Regularity and approximation of Gaussian random fields evolving temporally over compact two-point homogeneous spaces

Abstract

We consider Gaussian random fields on the product of a compact two-point homogeneous space cross the time, which are space isotropic and time stationary. We study regularity properties of these random fields in terms of function spaces whose elements have different smoothness in the space and time domain. Namely, we express the norm of the corresponding covariance kernel functions in terms of the summability of the associated spectral coefficients. Furthermore, we define an approximation method based on the truncation of the expansion related to the spectral representation of a given random field. The accuracy of this approximation is measured in the $L^p$ sense. Finally, we model a space–time dataset of ozone concentrations in Mexico City using a seasonal temporal covariance function constructed through an expansion of Jacobi polynomials. We find that we need relatively few Jacobi polynomials to get the best fit to the data in terms of the deviance information criterion. We discuss the characteristics of this model, including seasonality, decay and approximate conditional independencies.

A Appendix: Proofs

1.1 A.1 Mathematical background

We recall some properties of Jacobi polynomials (see Lemma 4.3 Cleanthous et al. 2020; Szegő 1939):

1. Orthogonality. For every $k,\ell \in {\mathbb N}_0$,

$$\begin{aligned} \Big \langle P_{k}^{(\alpha ,\beta )},P_{\ell }^{(\alpha ,\beta )}\Big \rangle _{\alpha ,\beta }=\delta _{k,\ell }h_{\ell }^{(\alpha ,\beta )}, \end{aligned}$$

(A.1)

where we set

$$\begin{aligned} h_{\ell }^{(\alpha ,\beta )}:=\frac{2^{\alpha +\beta +1}}{2\ell +\alpha +\beta +1}\frac{\Gamma (\ell +\alpha +1)\Gamma (\ell +\beta +1)}{\ell !\Gamma (\ell +\alpha +\beta +1)}\sim (\ell +1)^{-1}, \end{aligned}$$

(A.2)

where $\Gamma$ denotes the special function Gamma. Note that the constants are depending only on $\alpha$ and $\beta$.

From the above, it turns out that the sequence

$$\begin{aligned} \Big \{\tilde{P}_{\ell }^{(\alpha ,\beta )}:=\frac{P_{\ell }^{(\alpha ,\beta )}}{\sqrt{h_{\ell }^{(\alpha ,\beta )}}}:\ell \ge 0\Big \} \end{aligned}$$

(A.3)

forms an orthonormal basis for $L^2([-1,1],\omega _{\alpha ,\beta })$.

2. Maximum. Jacobi polynomials are maximized for $u=1$. Moreover, for every $\ell \in {\mathbb N}_0$,

$$\begin{aligned} \Vert P_{\ell }^{(\alpha ,\beta )}\Vert _{\infty }=P_{\ell }^{(\alpha ,\beta )}(1)\sim (\ell +1)^{\alpha }, \end{aligned}$$

(A.4)

provided that $\alpha \ge \beta$. Here, the constants depend only on $\alpha$.

3. Derivatives. The Jacobi polynomials $P_{\ell }^{(\alpha ,\beta )}$ have algebraic degree equal with $\ell$. Let $0\le n\le \ell$. We have the following formula for the derivatives of these polynomials:

$$\begin{aligned} \frac{\mathrm{d}^n}{ \mathrm{d} u^n}P^{(\alpha ,\beta )}_{\ell }(u)\sim (\ell +1)^{n}P^{(\alpha +n,\beta +n)}_{\ell -n}(u), \end{aligned}$$

(A.5)

where the similarity constants depend on n, $\alpha$ and $\beta$ and are independent of $\ell$ and u.

Let us list the corresponding properties here. For a comprehensive study of Hermite polynomials, the reader is referred to Szegő (1939).

