Adaptive Minimax Testing for Circular Convolution

Abstract

Given observations from a circular random variable contaminated by an additive measurement error, we consider the problem of minimax optimal goodness-of-fit testing in a non-asymptotic framework. We propose direct and indirect testing procedures using a projection approach. The structure of the optimal tests depends on regularity and ill-posedness parameters of the model, which are unknown in practice. Therefore, adaptive testing strategies that perform optimally over a wide range of regularity and ill-posedness classes simultaneously are investigated. Considering a multiple testing procedure, we obtain adaptive i.e. assumption-free procedures and analyse their performance. Compared with the non-adaptive tests, their radii of testing face a deterioration by a log-factor. We show that for testing of uniformity this loss is unavoidable by providing a lower bound. The results are illustrated considering Sobolev spaces and ordinary or super smooth error densities.

Appendices

AUXILIARY RESULTS USED IN SECTIONS 2 AND 4

The next two assertions, a concentration inequality for canonical U-statistics and a Bernstein inequality, provide our key arguments in order to control the deviation of the test statistics. The first assertion is a reformulation of Theorem 3.4.8 in [11].

Proposition A.1. Let $\{Y_{j}\}_{j=1}^{n}$ be independent and identically distributed $[0,1)$-valued random variables, and let $h:[0,1)^{2}\to\mathbb{R}$ be a bounded symmetric kernel, i.e., $h(y,\tilde{y})=h(\tilde{y},y)$ for all $y,\tilde{y}\in[0,1)$, fulfilling in addition

$$\mathbb{E}(h(Y_{1},y_{2}))=0\quad\forall y_{2}\in[0,1).$$

(A.1)

Then there are finite constantsA, B, C, and $\mathrm{d}$ such that

$$\sup_{y_{1},y_{2}\in[0,1)}|h(y_{1},y_{2})|\leq\textrm{A},\quad\sup_{y_{2}\in[0,1)}\mathbb{E}h^{2}(Y_{1},y_{2})\leq\textrm{B}^{2},\quad\mathbb{E}h^{2}(Y_{1},Y_{2})\leq\textrm{C}^{2},\quad{and}$$

$$\sup\{\mathbb{E}\left(h(Y_{1},Y_{2})\zeta(Y_{1})\xi(Y_{2})\right),\mathbb{E}\zeta^{2}(Y_{1})\leq 1,\mathbb{E}\xi^{2}(Y_{2})\leq 1\}\leq\mathrm{d}$$

(A.2)

and for all $n\geq 2$ the real-valued canonical U-statistic

$$\textrm{U}_{n}=\frac{1}{n(n-1)}\sum_{l,m\in[\![n]\!]\atop l\neq m}h(Y_{l},Y_{m})$$

satisfies for all $x\geq 0$

$$\mathbb{P}(\textrm{U}_{n}\geq 8\textrm{C}n^{-1}x^{1/2}+13\textrm{D}n^{-1}x+261\textrm{B}n^{-3/2}x^{3/2}+343\textrm{A}n^{-2}x^{2})\leq\exp(1-x).$$

The following version of Bernstein’s inequality can directly be deduced from Theorem 3.1.7 in [11].

Proposition A.2. Let $\{Z_{j}\}_{j=1}^{n}$ be independent with $|Z_{j}|\leq\textrm{b}$ almost surely and $\mathbb{E}(|Z_{j}|^{2})\leq\textrm{v}$ for all $j\in[\![n]\!]$, then for all $x>0$ and $n\geq 1$, we have

$$\mathbb{P}\left(\frac{1}{n}\sum_{j\in[\![n]\!]}(Z_{j}-\mathbb{E}Z_{j})\geq\sqrt{\frac{2\textrm{v}x}{n}}+\frac{\textrm{b}x}{3n}\right)\leq\exp(-x).$$

Preliminaries.We eventually calculate first A, B, and C satisfying (A.2) and exploit that $\textrm{D}:=\textrm{C}$ automatically also fulfils (A.2), which we briefly justify next. Throughout this section we assume that $\{Y_{j}\}_{j=1}^{n}$ are independent and identically distributed with Lebesgue-density $g=f{\bigcirc\hskip-8.5pt\ast}\varphi\in\mathcal{L}^{2}$. We denote by $\mathcal{L}_{\mathbb{R}}^{2}(g)$ the set of (Borel-measurable) functions $\zeta:[0,1)\to\mathbb{R}$ with $||\zeta||_{\mathcal{L}_{\mathbb{R}}^{2}(g)}^{2}:=\int_{[0,1)}\zeta^{2}(x)g(x)dx<\infty$. Let $h:[0,1)^{2}\to\mathbb{R}$ be a bounded kernel, i.e., $||h||_{\mathcal{L}^{\infty}}:=\sup_{y_{1},y_{2}\in[0,1)}|h(y_{1},y_{2})|<\infty$, and define the integral operator $H:\mathcal{L}_{\mathbb{R}}^{2}(g)\to\mathcal{L}_{\mathbb{R}}^{2}(g)$ with $\zeta\mapsto H\zeta$ and $H\zeta(s):=\int_{[0,1)}h(t,s)\zeta(t)g(t)dt$ for all $s\in[0,1)$. Then $H$ is linear and bounded, i.e., $||H||_{\mathcal{L}_{\mathbb{R}}^{2}(g)\to\mathcal{L}_{\mathbb{R}}^{2}(g)}:=\sup\{||H\zeta||_{\mathcal{L}_{\mathbb{R}}^{2}(g)}:||\zeta||_{\mathcal{L}_{\mathbb{R}}^{2}(g)}\leq 1\}\leq\mathbb{E}h^{2}(Y_{1},Y_{2})\leq C^{2}$ due to the Cauchy–Schwarz inequality. This shows the claim since its operator norm satisfies

$$||H||_{\mathcal{L}_{\mathbb{R}}^{2}(g)\to\mathcal{L}_{\mathbb{R}}^{2}(g)}=\sup\{\mathbb{E}(h(Y_{1},Y_{2})\zeta(Y_{1})\xi(Y_{2})),\mathbb{E}\zeta^{2}(Y_{1})\leq 1,\mathbb{E}\xi^{2}(Y_{2})\leq 1\}.$$

