A first-order limit law for functionals of two independent fractional Brownian motions in the critical case

Abstract

We prove a first-order limit law for functionals of two independent $d$-dimensional fractional Brownian motions with the same Hurst index $H=2/d\,(d\ge 4)$, using the method of moments and extending a result by LeGall in the case of Brownian motion.

Appendix

In this section, we prove a lemma which is important in the Proof of Theorem 1.1.

Recall that $\fancyscript{P}_m$ is the set of all permutations of $\{1,2,\ldots ,m\}$. For any $(\tau , \sigma )\in \fancyscript{P}_m\times \fancyscript{P}_m$, define

$$\begin{aligned} \phi _{\tau ,\sigma }(i)=\sup \left\{ \tau (j): j\in A^{\sigma }_i\right\} , \end{aligned}$$

where

$$\begin{aligned} A^{\sigma }_i=\big \{\sigma (i), \ldots , \sigma (m)\big \}\Delta \big \{\sigma (i)+1, \ldots , \sigma (m)+1\big \}. \end{aligned}$$

Let $\Omega _m$ be the set of all $(\tau , \sigma )\in \fancyscript{P}_m\times \fancyscript{P}_m$ such that $\phi _{\tau ,\sigma }$ is bijective. By Lemma 4 in [1], the number of elements in $\Omega _m$ is

$$\begin{aligned} \#\Omega _m=\prod ^{m}_{i=1}(2i-1)=(2m-1)!!. \end{aligned}$$

Recall the definition of $R^{n}_{m}(a_1,a_2,b_1,b_2)$ in (3.15). The following lemma shows that

$$\begin{aligned} \lim \limits _{n\rightarrow \infty } R^{n}_{m}(a_1,a_2,b_1,b_2) \end{aligned}$$

exists and does not depend on $a_1$ and $a_2$.

Lemma 3.4

$$\begin{aligned} \lim _{n\rightarrow \infty } R^{n}_{m}(a_1,a_2,b_1,b_2)=\left( \frac{2\pi ^{\frac{d}{2}}t\, \Gamma ^2\left( \frac{d+4}{4}\right) }{(b_1b_2)^{\frac{d}{4}}\, \Gamma \left( \frac{d+2}{2}\right) }\right) ^m\, (2m-1)!!. \end{aligned}$$

(3.17)

Proof

Recall that $Hd=2$. Then

$$\begin{aligned} R^{n}_{m}(a_1,a_2,b_1,b_2)&=\frac{m!}{n^m} \sum _{\sigma \in \fancyscript{P}_m}\int _{|y_1|<a_1e^{nHt}}\cdots \int _{|y_m|<a_1e^{nHt}} \int _{[0,a_2]^{2m}} \\&\quad \times \exp \bigg (-b_1\sum \limits ^m_{i=1} |y_i|^2 u_i^{2H}-b_2\sum \limits ^m_{i=1}\sup _{j\in A^{\sigma }_i} |y_j|^2 v_i^{2H}\bigg ) \hbox {d}u\, \hbox {d}v\, \hbox {d}y. \end{aligned}$$

It is easy to see that

$$\begin{aligned}&\limsup _{n\rightarrow \infty } R^{n}_{m}(a_1,a_2,b_1,b_2)\\&\quad =\limsup _{n\rightarrow \infty }\frac{m!}{n^m} \sum _{\sigma \in \fancyscript{P}_m}\int _{1<|y_1|<a_1e^{nHt}}\cdots \int _{1<|y_m|<a_1e^{nHt}} \int _{[0,a_2]^{2m}} \\&\qquad \times \exp \bigg (-b_1\sum \limits ^m_{i=1} |y_i|^2 u_i^{2H}-b_2\sum \limits ^m_{i=1}\sup _{j\in A^{\sigma }_i} |y_j|^2 v_i^{2H}\bigg ) \hbox {d}u\, \hbox {d}v\, \hbox {d}y\\&\quad \le \limsup _{n\rightarrow \infty }\frac{m!}{n^m} \int _{[0,+\infty )^{2m}} \exp \bigg (-b_1\sum \limits ^m_{i=1} u_i^{2H}-b_2\sum \limits ^m_{i=1}v_i^{2H}\bigg ) \hbox {d}u\, \hbox {d}v\\&\qquad \times \sum _{\sigma \in \mathcal {P}_m}\int _{1<|y_1|<a_1e^{nHt}}\cdots \int _{1<|y_m|<a_1e^{nHt}} \Big (\prod ^{m}_{i=1} |y_i|^{-\frac{d}{2}}\Big )\Big (\prod ^{m}_{i=1} (\sup _{j\in A^{\sigma }_i} |y_j|)^{-\frac{d}{2}}\Big )\, \hbox {d}y. \end{aligned}$$