Let $\ell \in {\mathbb N}_0$, the normalized $\ell$-th degree Hermite polynomial is defined as

$$\begin{aligned} H_\ell (t)=\big (\sqrt{\pi }2^{\ell } \ell !\big )^{-1/2}(-1)^{\ell } e^{x^2}\frac{\mathrm{d} ^{\ell }}{\mathrm{d}x^{\ell }}e^{-x^2}. \end{aligned}$$

(A.6)

Properties:

(i) Orthogonality. For every $n,m\in {\mathbb N}_0$

$$\begin{aligned} \langle H_n, H_m\rangle _{\gamma }=\delta _{n,m}. \end{aligned}$$

(A.7)

(ii) Derivatives. Note that the $H_n$ is a polynomial of degree n. Let $n,m\in {\mathbb N}_0$ with $n\ge m$. Then,

$$\begin{aligned} \frac{\mathrm{d}^m}{\mathrm{d}t^m} H_n (t)=\left( \frac{2^m n!}{(n-m)!}\right) ^{1/2}H_{n-m}(t)\sim (n+1)^{m/2} H_{n-m}(t) \end{aligned}$$

(A.8)

where the constants above depend on m, but are independent of n and t.

1.2 A.2 A convergence argument for identity (3.9)

Let us now consider the convergence of $K_{IS}(\cdot ,t)$ on $L^2_{(\alpha ,\beta )}$. Since $P_{k_1}^{(\alpha ,\beta )}=\tilde{P}_{k_1}^{(\alpha ,\beta )}\sqrt{h_{k_1}^{(\alpha ,\beta )}}$ and $\tilde{P}_{k_1}^{(\alpha ,\beta )}$ is an orthonormal basis for $L^2_{(\alpha ,\beta )}$, the convergence of the series

$$\begin{aligned} \sum _{k_1=0}^{\infty }\Big |B_{k_1}(t)\sqrt{h_{k_1}^{(\alpha ,\beta )}}\Big |^2 \end{aligned}$$

guarantees the $L^2_{(\alpha ,\beta )}$-convergence of (2.2), for every $t\in {\mathbb R}$.

But from (2.3) and (A.4), there exists a constant $c>0$ such that

$$\begin{aligned} B_{k_1}(0)\le c(k_1+1)^{-\alpha },\;\;\text {for every}\;k_1\in {\mathbb N}_0. \end{aligned}$$

(A.9)

Since now $\alpha =(d-2)/2\ge -1/2$ for any two-point homogeneous space $\mathcal {M}^d$, by invoking again (2.3) and using (A.2), we conclude

$$\begin{aligned} \sum _{k_1=0}^{\infty }\Big |B_{k_1}(t)\sqrt{h_{k_1}^{(\alpha ,\beta )}}\Big |^2&\le c\sum _{k_1=0}^{\infty }B_{k_1}(0)(k_1+1)^{-1-\alpha } \\&\le c\sum _{k_1=0}^{\infty }B_{k_1}(0)P_{k_1}^{(\alpha ,\beta )}(1)(k_1+1)^{-1-2\alpha } \\&\le c\sum _{k_1=0}^{\infty }B_{k_1}(0)P_{k_1}^{(\alpha ,\beta )}(1)<\infty . \end{aligned}$$

Inequality (2.4) implies that $B_{k_1}\in L^2({\mathbb R},\gamma )$, since here the measure is bounded. By the orthonormality of the set $\{H_{k_2}\}$, we derive that

$$\begin{aligned} B_{k_1}\sqrt{h_{k_1}^{(\alpha ,\beta )}}=\sum _{k_2=0}^{\infty } B_{\mathbf {k}} H_{k_2},\quad \text {convergence on}\;L^2({\mathbb R},\gamma ) \end{aligned}$$

(A.10)

and where we denoted $B_{\mathbf {k}}:=\Big \langle B_{k_1}\sqrt{h_{k_1}^{(\alpha ,\beta )}},H_{k_2}\Big \rangle _{\gamma }$, for every $k=(k_1,k_2)\in {\mathbb N}_0^2$.