(A.3)

However, an additional assumption allows us to determine a slightly different quantity D. For a symmetric kernel the operator $H$ is self-adjoint, i.e. $\langle H\zeta,\xi\rangle_{\mathcal{L}_{\mathbb{R}}^{2}(g)}=\langle\zeta,H\xi\rangle_{\mathcal{L}_{\mathbb{R}}^{2}(g)}$ for all $\zeta,\xi\in\mathcal{L}_{\mathbb{R}}^{2}(g)$ using the inner product $\langle\zeta,\xi\rangle_{\mathcal{L}_{\mathbb{R}}^{2}(g)}:=\int_{[0,1)}\zeta(s)\xi(s)g(s)ds$, and we have

$$||H||_{\mathcal{L}_{\mathbb{R}}^{2}(g)\to\mathcal{L}_{\mathbb{R}}^{2}(g)}=\sup\{|\langle H\zeta,\zeta\rangle_{\mathcal{L}_{\mathbb{R}}^{2}(g)}|:||\zeta||_{\mathcal{L}_{\mathbb{R}}^{2}(g)}\leq 1\}.$$

(A.4)

The last identity can further be reformulated in terms of a discrete convolution, which we briefly recall next. For $p\geq 1$ we denote by $\ell^{p}:=\ell^{p}(\mathbb{Z})$ the Banach space of complex sequences over $\mathbb{Z}$ endowed with its usual $\ell^{p}$-norm given by $||a_{\bullet}||_{\ell^{p}}:=\left(\sum_{j\in\mathbb{Z}}|a_{j}|^{p}\right)^{1/p}$ for $a_{\bullet}:=(a_{j})_{j\in\mathbb{Z}}\in\mathbb{C}^{\mathbb{Z}}$. In the case $p=2$, $\ell^{2}$ is a Hilbert space and the $\ell^{2}$-norm is induced by the inner product $\langle a_{\bullet},b_{\bullet}\rangle_{\ell^{2}}:=\sum_{j\in\mathbb{Z}}a_{j}\overline{b}_{j}$ for all $a_{\bullet},b_{\bullet}\in\ell^{2}$. For each sequence $a_{\bullet}\in\ell^{1}$ the discrete convolution operator $a_{\bullet}\star:\ell^{2}\to\ell^{2}$ with $b_{\bullet}\mapsto a_{\bullet}\star b_{\bullet}$ and $(a_{\bullet}\star b_{\bullet})_{j}:=\sum_{l\in\mathbb{Z}}a_{j-l}b_{l}$ for all $j\in\mathbb{Z}$, is linear and bounded by $||{a_{\bullet}}||_{\ell^{2}}$, i.e. $||a_{\bullet}\star||_{\ell^{2}\to\ell^{2}}:=\sup\{||a_{\bullet}\star b_{\bullet}||_{\ell^{2}}:||b_{\bullet}||_{\ell^{2}}\leq 1\}\leq||{a_{\bullet}}||_{\ell^{2}}$. Particularly, it holds

$$\left|\sum_{j\in\mathbb{Z}}\overline{b}_{j}\sum_{l\in\mathbb{Z}}a_{j-l}b_{l}\right|=|\langle a_{\bullet}\star b_{\bullet},b_{\bullet}\rangle_{\ell^{2}}|\leq||{a_{\bullet}}||_{\ell^{2}}||b_{\bullet}||_{\ell^{2}}^{2}\quad\text{for all}\quad a_{\bullet}\in\ell^{1}\quad\text{and}\quad b_{\bullet}\in\ell^{2}.$$

(A.5)

Note that the adjoint of $a_{\bullet}\star$ is a discrete convolution operator $a_{\bullet}^{\star}\star$ with $a_{j}^{\star}:=\bar{a}_{-j}$ for all $j\in\mathbb{Z}$. Hence, if in addition $a_{j}=\bar{a}_{-j}$ for all $j\in\mathbb{Z}$, then $a_{\bullet}\star$ is self-adjoint. Recall that the real density $g\in\mathcal{L}^{2}$ admits Fourier coefficients $g_{\bullet}=(g_{j})_{j\in\mathbb{Z}}$. The coefficients belong to both $\ell^{2}$ by Parseval’s identity, i.e. $||{g}||_{\mathcal{L}^{2}}=||g_{\bullet}||_{\ell^{2}}$, and also to $\ell^{1}$ due to the convolution theorem. Indeed, $(g_{j}=f_{j}\varphi_{j})_{j\in\mathbb{Z}}$ with $g=f{\bigcirc\hskip-8.5pt\star}\,\,\varphi$ and $f,\varphi\in\mathcal{L}^{2}$ implies $||{g_{\bullet}}||_{\ell^{1}}\leq||f_{\bullet}||_{\ell^{2}}||\varphi_{\bullet}||_{\ell^{2}}<\infty$ due to the Cauchy–Schwarz inequality. Consequently, the discrete convolution $g_{\bullet}\star:\ell^{2}\to\ell^{2}$ is linear, bounded and self-adjoint. Moreover, for all $\zeta\in\mathcal{L}^{2}$ with $|\zeta|\in\mathcal{L}_{\mathbb{R}}^{2}(g)$ and Fourier coefficients $\zeta_{\bullet}=(\zeta_{j})_{j\in\mathbb{Z}}\in\ell^{2}$ we note that $\langle g_{\bullet}\star\zeta_{\bullet},\zeta_{\bullet}\rangle_{\ell^{2}}=\sum_{j\in\mathbb{Z}}\bar{\zeta}_{j}\sum_{l\in\mathbb{Z}}g_{j-l}\zeta_{l}=\sum_{j\in\mathbb{Z}}\bar{\zeta}_{j}\sum_{l\in\mathbb{Z}}\mathbb{E}(\textrm{e}_{l}(Y)\textrm{e}_{j}(-Y))\zeta_{l}=\mathbb{E}|\zeta(Y)|^{2}=||\zeta||_{\mathcal{L}_{\mathbb{R}}^{2}(g)}^{2}\geq 0$. Thereby, if for all $\zeta=\sum_{j\in\mathbb{Z}}\zeta_{j}e_{l}\in\mathcal{L}^{2}$ we also have $|\zeta|\in\mathcal{L}_{\mathbb{R}}^{2}(g)$, then $g_{\bullet}\star$ is non-negative. As a result, there is a non-negative operator $(g_{\bullet}\star)^{1/2}$ with $||(g_{\bullet}\star)^{1/2}\zeta_{\bullet}||_{\ell^{2}}^{2}=\langle g_{\bullet}\star\zeta_{\bullet},\zeta_{\bullet}\rangle_{\ell^{2}}=||\zeta||_{\mathcal{L}_{\mathbb{R}}^{2}(g)}^{2}$ for all $\zeta_{\bullet}\in\ell^{2}$, which we use frequently in the proofs below.