Making the change of variables $r_i=|y_i|$ for $i=1,2,\ldots ,m$,

$$\begin{aligned}&\limsup _{n\rightarrow \infty } R^{n}_{m}(a_1,a_2,b_1,b_2)\\&\quad \le \bigg (\frac{\Gamma \left( \frac{d+4}{4}\right) }{b_1^{\frac{d}{4}}}\bigg )^m\bigg (\frac{\Gamma \left( \frac{d+4}{4}\right) }{b_2^{\frac{d}{4}}}\bigg )^m \bigg (\frac{d\pi ^{\frac{d}{2}}}{\Gamma \left( \frac{d+2}{2}\right) }\bigg )^m\\&\qquad \times \limsup _{n\rightarrow \infty }\frac{m!}{n^m} \sum _{\sigma \in \fancyscript{P}_m}\int _{(1,a_1e^{nHt})^m} \Big (\prod ^{m}_{i=1} r_i^{-\frac{d}{2}}\Big )\Big (\prod ^{m}_{i=1} \big (\sup _{j\in A^{\sigma }_i} r_j\big )^{-\frac{d}{2}}\Big )\, \hbox {d}r. \end{aligned}$$

Making another change of variables $r_i=e^{n\alpha _i}$ for $i=1,2,\ldots ,m$, the right-hand side of the above inequality is equal to

$$\begin{aligned}&\Bigg (\frac{d\pi ^{\frac{d}{2}}\, \Gamma ^2\left( \frac{d+4}{4}\right) }{(b_1b_2)^{\frac{d}{4}}\Gamma \left( \frac{d+2}{2}\right) }\Bigg )^m \limsup _{n\rightarrow \infty } m!\sum _{\sigma \in \fancyscript{P}_m}\int _{(0,\frac{\ln a_1}{n}+Ht)^m}\\&\quad \exp \bigg (\frac{d}{2}n\Big (\sum ^{m}_{i=1} \alpha _i-\sum ^{m}_{i=1}\sup _{j\in A^{\sigma }_i} \alpha _j \Big )\bigg )\, \hbox {d}\alpha \\&\quad =\left( \frac{d\pi ^{\frac{d}{2}}\, \Gamma ^2\left( \frac{d+4}{4}\right) }{(b_1b_2)^{\frac{d}{4}}\Gamma \left( \frac{d+2}{2}\right) }\right) ^m (Ht)^m\, \#\Omega _m, \end{aligned}$$

where in the last equality we used the dominated convergence theorem and the definition of the set $\Omega _m$.

Therefore,

$$\begin{aligned} \limsup _{n\rightarrow \infty } R^{n}_{m}(a_1,a_2,b_1,b_2)\le \Bigg (\frac{2\pi ^{\frac{d}{2}}t\, \Gamma ^2\left( \frac{d+4}{4}\right) }{(b_1b_2)^{\frac{d}{4}}\, \Gamma \left( \frac{d+2}{2}\right) }\Bigg )^m\, (2m-1)!!. \end{aligned}$$

(3.18)

