How Does Tempering Affect the Local and Global Properties of Fractional Brownian Motion?

Abstract

The present paper investigates the effects of tempering the power law kernel of the moving average representation of a fractional Brownian motion (fBm) on some local and global properties of this Gaussian stochastic process. Tempered fractional Brownian motion (TFBM) and tempered fractional Brownian motion of the second kind (TFBMII) are the processes that are considered in order to investigate the role of tempering. Tempering does not change the local properties of fBm including the sample paths and p-variation, but it has a strong impact on the Breuer–Major theorem, asymptotic behavior of the third and fourth cumulants of fBm and the optimal fourth moment theorem.

Appendices

Appendix A

This “Appendix” contains some notations, definitions and well-known results that we applied in the main text of this paper.

1.1 Special Functions $K_{\nu }$ and ${_2F_3}$

In this subsection we present definitions of two special functions $K_{\nu }$ and ${_2F_3}$ that we have used in Sect. 2.1. We also provide the proof of Lemma 5.1, see below, that we used in the proof of Proposition 2.4. First, we start with the definition of the modified Bessel function of the second kind that appears in the variance and covariance function of TFBM, see part (a) of Lemma 2.3. A modified Bessel function of the second kind $K_{\nu }(x)$ has the integral representation

$$\begin{aligned} K_{\nu }(x)=\int _0^\infty e^{-x \cosh t} \cosh {\nu t}\ \mathrm{d}t, \end{aligned}$$

where $\nu>0, x>0$. The function $K_{\nu }(x)$ also has the series representation

$$\begin{aligned} K_{\nu }(x) = \frac{1}{2}\pi \frac{ I_{-\nu }(x) - I_{\nu }(x) }{\sin (\pi \nu )}, \end{aligned}$$

where $I_{\nu }(x)=(\frac{1}{2}|x|)^{\nu } \sum _{n=0}^{\infty } \frac{ ( \frac{1}{2}x)^{2n} }{n! \varGamma (n+1+\nu )}$ is called the Bessel function. We refer the reader to see ([12, Section 8.43], pages 140–1414) for more information about the modified Bessel function of the second kind.

Next, we define the confluent Hypergeometric function ${_2F_3}$ that we used to obtain the variance and covariance of TFBMII, see part (b) of Lemma 2.3. In general, a generalized hypergeometric function ${_pF_q}$ is defined by

$$\begin{aligned} {_pF_q}(a_1,\cdots , a_p, b_1, \cdots , b_q, z)= \sum _{k=0}^{\infty } \frac{ (a_1)_{k} (a_2)_{k} \cdots (a_p)_{k} }{ (b_1)_{k} (b_2)_{k} \cdots (b_q)_{k} }\frac{z^k}{k!}, \end{aligned}$$

where $(c_i)_{k}=\frac{\varGamma (c_i + k)}{\varGamma (k)}$ is called Pochhammer symbol. Therefore,

$$\begin{aligned}&{_2F_3}( \{a_1, a_2\}, \{b_1, b_2, b_3\}, z)= {_2F_3}(a_1, a_2, b_1, b_2, b_3, z)\\&\quad = \sum _{k=0}^{\infty } \frac{ \varGamma (a_1 + k)\varGamma (a_2 + k) \varGamma (k) }{ \varGamma (b_1 + k)\varGamma (b_2 + k) \varGamma (b_3 + k) } \frac{z^k}{k!}, \end{aligned}$$

Lemma 5.1

Integral $I=\int _{0}^{\infty }\left( \int _{0}^{\infty }(s+x)^{H-3/2}e^{-\lambda (s+x)}\mathrm{d}s\right) ^2\mathrm{d}x$ is finite for any $H>0$.

Proof

Let $H<1/2$. Then,

$$\begin{aligned} I= & {} \int _{0}^{\infty }\left( \int _{x}^{\infty }s^{H-3/2}e^{-\lambda s}\mathrm{d}s\right) ^2\mathrm{d}x\le \int _{0}^{\infty }e^{-2\lambda x}\left( \int _{x}^{\infty }s^{H-3/2}\mathrm{d}s\right) ^2\mathrm{d}x\\= & {} \left( H-1/2\right) ^{-2}\int _{0}^{\infty }e^{-2\lambda x}x^{2H-1}\mathrm{d}x =\left( H-1/2\right) ^{-2}(2\lambda )^{-2H}\varGamma (2H)<\infty . \end{aligned}$$