A.1 Auxiliary Results Used in the Proof of Proposition 2.1

Lemma A.3. Consider $\{Y_{l}\}_{l=1}^{n}\stackrel{{\scriptstyle{iid}}}{{\sim}}g\in\mathcal{L}^{2}$ and for $k\in\mathbb{N}$ the kernel $h:[0,1)^{2}\to\mathbb{R}$ given by

$$h(y_{1},y_{2})=\sum_{|j|\in[\![k]\!]}\frac{(\textrm{e}_{j}(-y_{1})-g_{j})(\textrm{e}_{j}(y_{2})-\bar{g}_{j})}{|\varphi_{j}|^{2}},\quad\forall y_{1},y_{2}\in[0,1),$$

which is real-valued, bounded, symmetric and fulfils (A.1). Let $\nu_{k}$ and $m_{k}$ as in (2.4) then

$$\textrm{A}=4\nu_{k}^{4},\quad\textrm{B}={3||g_{\bullet}||_{\ell^{2}}\nu_{k}^{3}}\quad{and}\quad\textrm{D}=\textrm{C}=2||g_{\bullet}||_{\ell^{2}}\nu_{k}^{2}$$

(A.6)

satisfy the condition (A.2) in Proposition A.1. If, in addition, $\mathcal{L}_{\mathbb{R}}^{2}(g)=\mathcal{L}_{\mathbb{R}}^{2}$, then

$$\textrm{D}=4||{g_{\bullet}}||_{\ell^{1}}m_{k}^{2}$$

(A.7)

also satisfies the condition (A.2) in Proposition A.1.

Proof of Lemma A.3. We first calculate quantities $\textrm{A},\textrm{B}$, and C satisfying (A.2), then by the above discussion $\textrm{D}=\textrm{C}$ also satisfies (A.2). First, consider A. From

$$||(\textrm{e}_{j}-g_{j})(\textrm{e}_{-j}-\bar{g}_{l})||_{\mathcal{L}^{\infty}}\leq 4\quad\text{ and}\quad|\varphi_{j}|\leq 1\quad\text{for all}\quad j,l\in\mathbb{Z}$$

(A.8)

we immediately conclude that $||h||_{\mathcal{L}^{\infty}}\leq 4\sum_{|j|\in[\![k]\!]}|\varphi_{j}|^{-2}\leq 4\nu_{k}^{4}=\textrm{A}$. Next, consider B. Since $\mathbb{E}(\textrm{e}_{j}(-Y_{1})\textrm{e}_{l}(Y_{1}))=g_{j-l}$ for all $j,l\in\mathbb{Z}$, we deduce for arbitrary $y_{2}\in[0,1)$ that

$$\mathbb{E}\left(|h(Y_{1},y_{2})|^{2}\right)=\mathbb{V}\textrm{ar}\left(\sum_{|j|\in[\![k]\!]}\textrm{e}_{j}(-Y_{1})\frac{(\textrm{e}_{j}(y_{2})-\bar{g}_{j})}{|\varphi_{j}|^{2}}\right)\leq\mathbb{E}\left|\sum_{|j|\in[\![k]\!]}\textrm{e}_{j}(-Y_{1})\frac{(\textrm{e}_{j}(y_{2})-\bar{g}_{j})}{|\varphi_{j}|^{2}}\right|^{2}$$

$${}=\sum_{|j|\in[\![k]\!]}\frac{(\textrm{e}_{j}(y_{2})-\bar{g}_{j})}{|\varphi_{j}|^{2}}\sum_{|l|\in[\![k]\!]}g_{j-l}\frac{(\textrm{e}_{l}(-y_{2})-g_{l})}{|\varphi_{l}|^{2}}=\langle a_{\bullet}\star b_{\bullet},b_{\bullet}\rangle_{\ell^{2}},$$