We have

$$\begin{aligned} K_{IS}=\sum _{\mathbf {k}\in {\mathbb N}_0^2}B_{\mathbf {k}}R_{\mathbf {k}}^{(\alpha ,\beta )},\quad \text {convergence on}\;L^2_{\alpha ,\beta ,\gamma }. \end{aligned}$$

where $K_{IS}$ has been defined through (2.1). Indeed, by (2.4), (2.3), (A.2), (A.9) and (A.4)

$$\begin{aligned} \sum _{\mathbf {k}\in {\mathbb N}_0^2}\big |B_{\mathbf {k}}\big |^2&=\sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty }\big |B_{\mathbf {k}}\big |^2 =\sum _{k_1=0}^{\infty }\Big \Vert B_{k_1}\sqrt{h_{k_1}^{(\alpha ,\beta )}}\Big \Vert ^2_{L^2_{\gamma }} \\&\le \gamma ({\mathbb R})\sum _{k_1=0}^{\infty }B_{k_1}(t)^2 h_{k_1}^{(\alpha ,\beta )} \le c\sum _{k_1=0}^{\infty }B_{k_1}(0)P_{k_1}^{(\alpha ,\beta )}(1)<\infty . \end{aligned}$$

1.3 A.3 Proof of Theorem 1

The proof of Theorem 1 is based on the following equivalence:

Lemma 1

Let $n_1,n_2\in {\mathbb N}_0$ and $f\in L^2_{\alpha ,\beta ,\gamma }$. Then,

$$\begin{aligned} \big \Vert \partial ^{(n_1,n_2)}f\big \Vert _{L^2_{\alpha +n_1,\beta +n_1,\gamma }}^2\sim \sum _{k_1=n_1}^{\infty }\sum _{k_2=n_2}^{\infty } f_{\mathbf {k}}^2(k_1+1)^{2n_1}(k_2+1)^{n_2}, \end{aligned}$$

(A.11)

with similarity constants depending only on $\alpha ,\beta ,n_1$ and $n_2$.

Proof

Let $n_1,n_2\in {\mathbb N}_0$ and $f\in L^2_{\alpha ,\beta ,\gamma }$. Then, f can be represented as (3.8). Therefore,

$$\begin{aligned} \partial ^{(n_1,n_2)}f=\sum _{k_1=n_1}^{\infty }\sum _{k_2=n_2}^{\infty }\frac{f_{\mathbf {k}}}{\sqrt{h_{k_1}^{(\alpha ,\beta )}}}\big (P_{k_1}^{(\alpha ,\beta )}\big )^{(n_1)}H_{k_2}^{(n_2)}. \end{aligned}$$

(A.12)

Then, we get the inner product

$$\begin{aligned}&\langle \partial ^{(n_1,n_2)}f,\partial ^{(n_1,n_2)}f\rangle _{\alpha +n_1,\beta +n_1,\gamma } \\&=\sum _{k_1,\ell _1=n_1}^{\infty }\sum _{k_2,\ell _2=n_2}^{\infty }\frac{f_{\mathbf {k}}f_{\mathbf {\ell }}\langle H_{k_2}^{(n_2)},H_{\ell _2}^{(n_2)}\rangle _{\gamma }}{\sqrt{h_{k_1}^{(\alpha ,\beta )}h_{\ell _1}^{(\alpha ,\beta )}}}\Big \langle \big (P_{k_1}^{(\alpha ,\beta )}\big )^{(n_1)},\big (P_{\ell _1}^{(\alpha ,\beta )}\big )^{(n_1)}\Big \rangle _{\alpha +n_1,\beta +n_1}. \end{aligned}$$

Invoking (A.1) and (A.5), we infer

$$\begin{aligned}&\Big \langle \big (P_{k_1}^{(\alpha ,\beta )}\big )^{(n_1)},\big (P_{\ell _1}^{(\alpha ,\beta )}\big )^{(n_1)}\Big \rangle _{\alpha +n_1,\beta +n_1}\\&\sim (k_1+1)^{n_1}(\ell _1+1)^{n_1}\Big \langle P_{k_1-n_1}^{(\alpha +n_1,\beta +n_1)},P_{\ell _1-n_1}^{(\alpha +n_1,\beta +n_1)}\Big \rangle _{\alpha +n_1,\beta +n_1} \\&\sim \delta _{k_1,\ell _1}(k_1+1)^{2n_1}h_{k_1-n_1}^{(\alpha +n_1,\beta +n_1)}\\&\sim \delta _{k_1,\ell _1}(k_1+1)^{2n_1}(k_1+1)^{-1}, \end{aligned}$$

where the constants above depend only on $\alpha$, $\beta$ and $n_1$.