Let $H>1/2$. Then,

$$\begin{aligned} I= & {} \int _{0}^{\infty }\left( \int _{x}^{\infty }s^{H-3/2}e^{-\lambda s}\mathrm{d}s\right) ^2\mathrm{d}x\le \int _{0}^{\infty }e^{-\lambda x}\left( \int _{x}^{\infty }s^{H-3/2}e^{-\frac{ \lambda s}{2}}\mathrm{d}s\right) ^2\mathrm{d}x\\\le & {} \int _{0}^{\infty }e^{-\lambda x}\left( \int _{0}^{\infty }s^{H-3/2}e^{-\frac{ \lambda s}{2}}\mathrm{d}s\right) ^2\mathrm{d}x =2^{2H-1}\lambda ^{-2H}\varGamma ^2(H-1/2)<\infty . \end{aligned}$$

Finally, let $H=1/2$. Then,

$$\begin{aligned} I= & {} \int _{0}^{\infty }\left( \int _{x}^{\infty }s^{-1}e^{-\lambda s}\mathrm{d}s\right) ^2\mathrm{d}x\le \int _{0}^{1}x^{-1/2}\left( \int _{0}^{\infty }s^{ -3/4}e^{-\frac{ \lambda s}{2}}\mathrm{d}s\right) ^2\mathrm{d}x\\&+\int _{1}^{\infty }x^{-2}\left( \int _{0}^{\infty } e^{-\lambda s}\mathrm{d}s\right) ^2\mathrm{d}x<\infty , \end{aligned}$$

and the proof follows. $\square$

Appendix B

This “Appendix” section is devoted to the essential elements of Gaussian analysis and Malliavin calculus. For the sake of completeness, we also present some known results in Malliavin–Stein method that are used in this paper. For the first part, the reader can consult [24, 29, 30] for further details. A comprehensive reference on the Malliavin–Stein method is the excellent monograph [24].

1.1 Elements of Gaussian Analysis

Let $\mathfrak {H}$ be a real separable Hilbert space. For any $q\ge 1$, we write $\mathfrak {H}^{\otimes q}$ and $\mathfrak {H}^{\odot q}$ to indicate, respectively, the qth tensor power and the qth symmetric tensor power of $\mathfrak {H}$; we also set by convention $\mathfrak {H}^{\otimes 0} = \mathfrak {H}^{\odot 0} ={\mathbb {R}}$. When $\mathfrak {H}= L^2(A,{\mathcal {A}}, \mu ) {=:}L^2(\mu )$, where $\mu$ is a $\sigma$-finite and non-atomic measure on the measurable space $(A,{\mathcal {A}})$, then $\mathfrak {H}^{\otimes q} = L^2(A^q,{\mathcal {A}}^q,\mu ^q){=:}L^2(\mu ^q)$, and $\mathfrak {H}^{\odot q} = L_s^2(A^q,{\mathcal {A}}^q,\mu ^q) {:=} L_s^2(\mu ^q)$, where $L_s^2(\mu ^q)$ stands for the subspace of $L^2(\mu ^q)$ composed of those functions that are $\mu ^q$ almost everywhere symmetric. We denote by $W=\{W(h) : h\in \mathfrak {H}\}$ an isonormal Gaussian process over $\mathfrak {H}$. This means that W is a centered Gaussian family, defined on some probability space $(\varOmega ,{\mathcal {F}},P)$, with a covariance structure given by the relation $\mathbb E\left[ W(h)W(g)\right] =\langle h,g\rangle _{ \mathfrak {H}}$. We also assume that ${\mathcal {F}}=\sigma (W)$, that is, ${\mathcal {F}}$ is generated by W, and use the shorthand notation $L^2(\varOmega ) {:=} L^2(\varOmega , {\mathcal {F}}, P)$.