(A.9)

where $a_{l}:=g_{l}{\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}{\{|l|\in[\![{{2k}}]\!]\}}$ and $b_{l}:=(\textrm{e}_{l}(-y_{2})-g_{l})|\varphi_{l}|^{-2}{\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}{\{|l|\in[\![k]\!]\}}$ for all $l\in\mathbb{Z}$. Making use of (A.5), (A.8), $||g_{\bullet}||_{\ell^{2}}\geq 1$ and $(2k)^{1/2}\leq\nu_{k}^{2}$ it follows

$$\langle a_{\bullet}\star b_{\bullet},b_{\bullet}\rangle_{\ell^{2}}$$

$${}\leq\left(\sum_{|j|\in[\![{{2k}}]\!]}|g_{j}|\right)\sum_{|j|\in[\![k]\!]}\frac{|\textrm{e}_{j}(y_{2})-\bar{g}_{j}|^{2}}{|\varphi_{j}|^{4}}\leq(4k)^{1/2}\left(\sum_{|j|\in[\![{2k}]\!]}|g_{j}|^{2}\right)^{1/2}4\nu_{k}^{4}\leq 9||g_{\bullet}||_{\ell^{2}}^{2}\nu_{k}^{6}=\textrm{B}^{2}.$$

Combining the last bound and (A.9) we see that $\sup_{y_{2}\in[0,1)}\mathbb{E}|h(Y_{1},y_{2})|^{2}\leq\textrm{B}^{2}$. Next, consider C. Since $\mathbb{E}(\textrm{e}_{j}(-Y_{1})-g_{j})(\textrm{e}_{l}(Y_{1})-\bar{g}_{l})=g_{j-l}-g_{j}\bar{g}_{l}$ for all $j,l\in\mathbb{Z}$, applying the Cauchy–Schwarz inequality we obtain

$$\mathbb{E}|h(Y_{1},Y_{2})|^{2}=\sum_{|j|\in[\![k]\!]}\frac{1}{|\varphi_{j}|^{2}}\sum_{|l|\in[\![k]\!]}\frac{|g_{j-l}-g_{j}\bar{g}_{l}|^{2}}{|\varphi_{l}|^{2}}\leq 2\langle a_{\bullet}\star b_{\bullet},b_{\bullet}\rangle_{\ell^{2}}+2\nu_{k}^{4}||g_{\bullet}^{2}||_{\ell^{2}}^{2},$$

(A.10)

where $a_{l}:=|g_{l}|^{2}{\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}{\{|l|\in[\![{{2k}}]\!]\}}$ and $b_{l}:=|\varphi_{l}|^{-2}{\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}{\{|l|\in[\![k]\!]\}}$ for all $l\in\mathbb{Z}$. Moreover, from $\langle a_{\bullet}\star b_{\bullet},b_{\bullet}\rangle_{\ell^{2}}\leq||{a_{\bullet}}||_{\ell^{2}}||b_{\bullet}||_{\ell^{2}}^{2}\leq||g_{\bullet}||_{\ell^{2}}^{2}\nu_{k}^{4}$ due to (A.5) we conclude that (A.10) and $|g_{j}|\leq 1$, $j\in\mathbb{Z}$, together imply $\mathbb{E}|h(Y_{1},Y_{2})|^{2}\leq 4\nu_{k}^{4}||g_{\bullet}||_{\ell^{2}}^{2}=\textrm{C}^{2}$. Finally, consider D and assume in addition $\mathcal{L}_{\mathbb{R}}^{2}(g)=\mathcal{L}_{\mathbb{R}}^{2}$ which allows us to use the identities (A.3) and (A.4) formulated in terms of an operator $H$. Let $\zeta\in\mathcal{L}_{\mathbb{R}}^{2}(g)$, which implies $\zeta=\sum_{j\in\mathbb{Z}}\zeta_{j}\textrm{e}_{j}\in\mathcal{L}^{2}$. Exploiting $\mathbb{E}(\textrm{e}_{j}(-Y_{1})-g_{j})\zeta(Y_{1})=(g_{\bullet}\star\zeta_{\bullet})_{j}-g_{j}\mathbb{E}\zeta(Y_{1})$ and $|g_{j}|\leq 1$ for all $j\in\mathbb{Z}$ straightforward calculations show

$$|\langle H\zeta,\zeta\rangle_{\mathcal{L}_{\mathbb{R}}^{2}(g)}|=\sum_{|j|\in[\![k]\!]}\frac{1}{|\varphi_{j}|^{2}}|\mathbb{E}((\textrm{e}_{j}(-Y_{1})-g_{j})\zeta(Y_{1}))|^{2}\leq m_{k}^{2}\sum_{|j|\in[\![k]\!]}|(g_{\bullet}\star\zeta_{\bullet})_{j}-g_{j}\mathbb{E}\zeta(Y_{1})|^{2}$$

$${}\leq 2m_{k}^{2}\left(||g_{\bullet}\star\zeta_{\bullet}||_{\ell^{2}}^{2}+||{g_{\bullet}}||_{\ell^{1}}||\zeta||_{\mathcal{L}_{\mathbb{R}}^{2}(g)}^{2}\right).$$

(A.11)

Using the properties of the discrete convolution recalled above it follows

$$||g_{\bullet}\star\zeta_{\bullet}||_{\ell^{2}}^{2}\leq||(g_{\bullet}\star)^{1/2}||_{\ell^{2}\to\ell^{2}}^{2}||(g_{\bullet}\star)^{1/2}\zeta_{\bullet}||_{\ell^{2}}^{2}=||g_{\bullet}\star||_{\ell^{2}\to\ell^{2}}||\zeta||_{\mathcal{L}_{\mathbb{R}}^{2}(g)}^{2}\leq||{g_{\bullet}}||_{\ell^{1}}||\zeta||_{\mathcal{L}_{\mathbb{R}}^{2}(g)}^{2},$$

which together with (A.11) implies $|\langle H\zeta,\zeta\rangle_{\mathcal{L}_{\mathbb{R}}^{2}(g)}|\leq 4m_{k}^{2}||{g_{\bullet}}||_{\ell^{1}}||\zeta||_{\mathcal{L}_{\mathbb{R}}^{2}(g)}^{2}$ for all $\zeta\in\mathcal{L}_{\mathbb{R}}^{2}(g)$. We conclude from (A.4) that $||H||_{\mathcal{L}_{\mathbb{R}}^{2}(g)\to\mathcal{L}_{\mathbb{R}}^{2}(g)}\leq 4m_{k}^{2}||{g_{\bullet}}||_{\ell^{1}}=\textrm{D}$, and finally that D satisfies (A.2), by (A.4), which completes the proof. $\Box$