On the other hand,

$$\begin{aligned}&\liminf _{n\rightarrow \infty } R^{n}_{m}(a_1,a_2,b_1,b_2)\\&\quad =\liminf _{n\rightarrow \infty }\frac{m!}{n^m} \sum _{\sigma \in \fancyscript{P}_m}\int _{K^{H}<|y_1|<a_1e^{nHt}}\cdots \int _{K^H<|y_m|<a_1e^{nHt}} \int _{[0,a_2]^{2m}} \\&\qquad \times \exp \bigg (-b_1\sum \limits ^m_{i=1} |y_i|^2 u_i^{2H}-b_2\sum \limits ^m_{i=1}\sup _{j\in A^{\sigma }_i} |y_j|^2 v_i^{2H}\bigg ) \hbox {d}u\, \hbox {d}v\, \hbox {d}y\\&\quad \ge \liminf _{n\rightarrow \infty }\frac{m!}{n^m} \int _{[0,Ka_2]^{2m}} \exp \bigg (-b_1\sum \limits ^m_{i=1} u_i^{2H}-b_2\sum \limits ^m_{i=1}v_i^{2H}\bigg ) \hbox {d}u\, \hbox {d}v\\&\qquad \times \sum _{\sigma \in \mathcal {P}_m}\int _{K^H<|y_1|<a_1e^{nHt}}\cdots \int _{K^H<|y_m|<a_1e^{nHt}} \bigg (\prod ^{m}_{i=1} |y_i|^{-\frac{d}{2}}\bigg )\bigg (\prod ^{m}_{i=1} (\sup _{j\in A^{\sigma }_i} |y_j|)^{-\frac{d}{2}}\bigg )\, \hbox {d}y\\&\quad \ge \int _{[0,Ka_2]^{2m}} \exp \bigg (-b_1\sum \limits ^m_{i=1} u_i^{2H}-b_2\sum \limits ^m_{i=1}v_i^{2H}\bigg ) \hbox {d}u\, \hbox {d}v\, \bigg (\frac{d\pi ^{\frac{d}{2}}}{\Gamma (\frac{d+2}{2})}\bigg )^m\\&\qquad \times \liminf _{n\rightarrow \infty }\frac{m!}{n^m} \sum _{\sigma \in \fancyscript{P}_m}\int _{(K^H,a_1e^{nHt})^m} \Big (\prod ^{m}_{i=1} r_i^{-\frac{d}{2}}\Big )\Big (\prod ^{m}_{i=1} (\sup _{j\in A^{\sigma }_i} r_j)^{-\frac{d}{2}}\Big )\, dr\\&\quad =\int _{[0,Ka_2]^{2m}} \exp \bigg (-b_1\sum \limits ^m_{i=1} u_i^{2H}-b_2\sum \limits ^m_{i=1}v_i^{2H}\bigg ) \hbox {d}u\, \hbox {d}v\bigg (\frac{d\pi ^{\frac{d}{2}}}{\Gamma (\frac{d+2}{2})}\bigg )^m\\&\qquad \times \liminf _{n\rightarrow \infty } m!\sum _{\sigma \in \fancyscript{P}_m}\int _{(\frac{H\ln K}{n},\frac{\ln a_1}{n}+Ht)^m} \exp \bigg (\frac{d}{2}n\Big (\sum ^{m}_{i=1} \alpha _i-\sum ^{m}_{i=1}\sup _{j\in A^{\sigma }_i} \alpha _j \Big )\bigg )\, d\alpha \\&\quad =\int _{[0,Ka_2]^{2m}} \exp \bigg (-b_1\sum \limits ^m_{i=1} u_i^{2H}-b_2\sum \limits ^m_{i=1}v_i^{2H}\bigg ) \hbox {d}u\, \hbox {d}v\, \bigg (\frac{2\pi ^{\frac{d}{2}}t}{\Gamma \left( \frac{d+2}{2}\right) }\bigg )^m (2m-1)!!. \end{aligned}$$

Since the positive constant $K$ can be arbitrarily large,

$$\begin{aligned}&\liminf _{n\rightarrow \infty } R^{n}_{m}(a_1,a_2,b_1,b_2)\nonumber \\&\quad \ge \int _{[0,+\infty )^{2m}} \exp \bigg (-b_1\sum \limits ^m_{i=1} u_i^{2H}-b_2\sum \limits ^m_{i=1}v_i^{2H}\bigg ) \hbox {d}u\, \hbox {d}v\, \bigg (\frac{2\pi ^{\frac{d}{2}}t}{\Gamma \left( \frac{d+2}{2}\right) }\bigg )^m (2m-1)!!\nonumber \\&\quad =\Bigg (\frac{2\pi ^{\frac{d}{2}}t\, \Gamma ^2\left( \frac{d+4}{4}\right) }{(b_1b_2)^{\frac{d}{4}}\, \Gamma \left( \frac{d+2}{2}\right) }\Bigg )^m\, (2m-1)!!, \end{aligned}$$

(3.19)

where the last equality follows from a change of variables, the definition of Gamma function and the fact $Hd=2$.

Combining (3.18) and (3.19) gives the desired result (3.17).$\square$