For every $q\ge 1$, the symbol $C_{q}$ stands for the qth Wiener chaos of W, defined as the closed linear subspace of $L^2(\varOmega )$ generated by the family $\{H_{q}(W(h)) : h\in \mathfrak {H},\left\| h\right\| _{ \mathfrak {H}}=1\}$, where $H_{q}$ is the qth Hermite polynomial, defined as follows:

$$\begin{aligned} H_q(x) = (-1)^q e^{\frac{x^2}{2}} \frac{\mathrm{d}^q}{\mathrm{d}x^q} \big ( e^{-\frac{x^2}{2}} \big ). \end{aligned}$$

(6.1)

We write by convention $C_{0} = {\mathbb {R}}$. For any $q\ge 1$, the mapping $I_{q}(h^{\otimes q})=H_{q}(W(h))$ can be extended to a linear isometry between the symmetric tensor product $\mathfrak {H}^{\odot q}$ (equipped with the modified norm $\sqrt{q!}\left\| \cdot \right\| _{ \mathfrak {H}^{\otimes q}}$) and the qth Wiener chaos $C_{q}$. For $q=0$, we write by convention $I_{0}(c)=c$, $c\in {\mathbb {R}}$.

It is well known that $L^2(\varOmega )$ can be decomposed into the infinite orthogonal sum of the spaces $C_{q}$: this means that any square-integrable random variable $F\in L^2(\varOmega )$ admits the following Wiener–Itô chaotic expansion

$$\begin{aligned} F=\sum _{q=0}^{\infty }I_{q}(f_{q}), \end{aligned}$$

(6.2)

where the series converges in $L^2(\varOmega )$, $f_{0}=E[F]$, and the kernels $f_{q}\in \mathfrak {H}^{\odot q}$, $q\ge 1$, are uniquely determined by F. For every $q\ge 0$, we denote by $J_{q}$ the orthogonal projection operator on the qth Wiener chaos. In particular, if $F\in L^2(\varOmega )$ has the form (6.2), then $J_{q}F=I_{q}(f_{q})$ for every $q\ge 0$.

Let $\{e_{k},\,k\ge 1\}$ be a complete orthonormal system in $\mathfrak {H}$. Given $f\in \mathfrak {H}^{\odot p}$ and $g\in \mathfrak {H}^{\odot q}$, for every $r=0,\ldots ,p\wedge q$, the contraction of f and g of order r is the element of $\mathfrak {H}^{\otimes (p+q-2r)}$ defined by

$$\begin{aligned} f\otimes _{r}g=\sum _{i_{1},\ldots ,i_{r}=1}^{\infty }\langle f,e_{i_{1}}\otimes \ldots \otimes e_{i_{r}}\rangle _{ \mathfrak {H}^{\otimes r}}\otimes \langle g,e_{i_{1}}\otimes \ldots \otimes e_{i_{r}} \rangle _{ \mathfrak {H}^{\otimes r}}. \end{aligned}$$

(6.3)

Notice that the definition of $f\otimes _r g$ does not depend on the particular choice of $\{e_k,\,k\ge 1\}$, and that $f\otimes _{r}g$ is not necessarily symmetric; we denote its symmetrization by $f\widetilde{\otimes }_{r}g\in \mathfrak {H}^{\odot (p+q-2r)}$. Moreover, $f\otimes _{0}g=f\otimes g$ equals the tensor product of f and g while, for $p=q$, $f\otimes _{q}g=\langle f,g\rangle _{ \mathfrak {H}^{\otimes q}}$. When $\mathfrak {H}= L^2(A,{\mathcal {A}},\mu )$ and $r=1,...,p\wedge q$, the contraction $f\otimes _{r}g$ is the element of $L^2(\mu ^{p+q-2r})$ given by

$$\begin{aligned} f\otimes _{r}g (x_1,...,x_{p+q-2r})= & {} \int _{A^r} f(x_1,...,x_{p-r},a_1,...,a_r) \\&\times g(x_{p-r+1},...,x_{p+q-2r},a_1,...,a_r)\mathrm{d}\mu (a_1)...\mathrm{d}\mu (a_r). \end{aligned}$$

It is a standard fact of Gaussian analysis that the following multiplication formula holds: if $f\in \mathfrak {H}^{\odot p}$ and $g\in \mathfrak {H}^{\odot q}$, then

$$\begin{aligned} I_p(f) I_q(g) = \sum _{r=0}^{p \wedge q} r! {p \atopwithdelims ()r}{ q \atopwithdelims ()r} I_{p+q-2r} (f{\widetilde{\otimes }}_{r}g). \end{aligned}$$

(6.4)

We now introduce some basic elements of the Malliavin calculus with respect to the isonormal Gaussian process W. Let $\mathcal {S}$ be the set of all cylindrical random variables of the form

$$\begin{aligned} F=g\left( W(\phi _{1}),\ldots ,W(\phi _{n})\right) , \end{aligned}$$