Also by (A.2), it holds true

$$\begin{aligned} \sqrt{h_{k_1}^{(\alpha ,\beta )}h_{\ell _1}^{(\alpha ,\beta )}}\sim (k_1+1)^{-1/2}(\ell _1+1)^{-1/2}. \end{aligned}$$

On the other hand, (A.8) in concert with (A.7) implies

$$\begin{aligned} \langle H_{k_2}^{(n_2)},H_{\ell _2}^{(n_2)}\rangle _{\gamma }\sim \delta _{k_2,\ell _2}(k_2+1)^{n_2}. \end{aligned}$$

The combination of all the above clearly implies (A.11).

We proceed to prove the main regularity result:

Proof of Theorem 1

(i) Let $n_1,n_2\in {\mathbb N}_0$ and $f\in L^2_{\alpha ,\beta ,\gamma }$. By (4.3) and (A.11), we have the equivalence

$$\begin{aligned} \Vert f\Vert ^2_{DW^{(n_1,n_2)}}&\sim \sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty }f_{k_1,k_2}^2 +\sum _{k_1=n_1}^{\infty }\sum _{k_2=0}^{\infty }f_{k_1,k_2}^2(k_1+1)^{2n_1}\\&\quad +\sum _{k_1=0}^{\infty }\sum _{k_2=n_2}^{\infty }f_{k_1,k_2}^2(k_2+1)^{n_2}\\&\quad +\sum _{k_1=n_1}^{\infty }\sum _{k_2=n_2}^{\infty }f_{k_1,k_2}^2(k_1+1)^{2n_1}(k_2+1)^{n_2} \\&=:a_1+\cdots a_4. \end{aligned}$$

Apparently,

$$\begin{aligned} \Vert f\Vert ^2_{DW^{(n_1,n_2)}}\le 4\sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty }f_{k_1,k_2}^2(k_1+1)^{2n_1}(k_2+1)^{n_2}=:4a_0 \end{aligned}$$

and this proves that when $\{f_{k_1,k_2}\}\in \ell ^{(n_1,n_2)}$, then $f\in DW^{(n_1,n_2)}$.

We now assume that $f\in DW^{(n_1,n_2)}$. We will show that $\{f_{k_1,k_2}\}\in \ell ^{(n_1,n_2)}$ and there exists a constant $c=c_{\alpha ,\beta ,n_1,n_2}>0$, such that $a_0\le c\Vert f\Vert ^2_{DW^{(n_1,n_2)}}$. This will be sufficient for proving claim (i).

We decompose $a_0$ as follows:

$$\begin{aligned} a_0&=\left( \sum _{k_1=0}^{n_1-1}\sum _{k_2=0}^{n_2-1}+\sum _{k_1=n_1}^{\infty }\sum _{k_2=0}^{n_2-1}+\sum _{k_1=0}^{n_1-1}\sum _{k_2=n_2}^{\infty }+\sum _{k_1=n_1}^{\infty }\sum _{k_2=n_2}^{\infty }\right) g_{k_1,k_2} \\&=:a_5+\cdots +a_8, \end{aligned}$$

where we denoted by $g_{k_1,k_2}:=f_{k_1,k_2}^2(k_1+1)^{2n_1}(k_2+1)^{n_2}$.

