Limit Theorems for Quadratic Variations of the Lei–Nualart Process

Abstract

Let X be a Lei–Nualart process with Hurst index $H\in (0, 1)$, $Z_{1}$ be an Hermite random variable. For any $n \ge 1$, set

$$V_{n}=\sum _{k=0}^{n-1}\left[ n^{2H}(\varDelta _k X)^{2}-n^{2H}\mathrm {I\! E}(\varDelta _k X)^{2}\right] .$$

The aim of the current paper is to derive, in the case when the Hurst index verifies $H > 3/4$, an upper bound for the total variation distance between the laws $\mathcal {L}(Z_n)$ and $\mathcal {L}(Z_{1})$, where $Z_{n}$ stands for the correct renormalization of $V_{n}$ which converges in distribution towards $Z_{1}$. We derive also the asymptotic behavior of quadratic variations of process X in the critical case $H=3/4$, i.e. an upper bound for the total variation distance between the $\mathcal {L}(Z_n)$ and the Normal law.

Appendix

Proof

 (Proof of Lemma 5.2) (i) We have

$$ \alpha (x,x) = (2x+2)^{2H} - 2(2x+1)^{2H} + (2x)^{2H}. $$

Hence by applying Taylor’s theorem we observe

$$ \alpha (x,x) = 2H(2H-1)(2x)^{2H-2} + O(x^{2H-3}) $$

from which the claim follows.

(ii) We show that for any fixed x (resp. y), the function $x \mapsto \alpha (x,y)$ (resp. $y\mapsto \alpha (x,y)$) is decreasing provided that $H\in \left[ \frac{3}{4},1\right)$. For this, let y be fixed. Then

$$\begin{aligned} \begin{aligned} \partial _x \alpha (x,y) = 2H\left[ (x+y+2)^{2H-1}-(x+y+1)^{2H-1} \right. \\ \left. -(x+y+1)^{2H-1}+(x+y)^{2H-1}\right] \\ = 2H\left[ z_y(x+1) - z_y(x)\right] , \end{aligned} \end{aligned}$$

where $z_y(x) = (x+y+1)^{2H-1} - (x+y)^{2H-1}.$ Here

$$ z'_y(x) = (2H-1)\left[ (x+y+1)^{2H-2} - (x+y)^{2H-2}\right] \le 0, $$

and thus $z_y(x)$ is decreasing. This implies that $\alpha (x,y)$ is also decreasing as a function of x. Furthermore, by changing the roles of x and y we observe that for each fixed x, the function $y \mapsto \alpha (x,y)$ is also decreasing. Consequently, for every $0\le x \le y$ we have $\alpha (y,y) \le \alpha (x,y) \le \alpha (x,x).$    $\square$

Proof

 (Proof of Lemma 5.3) Using (5.3) and (5.4) we have, for any $H\in (0,1)\setminus \{\frac{1}{2}\}$,

$$\mathrm {I\! E}[V_{n}^{2}] =\frac{1}{2}\sum _{r,k=0}^{n-1}\alpha (k,r)^{2}=\sum _{r<k}^{n-1}\alpha (k,r)^{2}+\frac{1}{2}\sum _{k=0}^{n-1}\alpha (k,k)^{2} = I(n)+J(n).$$

The idea is to show that, depending on the value of H, the term I(n) behaves asymptotically correctly while the term J(n) is negligible.

(i) Case $H=\frac{3}{4}$: By Lemma 5.2 item (i) we have

$$ J(n) \le C \sum _{k=1}^{n-1} k^{-1} \le C\log n, $$

and hence, as $n\rightarrow \infty$, $\frac{J(n)}{n}\rightarrow 0.$ For the term I(n) we have $\alpha (r,k)^2 = (k+r)^{-1} + O\left( (k+r)^{-2}\right) .$ Here

$$ \frac{1}{n}\sum _{r<k}^{n-1} (k+r)^{-2} \le \frac{C}{n} \sum _{k=1}^{n-1} k^{-1} \le \frac{C \log n}{n} \rightarrow 0 $$

and consequently the term $O\left( (k+r)^{-2}\right)$ does not affect the asymptotic variance. For the term $(k+r)^{-1}$ we substitute $s = k+r$ to get

$$ \sum _{r<k}^{n-1} (k+r)^{-1} = \sum _{k=0}^{n-1} \sum _{s=k+1}^{2k-1} s^{-1}. $$

Changing the order of summation we get

$$ \sum _{k=0}^{n-1} \sum _{s=k+1}^{2k-1} s^{-1} = \sum _{s=1}^{2n-3} \sum _{\frac{s+1}{2}\le k \le s-1} s^{-1}. $$

Here $\sum _{\frac{s+1}{2}\le k \le s-1} 1 = \frac{s}{2} - \frac{1}{2}$ if s is odd and $\sum _{\frac{s+1}{2}\le k \le s-1} 1 = \frac{s}{2}$ if s is even. This implies $c n \le \sum _{r<k}^{n-1} (k+r)^{-1} \le Cn.$

(ii) Case $\frac{3}{4}< H<1$: By Lemma 5.2 item (i) we have $J(n) \le C \sum _{k=1}^{n-1} k^{4H-4} \le Cn^{4H-3}$ since $4H-4>-1$. Hence, as $n\rightarrow \infty$, $\frac{J(n)}{n^{4H-2}}\rightarrow 0.$ For the term I(n) we use Lemma 5.2 item (ii) to observe that

$$ \sum _{r<k}^{n-1}\alpha (k,k)^2 \le \sum _{r<k}^{n-1} \alpha (k,r)^2 \le \sum _{r<k}^{n-1}\alpha (r,r)^2. $$

Here

$$ \sum _{r<k}^{n-1}\alpha (k,k)^2 \ge C\sum _{k=1}^{n-1} \left[ k k^{4H-4} + O(k^{4H-4})\right] . $$

Here $\frac{1}{n^{4H-2}} \sum _{k=1}^{n-1}k^{4H-4} \le Cn^{-1} \rightarrow 0$ from which it follows that $\sum _{r<k}^{n-1}\alpha (k,k)^2 \ge c n^{4H-2}.$ On the other hand, we have

$$ \sum _{r<k}^{n-1}\alpha (r,r)^2 \le C\sum _{r=1}^{n-1}(n-r)r^{4H-4} \le Cn^{4H-2} $$

from which the claim follows.    $\square$