(6.5)

where $n\ge 1$, $g:{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}$ is an infinitely differentiable function such that its partial derivatives have polynomial growth, and $\phi _{i}\in \mathfrak {H}$, $i=1,\ldots ,n$. The Malliavin derivative of F with respect to W is the element of $L^2(\varOmega , \mathfrak {H})$ defined as

$$\begin{aligned} DF\;=\;\sum _{i=1}^{n}\frac{\partial g}{\partial x_{i}}\left( W(\phi _{1}),\ldots ,W(\phi _{n})\right) \phi _{i}. \end{aligned}$$

In particular, $DW(h)=h$ for every $h\in \mathfrak {H}$. By iteration, one can define the mth derivative $D^{m}F$, which is an element of $L^2(\varOmega , \mathfrak {H}^{\odot m})$, for every $m\ge 2$. For $m\ge 1$ and $p\ge 1$, ${{\mathbb {D}}}^{m,p}$ denotes the closure of $\mathcal {S}$ with respect to the norm $\Vert \cdot \Vert _{m,p}$, defined by the relation

$$\begin{aligned} \Vert F\Vert _{m,p}^{p}\;=\;\mathbb E\left[ |F|^{p}\right] +\sum _{i=1}^{m}\mathbb E\left[ \Vert D^{i}F\Vert _{ \mathfrak {H}^{\otimes i}}^{p}\right] . \end{aligned}$$

We often use the (canonical) notation ${\mathbb {D}}^{\infty } {:=} \bigcap _{m\ge 1} \bigcap _{p\ge 1}{\mathbb {D}}^{m,p}$.

The Malliavin derivative D obeys the following chain rule. If $\varphi :{\mathbb {R}}^{n}\rightarrow {\mathbb {R}}$ is continuously differentiable with bounded partial derivatives and if $F=(F_{1},\ldots ,F_{n})$ is a vector of elements of ${{\mathbb {D}}}^{1,2}$, then $\varphi (F)\in {{\mathbb {D}}}^{1,2}$ and

$$\begin{aligned} D\,\varphi (F)=\sum _{i=1}^{n}\frac{\partial \varphi }{\partial x_{i}}(F)DF_{i}. \end{aligned}$$

(6.6)

Note also that a random variable F as in (6.2) is in ${{\mathbb {D}}}^{1,2}$ if and only if $\sum _{q=1}^{\infty }q\Vert J_qF\Vert ^2_{L^2(\varOmega )}<\infty$ and in this case one has the following explicit relation:

$$\begin{aligned}\mathbb E\left[ \Vert DF\Vert _{ \mathfrak {H}}^{2}\right] =\sum _{q=1}^{\infty }q\Vert J_qF\Vert ^2_{L^2(\varOmega )}. \end{aligned}$$

If $\mathfrak {H}= L^{2}(A,{\mathcal {A}},\mu )$ (with $\mu$ non-atomic), then the derivative of a random variable F as in (6.2) can be identified with the element of $L^2(A \times \varOmega )$ given by

$$\begin{aligned} D_{t}F=\sum _{q=1}^{\infty }qI_{q-1}\left( f_{q}(\cdot ,t)\right) ,\quad t \in A. \end{aligned}$$

(6.7)

The operator L, defined as $L=-\sum _{q=0}^{\infty }qJ_{q}$, is the infinitesimal generator of the Ornstein–Uhlenbeck semigroup. The domain of L is

$$\begin{aligned} \mathrm {Dom}L=\{F\in L^2(\varOmega ):\sum _{q=1}^{\infty }q^{2}\left\| J_{q}F\right\| _{L^2(\varOmega )}^{2}<\infty \}={\mathbb {D}}^{2,2}\text {.} \end{aligned}$$

For any $F \in L^2(\varOmega )$, we define $L^{-1}F =-\sum _{q=1}^{\infty }\frac{1}{q} J_{q}(F)$. The operator $L^{-1}$ is called the pseudo-inverse of L. Indeed, for any $F \in L^2(\varOmega )$, we have that $L^{-1} F \in \mathrm {Dom}L = {\mathbb {D}}^{2,2}$, and

$$\begin{aligned} LL^{-1} F = F - \mathbb E(F). \end{aligned}$$

(6.8)

1.2 Malliavin–Stein Method: Selective Results

Next, we collect some known findings in the realm of Malliavin–Stein method that we have used in Sect. 3. We begin with the celebrated fourth moment theorem.