Lemma A.4. Let $\{Y_{l}\}_{l=1}^{n}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}g=f{\bigcirc\hskip-8.5pt\star}\,\,\varphi\in\mathcal{L}^{2}$ with joint distribution $\mathbb{P}_{f}$ and $g^{\circ}=f^{\circ}{\bigcirc\hskip-8.5pt\ast}\varphi\in\mathcal{L}^{2}$. For each $k\in\mathbb{N}$ consider $\mathrm{q}^{2}_{k}(f-f^{\circ})$ and $m_{k}$ as in (2.1) and (2.4), respectively. Then the linear centred statistic $\textrm{V}_{n}$ defined in (2.3) satisfies for all $x\geq 1$ and $n\geq 1$

$$\mathbb{P}_{f}(2\textrm{V}_{n}\leq-cx^{2}(1\vee m_{k}^{2}n^{-1})m_{k}^{2}n^{-1}-\frac{1}{2}\textrm{q}^{2}_{k}(f-f^{\circ}))\leq\exp(-x),$$

where $c=8||{g_{\bullet}}||_{\ell^{1}}+||\varphi_{\bullet}||_{\ell^{2}}^{2}$.

Proof of Lemma A.4. Introducing the real function $\psi:=\sum_{|j|\in[\![k]\!]}(g_{j}-g_{j}^{\circ})|\varphi_{j}|^{-2}\textrm{e}_{j}$ and independent and identically distributed random variables $Z_{j}:=2\psi(Y_{j})$, $j\in[\![n]\!]$, we intend to apply Proposition A.2 to $\textrm{V}_{n}=\frac{1}{n}\sum_{j\in[\![n]\!]}(Z_{j}-\mathbb{E}_{f}(Z_{j}))$. Therefore, we compute the required quantities v and b. First consider b. Using subsequently the identity $g_{l}-g_{l}^{\circ}=(f_{l}-f_{l}^{\circ})\varphi_{l}$, $l\in\mathbb{Z}$, and the Cauchy–Schwarz inequality we deduce that

$$|Z_{1}|\leq 2||{\psi}||_{\mathcal{L}^{\infty}}\leq 2m_{k}^{2}\sum_{|l|\in[\![k]\!]}|g_{l}-g_{l}^{\circ}|\leq 2m_{k}^{2}\textrm{q}_{k}(f-f^{\circ})||\varphi_{\bullet}||_{\ell^{2}}=:\textrm{b}.$$

(A.12)

Secondly, consider v. Since $\mathbb{E}_{f}(\mathrm{e}_{j}(-Y_{1})e_{l}(Y_{1}))=g_{j-l}$ for all $j,l\in\mathbb{Z}$, we see that

$$\mathbb{E}_{f}|Z_{1}|^{2}=4\mathbb{E}_{f}|\psi(Y_{1})|^{2}=4\sum_{|j|\in[\![k]\!]}\frac{\bar{g}_{j}-\bar{g}_{j}^{\circ}}{|\varphi_{j}|^{2}}\sum_{|l|\in[\![k]\!]}g_{j-l}\frac{g_{l}-g_{l}^{\circ}}{|\varphi_{l}|^{2}}=4\langle a_{\bullet}\star b_{\bullet},b_{\bullet}\rangle_{\ell^{2}},$$

where $a_{l}:=g_{l}{\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}{\{|l|\in[\![{2k}]\!]\}}$ and $b_{l}:=(g_{l}-g_{l}^{\circ})|\varphi_{l}|^{-2}{\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}{\{|l|\in[\![k]\!]\}}$ for all $l\in\mathbb{Z}$. Successively exploiting further (A.5) and the identity $g_{l}-g_{l}^{\circ}=(f_{l}-f_{l}^{\circ})\varphi_{l}$, $l\in\mathbb{Z}$, we conclude that

$$\mathbb{E}_{f}|Z_{1}|^{2}\leq 4||b_{\bullet}||_{\ell^{2}}^{2}||{a_{\bullet}}||_{\ell^{1}}=4\sum_{|j|\in[\![k]\!]}|\varphi_{j}|^{-4}|g_{j}-g_{j}^{\circ}|^{2}\sum_{|j|\in[\![k]\!]}|g_{j}|$$

$${}\leq 4m_{k}^{2}\textrm{q}^{2}_{k}(f-f^{\circ})||{g_{\bullet}}||_{\ell^{1}}=:\textrm{v}.$$

(A.13)

The claim of Lemma A.4 now follows from Proposition A.2 with b and v as in (A.12) and (A.13), respectively. Indeed, making use of $2ac\leq\frac{a^{2}}{\varepsilon}+c^{2}\varepsilon$ for any $\varepsilon,a,c>0$, we have

$$\frac{\textrm{b}x}{3n}\leq\varepsilon_{1}\textrm{q}^{2}_{k}(f-f^{\circ})+\frac{x^{2}}{9\varepsilon_{1}}||\varphi_{\bullet}||_{\ell^{2}}^{2}\frac{m_{k}^{4}}{n^{2}}\quad\text{and}\quad\sqrt{\frac{2\textrm{v}x}{n}}\leq\varepsilon_{2}\textrm{q}^{2}_{k}(f-f^{\circ})+\frac{2x}{\varepsilon_{2}}\frac{m_{k}^{2}}{n}||{g_{\bullet}}||_{\ell^{1}}.$$

