1+u)2H−2du)=2H2H(1−θ)+1(2H−1)[β(Hθ+1−H,2H−1)−k(θ,H)]×∫a0aty2Hθ−1dy=H2H(2H−1)θ[β(Hθ+1−H,2H−1)−k(θ,H)](e2θt−1).
Moreover,
1n∫0ne−2θtAtdt=H2H(2H−1)θ[β(Hθ+1−H,2H−1)−k(θ,H)]×1+e−2θn−12θn.
Thus
1nρθ,H∫0ne−2θtAtdt−1=1−e−2θn2θn≤12θn.
On the other hand,
|Bt|≤2H2H(1−θ)+1(2H−1)∫a0aty2Hθ−1×∫0a0/yuHθ−H||1−u|2H−2−(1+u)2H−2|dudy≤2H2H(1−θ)+1(2H−1)∫a0aty2Hθ−1∫0a0/yuHθ−H|1−u|2H−2dudy≤2H2H(1−θ)+1(2H−1)∫a0aty2Hθ−1(a0/y)Hθ∫0a0/yu−H(1−u)2H−2dudy≤2H2H−θH+1(2H−1)∫a0atyHθ−1dy∫01u−H(1−u)2H−2du=2H2H+1(2H−1)β(1−H,2H−1)eθt−1Hθ≤Ceθt.
Hence,
1nρθ,H∫0ne−2θt|Bt|dt≤Cn∫0ne−θtdt≤Cn.
Therefore, combining (24), (25) and (26), we deduce (22).