Theorem 6.1

(Fourth Moment Theorem and Ramifications, see [25,26,27, 31]) Fix $q \ge 2$. Let $F_n = I_q (f_n), n\ge 1$ be a sequence of elements belonging to the qth Wiener chaos of some isonormal Gaussian process $W = \{ W(h) : h \in \mathfrak {H}\}$ such that $\mathbb E[F^2_n]= q! \Vert f_n \Vert ^2_{\mathfrak {H}^{\otimes q}} =1$ for every $n \ge 1$.

• (a) Then, the following asymptotic statements are equivalent as $n \rightarrow \infty$:

  • (a) $F_n$ converges in distribution toward ${\mathcal {N}}(0,1)$.

  • (b) $\mathbb E[F^4_n] \rightarrow 3$.

  • (c) $\Vert f_n \otimes _r f_n \Vert _{\mathfrak {H}^{\otimes (2q-2r)}} \rightarrow 0$ for $r=1,...,q-1$.

  • (d) $\Vert DF_n \Vert ^2_{\mathfrak {H}} \rightarrow q$ in $L^2$.

• (b) Furthermore, whenever one of the equivalent statements at item (a) take place, then there exist two constants $C_1$ and $C_2$ (independent of n) such that the following optimal rate of convergence in total variation distance holds:

  $$\begin{aligned} C_1 \, \max \{ \left| \kappa _3(F_n)\right| , \kappa _4(F_n) \} \le d_{TV}(F_n,N) \le C_2 \, \max \{ \left| \kappa _3(F_n)\right| , \kappa _4(F_n) \}.\end{aligned}$$

• (c) Assume one of the equivalent statements at item (a) take place. Let $G_n$, $n\ge 1$ be a sequence of the form

  $$\begin{aligned} G_n = \sum _{p=1}^{M} I_p (g^{(p)}_n) \end{aligned}$$

  for $M\ge 1$ (independent of n) and some kernels $g^{(p)}_n \in \mathfrak {H}^{ \odot p}, p=1,...,M$. Suppose that as n tends to infinity,

  $$\begin{aligned} \mathbb E[G^2_n]= & {} \sum _{p=1}^{M} p! \Vert g^{(p)}_n \Vert ^2_{\mathfrak {H}^{\otimes p}} \rightarrow c^2 >0, \quad \\&\Vert g^{(p)}_n \otimes _r g^{(p)}_n \Vert _{\mathfrak {H}^{\otimes (2p-2r)}} \rightarrow 0, \quad \forall \, r=1,...,p-1 \end{aligned}$$

  and every $p=1,...,M$. If furthermore, sequence $\mathbb E[F_nG_n] \rightarrow \rho$, then sequence $(F_n,G_n)$ converges in distribution toward a two-dimensional centered Gaussian vector $(N_1,N_2)$ with $\mathbb E[N^2_1]=1$, $\mathbb E[N^2_2] =c^2$, and $\mathbb E[N_1 N_2] =\rho$.

Theorem 6.2

(Peccati–Tudor Multidimensional Fourth Moment Theorem [24] Theorem 6.2.3) Fix $d \ge 2$, and $q_1,...,q_d \ge 1$. Let $F_n = ( F_{1,n},...,F_{d,n} ) = ( I_{q_1}(f_{1,n}),...,I_{q_d} (f_{d,n}) ), n\ge 1$ with the kernels $f_{n,j} \in \mathfrak {H}^{\odot j}$ for $j=1,...,d$ and every n. Let $N \sim {\mathcal {N}}_d (0,C)$ denote a d-dimensional centered Gaussian vector with a symmetric, nonnegative covariance matrix C. Assume that $\mathbb E[F_{i,n} F_{n,j}] \rightarrow C_{i,j}$ as $n \rightarrow \infty$. Then, the following asymptotic statements are equivalent:

• (a) $F_n \rightarrow N$ is distribution.

• (b) for every $j=1,...,d$, sequence $F_{j,n} \rightarrow {\mathcal {N}}(0,C_{j,j})$ in distribution.

Now we recall Breuer–Major Theorem (see [8] or Theorem 7.2.4 in [24] for a modern treatment) that is the cornerstone piece in Sect. 3.