Combining both bounds (with $\varepsilon_{1}=\varepsilon_{2}=\frac{1}{4}$) yields for all $x\geq 1$

$$\sqrt{\frac{2\textrm{v}x}{n}}+\frac{\textrm{b}x}{3n}\leq\frac{1}{2}\textrm{q}^{2}_{k}(f-f^{\circ})+cx^{2}\left(1\vee\frac{m_{k}^{2}}{n}\right)\frac{m_{k}^{2}}{n}\quad\text{with}\quad c=8||{g_{\bullet}}||_{\ell^{1}}+||\varphi_{\bullet}||_{\ell^{2}}^{2}.$$

Hence, the assertion follows from Proposition A.2 by the usual symmetry argument. $\Box$

A.2 Auxiliary Results Used in the Proof of Proposition 4.1

Corollary A.5. Consider $\{Y_{l}\}_{l=1}^{n}\stackrel{{\scriptstyle\mathrm{iid}}}{{\sim}}g\in\mathcal{L}^{2}$ and for $k\in\mathbb{N}$ the kernel $h:[0,1)^{2}\to\mathbb{R}$ given by

$$h(y_{1},y_{2})=\sum_{|j|\in[\![k]\!]}(\mathrm{e}_{j}(-y_{1})-g_{j})(\mathrm{e}_{j}(y_{2})-\bar{g}_{j}),\quad\forall y_{1},y_{2}\in[0,1),$$

which is real-valued, bounded, symmetric and fulfils (A.1). Then the quantities

$$\textrm{A}=8k,\quad\textrm{B}=3||g_{\bullet}||_{\ell^{2}}(2k)^{3/4},\quad\text{and}\quad\textrm{D}=\textrm{C}=2||g_{\bullet}||_{\ell^{2}}(2k)^{1/2}$$

satisfy the condition (A.2) in Proposition A.1. If, in addition, $\mathcal{L}_{\mathbb{R}}^{2}(g)=\mathcal{L}_{\mathbb{R}}^{2}$, then

$$\textrm{D}=4||{g_{\bullet}}||_{\ell^{1}}$$

also satisfies the condition (A.2) in Proposition A.1.

Proof of Corollary A.5. Setting $|\varphi_{j}|^{2}=1$ for all $|j|\in[\![k]\!]$ the assertion immediately follows from Lemma A.3. $\Box$

Lemma A.6. Let $\{Y_{l}\}_{l=1}^{n}\stackrel{{\scriptstyle mathrm{iid}}}{{\sim}}g=f{\bigcirc\hskip-8.5pt\star}\,\,\varphi\in\mathcal{L}^{2}$ with joint distribution $\mathbb{P}_{f}$ and $g^{\circ}=f^{\circ}{\bigcirc\hskip-8.5pt\ast}\varphi\in\mathcal{L}^{2}$. For each $k\in\mathbb{N}$ consider $\mathrm{q}^{2}_{k}(g-g^{\circ})$ as in (4.1). Then the linear centred statistic $\textrm{V}_{n}^{\mathrm{d}}$ defined in (4.3) satisfies for all $x\geq 1$ and $n\geq 1$

$$\mathbb{P}_{f}(2\textrm{V}_{n}^{\textrm{d}}\leq-cx^{2}(1\vee(2k)^{1/2}n^{-1})(2k)^{1/2}n^{-1}-\frac{1}{2}\textrm{q}^{2}_{k}(g-g^{\circ}))\leq\exp(-x),$$

where $c=12||g_{\bullet}||_{\ell^{2}}+1$.

Proof of Lemma A.6. Introducing the real function $\psi:=\sum_{|j|\in[\![k]\!]}(g_{j}-g_{j}^{\circ})\textrm{e}_{j}$ and independent and identically distributed random variables $Z_{j}:=2\psi(Y_{j})$, $j\in[\![n]\!]$, we intend to apply Proposition A.2 to $\textrm{V}_{n}^{\textrm{d}}=\frac{1}{n}\sum_{j\in[\![n]\!]}(Z_{j}-\mathbb{E}_{f}(Z_{j}))$. Therefore, we compute the required quantities v and b. First consider b. Using the Cauchy–Schwarz inequality we see that

$$|Z_{1}|\leq 2||{\psi}||_{\mathcal{L}^{\infty}}\leq 2\sum_{|l|\in[\![k]\!]}|g_{l}-g_{l}^{\circ}|\leq 2(2k)^{1/2}\textrm{q}_{k}(g-g^{\circ})=:\textrm{b}.$$

(A.14)

Secondly, consider v. Since $\mathbb{E}_{f}(\mathrm{e}_{j}(-Y_{1})e_{l}(Y_{1}))=g_{j-l}$ for all $j,l\in\mathbb{Z}$, we deduce that

$$\mathbb{E}_{f}|Z_{1}|^{2}=4\mathbb{E}_{f}|\psi(Y_{1})|^{2}=4\sum_{|j|\in[\![k]\!]}(\bar{g}_{j}-\bar{g}_{j}^{\circ})\sum_{|l|\in[\![k]\!]}g_{j-l}(g_{l}-g_{l}^{\circ})=4\langle a_{\bullet}\star b_{\bullet},b_{\bullet}\rangle_{\ell^{2}},$$

where $a_{l}:=g_{l}{\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}{\{|l|\in[\![{{2k}}]\!]\}}$ and $b_{l}:=(g_{l}-g_{l}^{\circ}){\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}{\{|l|\in[\![k]\!]\}}$ for all $l\in\mathbb{Z}$. Hence, (A.5) shows that

$$\mathbb{E}_{f}|Z_{1}|^{2}\leq 4||b_{\bullet}||_{\ell^{2}}^{2}||{a_{\bullet}}||_{\ell^{1}}=4\textrm{q}^{2}_{k}(g-g^{\circ})\sum_{|j|\in[\![{{2}k}]\!]}|g_{j}|=:\textrm{v}.$$