Now let us prove (23). First we decompose the integral ∫[0,n]4 into
∫[0,n]4=∫∪i=15Ai,n=∑i=15∫Ai,n,
where
A1,n=∪i=18Di,n,A2,n=∪i=912Di,n,A3,n=∪i=1316Di,n,A4,n=∪i=1720Di,n,A5,n=∪i=2124Di,n,
with
D1,n:={0<x1<x2<x3<x4<n}D2,n:={0<x1<x2<x4<x3<n}D3,n:={0<x2<x1<x3<x4<n}D4,n:={0<x2<x1<x4<x3<n}D5,n:={0<x3<x4<x1<x2<n}D6,n:={0<x3<x4<x2<x1<n}D7,n:={0<x4<x3<x1<x2<n}D8,n:={0<x4<x3<x2<x1<n},D9,n:={0<x1<x3<x2<x4<n}D10,n:={0<x3<x1<x4<x2<n}D11,n:={0<x2<x4<x1<x3<n}D12,n:={0<x4<x2<x3<x1<n},D13,n:={0<x1<x3<x4<x2<n}D14,n:={0<x3<x1<x2<x4<n}D15,n:={0<x2<x4<x3<x1<n}D16,n:={0<x4<x2<x1<x3<n},D17,n:={0<x1<x4<x2<x3<n}D18,n:={0<x4<x1<x3<x2<n}D19,n:={0<x2<x3<x1<x4<n}D20,n:={0<x3<x2<x4<x1<n},D21,n:={0<x1<x4<x3<x2<n}D22,n:={0<x4<x1<x2<x3<n}D23,n:={0<x3<x2<x1<x4<n}D24,n:={0<x2<x3<x4<x1<n}.
Therefore, using (9), (27), and setting
mH(x1,x2,x3,x4):=e−θ|x1−x3|e−θ|x2−x4|rH(x1,x2)rH(x3,x4),
we have
hnH⊗22=14n∫[0,n]4mH(x1,x2,x3,x4)dx1…dx4=14n∫A1,n+∫A2,n+∫A3,n+∫A4,n+∫A5,n×mH(x1,x2,x3,x4)dx1…dx4=14n8∫D1,n+4∫D9,n+4∫D13,n+4∫D17,n+4∫D21,n×mH(x1,x2,x3,x4)dx1…dx4=:2I1,n+I2,n+I3,n+I4,n+I5,n,
where we used the fact that
∫D1,nmH(x1,x2,x3,x4)dx1dx2dx3dx4=⋯=∫D8,nmH(x1,x2,x3,x4)dx1dx2dx3dx4,∫D9,nmH(x1,x2,x3,x4)dx1dx2dx3dx4=⋯=∫D12,nmH(x1,x2,x3,x4)dx1dx2dx3dx4,∫D13,nmH(x1,x2,x3,x4)dx1dx2dx3dx4=⋯=∫D16,nmH(x1,x2,x3,x4)dx1dx2dx3dx4,∫D17,nmH(x1,x2,x3,x4)ddx1dx2dx3dx4=⋯=∫D20,nmH(x1,x2,x3,x4)dx1dx2dx3dx4,∫D21,nmH(x1,x2,x3,x4)dx1dx2dx3dx4=⋯=∫D24,nmH(x1,x2,x3,x4)dx1dx2dx3dx4.
Let us now estimate I1,n. Making the change of variables y3=x4−x1, y2=x4−x2, y1=x4−x3, and y4=x4, we obtain that 1H4H−2(2H−1)2I1,n is equal to
1n∫0n∫0<x1<x2<x3<x4e−θ|x1−x3|e−θ|x2−x4|e(1/H−1)(x1+x2+x3+x4)×ex1/H−ex2/H2H−2−ex1/H+ex2/H2H−2×ex3/H−ex4/H2H−2−ex3/H+ex4/H2H−2dx1dx2dx3dx4=1n∫0n∫0<y1<y2<y3<y4F(y1,y2,y3)dy1dy2dy3dy4=1n∫0n∫0<y1<y2<y3<∞−∫0n∫y4∞∫0y3∫0y2F(y1,y2,y3)dy1dy2dy3dy4=∫0<y1<y2<y3<∞F(y1,y2,y3)dy1dy2dy3−1n∫0n∫y4∞∫0y3∫0y2F(y1,y2,y3)dy1dy2dy3dy4,
where the function F is given by (7). Moreover, 1n∫0n∫y4∞∫0y3∫0y2F(y1,y2,y3)dy1dy2dy3dy4 is equal to
1n∫0n∫y4∞∫0y3∫0y2e−θ|y1−y3|e−θy2e(1−1H)(y1+y2+y3)e−y2H−e−y3H2H−2−e−y2H+e−y3H2H−2×1−e−y1H2H−2−1+e−y1H2H−2dy1dy2dy3dy4=1n∫0n∫y4∞∫0y3∫0y2e−θ|y1−y3|e−θy2e(1−1H)(y1+y2+y3)(e−y2H−e−y3H2H−21−e−y1H2H−2−e−y2H−e−y3H2H−21+e−y1H2H−2−e−y2H+e−y3H2H−21−e−y1H2H−2+e−y2H+e−y3H2H−21+e−y1H2H−2)dy1dy2dy3dy4:=An(1)−An(2)−An(3)+An(4),
where
An(1)=1n∫0n∫y4∞∫0y3∫0y2e−θ|y1−y3|e−θy2e(1−1H)(y1+y2+y3)×e−y2H−e−y3H2H−21−e−y1H2H−2dy1dy2dy3dy4≤1n∫0n∫y4∞∫0y3∫0y2e−θy3e(1−1H)(y1+y2+y3)×e−y2H−ey3H2H−21−e−y1H2H−2dy1dy2dy3dy4≤Hβ(1−H,2H−1)n∫0n∫y4∞∫0y3e−θy3e(1−1H)(y2+y3)×e−y2H−e−y3H2H−2dy2dy3dy4=Hβ(1−H,2H−1)n∫0n∫y4∞∫0y3e−θy3e(1−1H)(y3−y2)×1−e−(y3−y2)/H2H−2dy2dy3dy4=Hβ(1−H,2H−1)n∫0n∫y4∞∫0y3e−θy3e(1−1H)x2×1−e−x2H2H−2dx2dy3dy4≤Hβ(1−H,2H−1)2n∫0n∫y4∞e−θy3dy3dy4≤Hβ(1−H,2H−1)2θ2n.
Since
|e−y2H+e−y3H|2H−2≤|e−y2H−e−y3H|2H−2and|1+e−y1H|2H−2≤|1−e−y1H|2H−2,
we have
|An(2)|≤|An(1)|≤Cn,|An(3)|≤|An(1)|≤Cnand|An(4)|≤|An(1)|≤Cn.
Consequently,
1n∫0n∫y4∞∫0y3∫0y2F(y1,y2,y3)dy1dy2dy3dy4≤Cn.
Combining (29) and (31) we deduce
I1,n−H4H−2(2H−1)2∫0<y1<y2<y3<∞F(y1,y2,y3)dy1dy2dy3≤Cn.
Moreover,
I1,n−H4H−2(2H−1)2∫0<y1<y3<y2<∞F(y1,y2,y3)dy1dy2dy3≤Cn,
since
∫0<y1<y3<y2<∞F(y1,y2,y3)dy1dy3dy2=∫0<y1<y2<y3<∞F(y1,y2,y3)dy1dy2dy3.
Using similar arguments as above, we can conclude I2,n−H4H−2(2H−1)2∫0<y2<y1<y3<∞F(y1,y2,y3)dy2dy1dy3≤Cn,I3,n−H4H−2(2H−1)2∫0<y2<y3<y1<∞F(y1,y2,y3)dy2dy3dy1≤Cn,I4,n−H4H−2(2H−1)2∫0<y3<y1<y2<∞F(y1,y2,y3)dy3dy1dy2≤Cn,I5,n−H4H−2(2H−1)2∫0<y3<y2<y1<∞F(y1,y2,y3)dy3dy2dy1≤Cn. Combining (28), (32)–(37) and the fact that
(0,∞)3=⋃σ∈S{0<yσ(1)<yσ(2)<yσ(3)<∞},
where S is a set of permutations on a set {1,2,3}, we deduce that
hnH⊗22−H4H−2(2H−1)2∫(0,∞)3F(y1,y2,y3)dy1dy2dy3≤Cn,
which proves (23). □

SupposeH∈(12,1). Letgnandfnbe the functions given by (19) and (21), respectively. Then, for alln≥1,‖fn⊗1fn‖H⊗2≤Cn,‖gn‖H⊗2≤Cn,‖gn⊗1gn‖H⊗2≤Cn3/2,‖fn⊗1gn‖H⊗2≤Cn,⟨fn,gn⟩H⊗2≤Cn.