Theorem 6.3

Let $X =\{X_k, k\in {\mathbb {Z}}\}$ be a centered Gaussian stationary sequence with unit variance and set $r(k)={\mathbb {E}}[X_{0}X_{k}]$ for every $k \in {\mathbb {Z}}$. Let $\gamma$ be the standard normal ${\mathcal {N}}(0,1)$ distribution and $f\in L^{2}({\mathbb {R}}, \gamma )$ be a fixed deterministic function such that ${\mathbb {E}}[f(X_1)]=0$ and f has Hermite rank $d\ge 1$, which means that f admits the Hermite expansion

$$\begin{aligned} f(x)=\sum _{j=d}^{\infty }a_j {H_j}(x), \end{aligned}$$

where $H_{j}$ is the j-Hermite polynomial, and $a_d \not =0$. Define $V_{n}=\frac{1}{\sqrt{n}}\sum _{k=1}^{n}f(X_k)$. Suppose that $\sum _{\nu \in {\mathbb {Z}}}|r(\nu )|^{d}<\infty$. Then,

$$\begin{aligned} \sigma ^2 : =\sum _{j=d}^{\infty }j! a^{2}_{j}\sum _{\nu \in {\mathbb {Z}}}r(\nu )^{j}\in [0,\infty ), \end{aligned}$$

and the convergence

$$\begin{aligned} V_{n} \begin{array}[t]{c} {\mathop {\longrightarrow }\limits ^\mathrm{d}} \\ \end{array}{\mathcal {N}}(0,\sigma ^2) \end{aligned}$$

holds as $n\rightarrow \infty$.

Theorem 6.4

(See [28]) Let $N \sim {\mathcal {N}}(0,1)$, and $X =\{X_k, k\in {\mathbb {Z}}\}$ be a centered Gaussian stationary sequence with unit variance and covariance function $r(k)={\mathbb {E}}[X_{0}X_{k}]$. Let $\gamma$ be the standard normal ${\mathcal {N}}(0,1)$ distribution and $f\in {\mathbb {D}}^{1,4} \subseteq L^2({\mathbb {R}}, \gamma )$ be a fixed deterministic function such that ${\mathbb {E}}[f(X_1)]=0$. Let $V_{n}=\frac{1}{\sqrt{n}}\sum _{k=1}^{n}f(X_k)$, and $\sigma ^2_n = {{\text {Var}}}\left( V_n \right)$. Define $F_n : = \frac{V_n}{\sigma _n}$. Then, there exists an explicit constant $C=C(f)$ such that for every $n \in \mathbb {N}$,

$$\begin{aligned} d_{TV} (F_n,N) \le \frac{C(f)}{\sigma ^2_n} n^{- \frac{1}{2}} \, \left( \sum _{\vert k \vert < n} \vert r (k) \vert \right) ^{\frac{3}{2}} \end{aligned}$$

(6.9)

Theorem 6.5

(See [14]) Let $N \sim {\mathcal {N}}(0,1)$. Assume that $X =\{X_k, k\in {\mathbb {Z}}\}$ is a centered Gaussian stationary sequence with unit variance and covariance function $r(k)={\mathbb {E}}[X_{0}X_{k}]$ whose spectral density function $f_r$ satisfies in $\log (f_r )\in L^1[-\pi ,\pi ]$. Fix $2 \le d \le q$. Let $V_n = \frac{1}{\sqrt{n}} \sum _{k=1}^{n} \sum _{j=d}^{q} a_j H_j (X_k)$ where $a_j \in {\mathbb {R}}$ for $d \le j \le q$, and that $\sigma ^2_n : = {{\text {Var}}}\left( V_n \right)$. Define $F_n {:=} \frac{V_n}{\sigma _n}$.

• (a) Assume further that $\sigma ^2 {:=} \sum _{j=d}^{q} j! a^2_j \sum _{\nu \in \mathbb {Z}} r(\nu ) ^j \in (0,\infty )$. Then, for every $m \ge 0$ as n tends to infinity

  $$\begin{aligned} \Big \Vert p^{(m)}_n - p^{(m)}_N \Big \Vert _{L^{\infty }({\mathbb {R}})}{:=} \sup _{x \in {\mathbb {R}}} \Big \vert p^{(m)}_n (x) - p^{(m)}_N (x) \Big \vert \longrightarrow 0 \end{aligned}$$

  where here $p^{(m)}_n$ and $p^{(m)}_N$ denote the mth derivative of density function of random variables $F_n$ and N, respectively.