(A.15)

The claim of Lemma A.6 follows now from Proposition A.2 with b and v as in (A.14) and (A.15), respectively. Indeed, exploiting $2ac\leq\frac{a^{2}}{\varepsilon}+c^{2}\varepsilon$ for any $\varepsilon,a,c>0$ we see that

$$\frac{\textrm{b}x}{3n}\leq\varepsilon_{1}\textrm{q}^{2}_{k}(g-g^{\circ})+\frac{x^{2}}{9\varepsilon_{1}}\frac{2k}{n^{2}}\quad\text{and }$$

$$\sqrt{\frac{2\textrm{v}x}{n}}\leq\varepsilon_{2}\textrm{q}^{2}_{k}(g_{\bullet}-g^{\circ})+\frac{2x}{\varepsilon_{2}n}\sum_{|j|\in[\![{{2k}}]\!]}|g_{j}|\leq\varepsilon_{2}\textrm{q}^{2}_{k}(g-g^{\circ})+\frac{2x}{\varepsilon_{2}}\frac{(4k)^{1/2}}{n}||g_{\bullet}||_{\ell^{2}}.$$

Combining both bounds (with $\varepsilon_{1}=\varepsilon_{2}=\frac{1}{4}$) we get for all $x\geq 1$

$$\sqrt{\frac{2\textrm{v}x}{n}}+\frac{\textrm{b}x}{3n}\leq\frac{1}{2}\textrm{q}^{2}_{k}(g-g^{\circ})+cx^{2}(1\vee(2k)^{1/2}n^{-1})(2k)^{1/2}n^{-1}\quad\text{with}\quad c=12||g_{\bullet}||_{\ell^{2}}+1.$$

Hence, the assertion follows from Proposition A.2 by the usual symmetry argument. $\Box$

CALCULATIONS FOR THE ILLUSTRATIONS

B.1 Calculations for the Radius Bounds in Illustration 3.5

Firstly, we determine the order of the term $\rho_{\mathcal{K},a_{\bullet}}^{2}(\delta n)$ by showing that $\rho_{\mathcal{K},a_{\bullet}}^{2}(n)\sim\rho_{a_{\bullet}}^{2}(n)$ and replacing $n$ with $\delta n$. Indeed, we trivially have $\rho_{a_{\bullet}}^{2}(n)\leq\rho_{\mathcal{K},a_{\bullet}}^{2}(n)$. By defining $j_{\star}:=\lceil\frac{2}{4p+4s+1}\log_{2}n\rceil\lesssim\log(n^{2}/2)$ (in the ordinary smooth case) respectively $j_{\star}:=\lceil\frac{1}{s}\log_{2}\log n\rceil\lesssim\frac{1}{s_{\star}}\log\log n$ (in the super smooth case), straightforward calculations then show that $\rho_{\mathcal{K},a_{\bullet}}^{2}(n)\lesssim\rho_{2^{j\star},a_{\bullet}}^{2}(n)\lesssim\rho_{a_{\bullet}}^{2}(n)$. Next, we determine the order of the remainder term $r_{\mathcal{K},a_{\bullet}}^{2}(\delta^{2}n)$ by first calculating $r_{\mathbb{N},a_{\bullet}}^{2}(n):=\min_{k\in\mathbb{N}}a_{k}^{2}\vee\frac{m_{k}^{2}}{n}$, showing $r_{\mathbb{N},a_{\bullet}}^{2}(n)\sim r_{\mathcal{K}_{g},a_{\bullet}}^{2}(n)$ and then replacing $n$ with $\delta^{2}n$. The variance term $\frac{m_{k}^{2}}{n}$ is of order $\frac{k^{2p}}{n}$. In the ordinary smooth case the bias term $a_{k}^{2}$ is of order $k^{-2s}$. Hence, the minimising $k_{\star}$ satisfies $k_{\star}\sim n^{\frac{1}{2s+2p}}$, which yields $r_{\mathbb{N},a_{\bullet}}^{2}(n)\sim n^{-\frac{s}{s+p}}$. We define $j_{\star}:=\lceil\frac{1}{2p+2s}\log_{2}n\rceil\lesssim\log(n^{2}/2)$. Straightforward calculations show that $r_{\mathcal{K}_{2},a_{\bullet}}^{2}(n)\lesssim r_{2^{j\star},a_{\bullet}}^{2}(n)\lesssim r_{\mathbb{N},a_{\bullet}}^{2}(n)$. Since, trivially $r_{\mathbb{N},a_{\bullet}}^{2}(n)\leq r_{\mathcal{K}_{g},a_{\bullet}}^{2}(n)$, we obtain the assertion. In the super smooth case the bias term $a_{k}^{2}$ is of order $e^{-2k^{s}}$. Hence, the minimising $k_{\star}$ satisfies $k_{\star}\sim\log(n)^{\frac{1}{s}}$, which yields $r_{\mathbb{N},a_{\bullet}}^{2}(n)\sim\frac{1}{n}(\log n)^{\frac{2p}{s}}$. We define $j_{\star}:=\lceil\frac{1}{s}\log_{2}\log n\rceil\lesssim\frac{1}{s_{\star}}\log\log n$. Straightforward calculations show that $r_{\mathcal{K}_{s_{\star},a_{\bullet}}}^{2}(n)\lesssim r_{2^{j\star},a_{\bullet}}^{2}(n)\lesssim r_{\mathbb{N},a_{\bullet}}^{2}(n)$. Since, trivially $r_{\mathbb{N},a_{\bullet}}^{2}(n)\leq r_{\mathcal{K}_{s_{\star},a_{\bullet}}}^{2}(n)$, we obtain the assertion.