By (A.11), we derive the followings:

$$\begin{aligned}&a_5\le (n_1+1)^{2n_1} (n_2+1)^{n_2}\sum _{k_1=0}^{\infty }\sum _{k_2=0}^{\infty }f_{k_1,k_2}^2\le c\Vert f\Vert ^2_{L^2_{(\alpha ,\beta ,\gamma )}},\\&a_6\le (n_2+1)^{n_2}\sum _{k_1=n_1}^{\infty }\sum _{k_2=0}^{\infty }f_{k_1,k_2}^2(k_1+1)^{2n_1}\le c\big \Vert \partial ^{n_1}_1 f\big \Vert ^2_{L^2_{\alpha +n_1,\beta +n_1,\gamma }},\\&a_7\le c\Vert \partial ^{n_2}_2 f\Vert ^2_{L^2_{\alpha ,\beta ,\gamma }} \end{aligned}$$

and

$$\begin{aligned} a_8\le c\Vert \partial ^{(n_1,n_2)}f\Vert ^2_{L^2_{\alpha +n_1,\beta +n_1,\gamma }}. \end{aligned}$$

All the above complete the proof of (i).

(ii) We start by observing that $W^{(n_1,n_2)}=DW^{(n_1,0)}\cap DW^{(0,n_2)}$. Then, by claim (i) it holds that $f\in W^{(n_1,n_2)}$ if and only if $(f_{\mathbf {k}})\in \ell ^{(n_1,0)}\cap \ell ^{(0,n_2)}$. On the other hand by (A.11) and (4.2), we get that

$$\begin{aligned} \Vert f\Vert ^2_{W^{(n_1,n_2)}}\le c\sum _{\mathbf {k}\in {\mathbb N}_0^2}f_{\mathbf {k}}^2\big ((k_1+1)^{2n_1}+(k_2+1)^{n_2}\big )=:a_9. \end{aligned}$$

Conversely,

$$\begin{aligned} a_9&=\sum _{k_1<n_1}\sum _{k_2\ge 0}f_{\mathbf {k}}^2(k_1+1)^{2n_1}+\sum _{k_1\ge n_1}\sum _{k_2\ge 0}f_{\mathbf {k}}^2(k_1+1)^{2n_1} \\&\quad +\sum _{k_2<n_2}\sum _{k_1\ge 0}f_{\mathbf {k}}^2(k_2+1)^{n_2}+\sum _{k_2\ge n_2}\sum _{k_1\ge 0}f_{\mathbf {k}}^2(k_2+1)^{n_2} \\&\le c\big ((n_1+1)^{2n_1}+(n_2+1)^{n_2}\big )\Vert f\Vert ^2_{L^2_{\alpha ,\beta ,\gamma }}+c\Vert \partial ^{(n_1,0)}f\Vert ^2_{L^2_{\alpha +n_1,\beta +n_1,\gamma }} \\&\quad +c\Vert \partial ^{(0,n_2)}f\Vert ^2_{L^2_{\alpha ,\beta ,\gamma }} \\&\le c\Vert f\Vert ^2_{W^{(n_1,n_2)}} \end{aligned}$$

and the proof of claim (ii) is complete.

(iii) Let $N\in {\mathbb N}_0$ and $f\in L^2_{\alpha ,\beta ,\gamma }$. Note that the following connection between Sobolev-type spaces is true:

$$\begin{aligned} W^{N}=\bigcap _{\nu =0}^{N}DW^{(\nu ,N-\nu )}. \end{aligned}$$

Then, f belongs to $W^N$ if and only if it belongs to the intersection of $W^{(\nu ,N-\nu )}$, for every $\nu =0,\dots ,N$, which equivalently means that its Fourier coefficients belong to $\bigcap _{\nu =0}^{N}\ell ^{(\nu ,N-\nu )}$, in the light of claim (i) and the proof is complete.

We close by proving Corollary 1.