• (b) In particular, if $q=d$ (in other words the sequence $F_n$ belongs to the fixed Wiener chaos of order d), then for all $m \ge 0$ there exist $n_0 \in \mathbb {N}$ and a constant C (depending only on m and q) such that for all $n \ge n_0$ we have

  $$\begin{aligned} \Big \Vert p^{(m)}_n - p^{(m)}_N \Big \Vert _{L^{\infty }({\mathbb {R}})} \le C \, \sqrt{\mathbb E\left[ F^4_n \right] - 3}. \end{aligned}$$

  (6.10)

Theorem 6.6

(See [25]) Let $(F_n : n\ge 1)$ be a sequence of centered square integrable functionals of some isonormal Gaussian process $W =\{ W(h) : h \in \mathfrak {H}\}$ such that $\mathbb E[F^2_n] \rightarrow 1$ as n tends to infinity. Assume further the following assumptions hold:

• (a) for every n, the random variable $F_n \in {\mathbb {D}}^{1,2}$, and that the law of $F_n$ is absolutely continuous with respect to the Lebesgue measure.

• (b) the quantity $\varphi (n){:=} \sqrt{ \mathbb E\left[ (1 - \langle DF_n , -DL^{-1}F_n \rangle _{\mathfrak {H}} )^2 \right] }$ is such that: (i) $\varphi (n) < \infty$ for every n, (ii) $\varphi (n) \rightarrow 0$ as n tends to infinity, and (iii) there exists $m \in \mathbb {N}$ such that $\varphi (n) > 0$ for all $n \ge m$.

• (c) as n tends to infinity,

  $$\begin{aligned} \left( F_n ,\frac{1 - \langle DF_n , -DL^{-1}F_n \rangle _{\mathfrak {H}} }{\varphi (n)} \right) {\mathop {\longrightarrow }\limits ^{d}} (N_1,N_2) \end{aligned}$$

  where $(N_1,N_2)$ is a two-dimensional centered Gaussian vector with $\mathbb E[N^2_1]= \mathbb E[N^2_2]=1$, and $\mathbb E[N_1 N_2] = \rho$.

Then, we have $d_{Kol} (F_n, N) \le \varphi (n)$, and moreover, for every $z \in {\mathbb {R}}$ as $n \rightarrow \infty$:

$$\begin{aligned} \frac{ {\mathbb {P}} \left( F_n \le z \right) - {\mathbb {P}} (N \le z)}{\varphi (n)} \longrightarrow \frac{\rho }{3 \sqrt{2\pi }} (z^2 -1) e ^{- \frac{z^2}{2}}. \end{aligned}$$

Theorem 6.7

(See [3]) Let $N \sim {\mathcal {N}}(0,1)$. Assume that $X =\{X_k, k\in {\mathbb {Z}}\}$ is a centered Gaussian stationary sequence with unit variance and covariance function $r(k)={\mathbb {E}}[X_{0}X_{k}]$ such that $\sum _{\nu \in \mathbb {Z}} \vert r(\nu ) \vert < \infty$. Assume that $f \in L^2({\mathbb {R}},\gamma )$ is a non-constant function of the class $C^2({\mathbb {R}})$ so that $\mathbb E[f''(N)^4]< \infty$ and that $\mathbb E_\gamma [f]=0$. Let $V_n = \frac{1}{\sqrt{n}} \sum _{k=1}^{n} f (X_k)$, and that $\sigma ^2_n : = {{\text {Var}}}\left( V_n \right)$. Define $F_n {:=} \frac{V_n}{\sigma _n}$. If as n tends to infinity, $\sigma ^2_n \rightarrow \sigma ^2 > 0$, then the sequence $(F_n : n \ge 1)$ converges in distribution toward N, and moreover, it satisfies an ASCLT meaning that, almost surely, for every bounded continuous function $\varphi : {\mathbb {R}}\rightarrow {\mathbb {R}}$ it holds that

$$\begin{aligned} \frac{1}{\log n} \sum _{k=1}^{n} \frac{1}{k} \varphi (F_n) \longrightarrow \mathbb E\left[ \varphi (N) \right] , \quad \text { as } \quad n \rightarrow \infty . \end{aligned}$$