Taking into account that
8‖fn⊗1fn‖H⊗22=Var12‖DFn‖H⊗22,forFn:=I2(fn),
in order to obtain (39), it is sufficient to show that
Var‖DFn‖H⊗22≤Cn.
Now using (9), we can write
‖DFn‖H⊗22=4∫0n∫0nI1(fn(u,.))I1(fn(v,.))rH(u,v)dudv
Using the multiplicative formula (12), we see that
I1(fn(u,.))I1(fn(v,.))=⟨fn(u,.),fn(v,.)⟩H⊗2+I2(fn(u,.)⊗˜fn(v,.))=:A1(u,v)+A2(u,v).
Here A1 is deterministic and A2 has expectation zero. Thus, we obtain that
‖DFn‖H⊗22−E(‖DFn‖H⊗22)=4∫0n∫0nA2(u,v)rH(u,v)dudv
Hence, in order to have (44), we need to show that
E∫0n∫0nA2(u,v)rH(u,v)dudv2≤Cn.
We have
E∫0n∫0nA2(u,v)rH(u,v)dudv2=∫[0,n]4EA2(u1,v1)A2(u2,v2)rH(u1,v1)rH(u2,v2)du1dv1du2dv2=∫[0,n]4(∫[0,n]4fn(u1,.)⊗˜fn(v1,.)(x1,y1)×fn(u2,.)⊗˜fn(v2,.)(x2,y2)rH(x1,y1)rH(x2,y2)dx1dy1dx2dy2)×rH(u1,v1)rH(u2,v2)du1dv1du2dv2=∫[0,n]8fn(u1,.)⊗˜fn(v1,.)(x1,y1)fn(u2,.)⊗˜fn(v2,.)(x2,y2)×rH(x1,y1)rH(x2,y2)rH(u1,v1)×rH(u2,v2)dx1dy1dx2dy2du1dv1du2dv2.
Note that for every 0≤x, y≤TrH(x,y)≤H2H−1(2H−1)e(1H−1)(x+y)exH−eyH2H−2:=K(x,y).
Thus,
E∫0n∫0nA2(u,v)rH(u,v)dudv2≤∫[0,n]8fn(u1,.)⊗˜fn(v1,.)(x1,y1)fn(u2,.)⊗˜fn(v2,.)(x2,y2)×K(x1,y1)K(x2,y2)K(u1,v1)K(u2,v2)dx1dy1dx2dy2du1dv1du2dv2≤Cn,
where the last inequality comes from [3, Lemma 5.1]. Hence the inequality (39) is obtained.

Since for every (u,v)∈[0,n]2, gn(u,v)≥0, hn(u,v)≥0 and ln(u,v)≥0, using (19), we get
‖gn‖H⊗2≤1θρθ,Hn‖hn‖H⊗2.
Combining this with (38), we obtain (40). Similarly, using (19), (21) and (39), we have
‖gn⊗1gn‖H⊗22≤Cn2‖hn⊗1hn‖H⊗22≤Cn2‖fn⊗1fn‖H⊗22≤Cn3,
which implies (41).

It is well known that
‖fn⊗1gn‖H⊗22=⟨fn⊗1fn,gn⊗1gn⟩H⊗2,
due to a straightforward application of the definition of contractions and the Fubini theorem.

Thus, from (39) and (41), we obtain
‖fn⊗1gn‖H⊗22≤‖fn⊗1fn‖H⊗2‖gn⊗1gn‖H⊗2≤Cn3,
which leads to (42).

Finally, the inequality (43) is a direct consequence of (23) and (40). The proof of the lemma is thus complete. □

First we prove the almost sure convergence (4). From (20) we can write
θ−θˆn=σθ,HnI2(fn)I2(gn)+bn.
Furthermore, using (23), and (40), we have
E1nI2(fn)2=2n‖fn‖H⊗22≤Cn,andEI2(gn)2=2‖gn‖H⊗22≤Cn.
Combining this with (13) and Lemma 1, we obtain that, as n→∞,
1nI2(fn)⟶0,andI2(gn)⟶0
almost surely. Moreover, it follows from (22) that bn⟶0, as n→∞. Therefore (4) is obtained.

Let us now prove the convergence in law (5). It follows from (20), Proposition 1, Lemma 2 and Lemma 3 that for every n large,
supz∈RPnσθ,Hθ−θˆn≤z−PZ≤z≤C×max{bn2−2‖fn‖H⊗22,‖fn⊗1fn‖H⊗2,⟨fn,gn⟩H⊗2,‖fn⊗1gn‖H⊗212,‖gn⊗1gn‖H⊗2,‖gn‖H⊗22}≤Cn,
which implies the desired conclusion. □

Acknowledgments

We thank the two anonymous reviewers for their very careful reading and suggestions, which have led to significant improvements in the presentation of our results.

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