Proof

Let $n_1,n_2\in {\mathbb N}_0$. By (3.9), we have that $K_{IS}\in L^2_{\alpha ,\beta ,\gamma }$. Using Lemma 1, we derive that

$$\begin{aligned}&(1-u)^{n_1/2}(1+u)^{n_1/2}\partial ^{(n_1,n_2)}K_{IS}(u,t)\in L^2_{\alpha ,\beta ,\gamma } \\&\Leftrightarrow \partial ^{(n_1,n_2)}K_{IS}\in L^2_{\alpha +n_1,\beta +n_1,\gamma } \\&\Leftrightarrow \sum _{k_1=n_1}^{\infty }\sum _{k_2=n_2}^{\infty }B_{\mathbf {k}}^2(k_1+1)^{2n_1}(k_2+1)^{n_2}<\infty . \end{aligned}$$

1.4 A.4 Proof of Theorem 2

Proof

We have

$$\begin{aligned} Z(x,t)-Z^R(x,t)=\sum _{n>R}V_n(t) P_n^{(\alpha ,\beta )}\big (\cos (\rho (x,U))\big ), \end{aligned}$$

for every $(x,t)\in \mathcal {M}^d\times {\mathbb R}$.

We replace the last in the norm (3.10) and by (3.11), we extract

$$\begin{aligned}&\big \Vert Z-Z^R\big \Vert _{L^2(\mathcal {M}^d\times {\mathbb R})}^2=\big \langle Z-Z^R,Z-Z^R\big \rangle _{L^2(\mathcal {M}^d\times {\mathbb R})} \\&=\sum _{n,m>R}\big \langle V_n(t) P_n^{(\alpha ,\beta )}\big (\cos (\rho (x,U))\big ),V_m(t) P_m^{(\alpha ,\beta )}\big (\cos (\rho (x,U))\big )\big \rangle _{L^2(\mathcal {M}^d\times {\mathbb R})} \\&=\sum _{n,m>R}\int _{\mathcal {M}^d}P_n^{(\alpha ,\beta )}\big (\cos (\rho (x,U))\big ) P_m^{(\alpha ,\beta )}\big (\cos (\rho (x,U))\big )\mathrm{d}x \\&\quad \times \int _{{\mathbb R}}V_n (t)V_m (t)\gamma (t)\mathrm{d}t. \end{aligned}$$

By Ma and Malyarenko (2020), it holds true that

$$\begin{aligned} \int _{\mathcal {M}^d}P_n^{(\alpha ,\beta )}\big (\cos (\rho (x,U))\big ) P_m^{(\alpha ,\beta )}\big (\cos (\rho (x,U))\big )\mathrm{d}x=\frac{\delta _{nm}\omega _{d}}{a_n^2}P_n^{(\alpha ,\beta )}(1). \end{aligned}$$

Therefore,

$$\begin{aligned} \big \Vert Z-Z^R\big \Vert _{L^2(\mathcal {M}^d\times {\mathbb R})}^2=\omega _d \sum _{n>R}\frac{\Vert V_n\Vert _{L^2_{\gamma }}^2}{a_n^2}P_n^{(\alpha ,\beta )}(1). \end{aligned}$$

Note now that $P_n^{(\alpha ,\beta )}(1)>0$. By Fubini–Tonelli theorem, the mixed-norm (3.13) takes the form

$$\begin{aligned} \big \Vert Z-Z^{R}\big \Vert _{L^2\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))}&={\mathbb E}\left( \Vert Z-Z^{R}\big \Vert _{L^2(\mathcal {M}^d\times {\mathbb R})}^2\right) \\&={\mathbb E}\left( \omega _d \sum _{n>R}\frac{\Vert V_n\Vert _{L^2_{\gamma }}^2}{a_n^2}P_n^{(\alpha ,\beta )}(1)\right) \\&=\omega _d \sum _{n>R}\frac{{\mathbb E}(\Vert V_n\Vert _{L^2_{\gamma }}^2)}{a_n^2}P_n^{(\alpha ,\beta )}(1) \\&=\omega _d \sum _{n>R}\frac{\Vert {\mathbb E}(V_n(t)V_n(t))\Vert _{L^2_{\gamma }}^2}{a_n^2}P_n^{(\alpha ,\beta )}(1) \end{aligned}$$

Since now ${\mathbb E}(V_n(t)V_n(t))=a_n^2 B_n(0)$ by (5.2), we extract

$$\begin{aligned} \frac{\Vert {\mathbb E}(V_n(t)V_n(t))\Vert _{L^2_{\gamma }}^2}{a_n^2}=B_n(0)\gamma ({\mathbb R})\le c(n+1)^{-\varepsilon }, \end{aligned}$$

where we used the condition (5.5) and the finiteness of the measure $\gamma$. Moreover, $P_n^{(\alpha ,\beta )}(1)\le c(n+1)^{\alpha }$ and combing all the above, we conclude

$$\begin{aligned} \big \Vert Z-Z^{R}\big \Vert _{L^2(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))}&\le c\sum _{n>R}n^{-(\varepsilon -\alpha )} \\&\le c\int _{R}^{\infty }\frac{\mathrm{d}x}{x^{\varepsilon -\alpha }}=c R^{-(\varepsilon -\alpha -1)}, \end{aligned}$$

since $\varepsilon >\alpha +1$.

To show (5.7), we begin with the case $p<2$. Applying the Hölder inequality on a probability space for the index $2/p>1$ directly gives

$$\begin{aligned} \big \Vert Z-Z^{R}\big \Vert _{L^p(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))}&\le \big \Vert Z-Z^{R}\big \Vert _{L^2(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))} \le cR^{-(\varepsilon -\alpha -1)}. \end{aligned}$$

We consider now $p>2$. We can choose $\nu =\nu _p\in {\mathbb N}$ such that $2(\nu -1)<p\le 2\nu$. Similarly to the case $p<2$, the Hölder inequality for the index $2\nu /p$ yields

$$\begin{aligned} \big \Vert Z-Z^{R}\big \Vert _{L^p(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))}&\le \big \Vert Z-Z^{R}\big \Vert _{L^{2\nu }(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R})}. \end{aligned}$$

(A.13)

By Corollary 2.17 of Da Prato and Zabczyk (1992), there exists a constant $c=c_\nu >0$ such that

$$\begin{aligned} \Vert Z-Z^{R}\big \Vert _{L^{2\nu }(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))}\le c_{\nu }\Vert Z-Z^{R}\big \Vert _{L^{2}(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))}. \end{aligned}$$

(A.14)

The combination of (A.13), (A.14) and (5.6) leads to  (5.7).

For the last claim, we have to prove that

$$\begin{aligned} \mathbb {P}\big (\big \Vert Z-Z^{R}\big \Vert _{L^2(\mathcal {M}^d\times {\mathbb R})}\ge R^{-\gamma },\;\text {for infinitely many}\;R\in {\mathbb N}\big )=0. \end{aligned}$$

(A.15)

This will be a consequence of the Borel–Cantelli lemma Klenke (2013), Theorem  2.7. It suffices to prove that

$$\begin{aligned} \sum _{R=1}^{\infty }\mathbb {P}\left( \big \Vert Z-Z^{R}\big \Vert _{L^2(\mathcal {M}^d\times {\mathbb R})}\ge R^{-\gamma }\right) <\infty . \end{aligned}$$

(A.16)

Let $p>0$. By Chebyshev’s inequality, (5.7) and using the mixed-norm (3.12), we derive

$$\begin{aligned} \mathbb {P}\left( \big \Vert Z-Z^{R}\big \Vert _{L^2(\mathcal {M}^d\times {\mathbb R})}\ge R^{-\gamma }\right)&\le R^{\gamma p} {\mathbb E}\left( \big \Vert Z-Z^{R}\big \Vert _{L^2(\mathcal {M}^d\times {\mathbb R})}^p\right) \nonumber \\&=R^{\gamma p} \big \Vert Z-Z^{R}\big \Vert _{L^{p}(\Omega ,L^2(\mathcal {M}^d\times {\mathbb R}))}^p \nonumber \\&\le cR^{-(\varepsilon -\alpha -1-\gamma )p}. \end{aligned}$$

(A.17)

We choose now $p>1/(\varepsilon -\alpha -1-\gamma )$, and this is allowed since $0<\gamma <\varepsilon -\alpha -1$. For this choice, we obtain

$$\begin{aligned} \sum _{R=1}^{\infty }R^{-(\varepsilon -\alpha -1-\gamma )p}<\infty , \end{aligned}$$

which together with (A.17) leads to (A.16) and completes the proof.