B.2 Calculations for the Radius Bounds in Illustration 4.5

Ordinary smooth—mildly ill-posed. Since $\frac{(2k)^{1/2}}{n}m_{k}^{2}\sim\frac{1}{n}k^{2p+1/2}$ and $a_{k}^{2}\sim k^{-2s}$, the optimal $k_{a_{\bullet}}^{\textrm{d}}$ satisfies $k_{a_{\bullet}}^{\textrm{d}}\sim n^{\frac{2}{4p+4s+1}}$, which yields an upper bound of order $(\rho_{a_{\bullet}}^{\textrm{d}})^{2}\sim(k_{a_{\bullet}}^{\textrm{d}})^{-2s}\sim n^{-\frac{4s}{4p+4s+1}}$. Ordinary smooth—severely ill-posed. Since $\frac{(2k)^{1/2}}{n}m_{k}^{2}\sim\frac{1}{n}k^{1/2}e^{2k^{p}}$ and $a_{k}^{2}\sim k^{-2s}$, we obtain $k_{a_{\bullet}}^{\textrm{d}}\sim(\log n)^{\frac{1}{p}}$, which yields an upper bound of order $(\rho_{a_{\bullet}}^{\textrm{d}})^{2}\sim(k_{a_{\bullet}}^{\textrm{d}})^{-2s}\sim(\log n)^{-\frac{2s}{p}}$. Super smooth—mildly ill-posed. Since $\frac{(2k)^{1/2}}{n}m_{k}^{2}\sim\frac{1}{n}k^{2p+1/2}$ and $a_{k}^{2}\sim e^{-2k^{s}}$, we obtain $k_{a_{\bullet}}^{\textrm{d}}\sim(\log n)^{\frac{1}{s}}$, which yields an upper bound of order $(\rho_{a_{\bullet}}^{\textrm{d}})^{2}\sim\frac{1}{n}(k_{a_{\bullet}}^{\textrm{d}})^{2p+1/2}\sim\frac{1}{n}(\log n)^{\frac{2p+1/2}{s}}$.

B.3 Calculations for the Radius Bounds in Illustration 5.4

Firstly, we determine the order of the terms $(\rho_{\mathcal{K},a_{\bullet}}^{\textrm{d}}(\delta n))^{2}$ by showing that $(\rho_{\mathcal{K}_{2},a_{\bullet}}^{\textrm{d}}(n))^{2}\sim(\rho_{a_{\bullet}}^{\textrm{d}}(n))^{2}$ and replacing $n$ with $\delta n$. Indeed, we trivially have $(\rho_{\mathcal{K},a_{\bullet}}^{\textrm{d}}(\delta n))^{2}\leq(\rho_{a_{\bullet}}^{\textrm{d}}(n))^{2}$. Define $j_{\star}:=\lceil\frac{2}{4p+4s+1}\log_{2}n\rceil\lesssim\log(n^{2}/2)$ (ordinary smooth—mildly ill-posed case), $j_{\star}:=\lceil\frac{1}{s}\log_{2}\log n\rceil\lesssim\frac{1}{s_{\star}}\log\log n$ (super smooth—mildly ill-posed case) respectively $j_{\star}:=\lceil\frac{1}{s}\log_{2}\log n\rceil\lesssim\frac{1}{s_{\star}}\log\log n$ (ordinary smooth—severely ill-posed case). Straightforward calculations then show that $\rho_{\mathcal{K},a_{b}ullet}^{2}(n)\lesssim\rho_{2^{j\star},a_{b}ullet}^{2}(n)\lesssim\rho_{a_{b}ullet}^{2}(n)$. Next, we determine the order of the remainder term $r_{\mathcal{K},a_{\bullet}}^{2}(\delta^{2}n)$ by first calculating $r_{\mathbb{N},a_{\bullet}}^{2}(\delta^{2}n):=\min_{k\in\mathbb{N}}a_{k}^{2}\vee\frac{m_{k}^{2}}{n}$ and then showing that minimisation over $\mathcal{K}_{g}$ approximates the minimisation over $\mathbb{N}$ well enough. The calculations in the (ordinary smooth—mildly ill-posed) and (super smooth—mildly ill-posed) cases have already been done in Illustration 3.5. It remains to consider the third case (ordinary smooth—severely ill-posed). Since $\frac{m_{k}^{2}}{n}\sim\frac{e^{2k^{p}}}{n}$ and $a_{k}^{2}\sim k^{-2s}$, the minimising $k_{\star}$ satisfies $k_{\star}^{-2s}\sim\frac{e^{2k_{\star}^{p}}}{n}$ and thus $k_{\star}\sim(\log n)^{\frac{1}{p}}$, which yields $r_{\mathbb{N},a_{\bullet}}^{2}(n)\sim(\log n)^{\frac{2s}{p}}$. Next, we show $r_{\mathcal{K}_{g},a_{\bullet}}^{2}(n)\sim r_{\mathbb{N},a_{\bullet}}^{2}(n)$. We define $j_{\star}:=\lceil\frac{1}{p}\log_{2}\log n\rceil\lesssim\log(n^{2}/2)$. Straightforward calculations show that $r_{\mathcal{K}_{g},a_{\bullet}}^{2}(n)\lesssim r_{2^{j\star},a_{\bullet}}^{2}(n)\lesssim r_{\mathbb{N},a_{\bullet}}^{2}(n)$. Since, trivially $r_{\mathbb{N},a_{\bullet}}^{2}(n)\leq r_{\mathcal{K}_{g},a_{\bullet}}^{2}(n)$, we obtain the assertion by replacing $n$ with $\delta^{2}n$.

CALCULATIONS FOR THE $\chi^{2}$-DIVERGENCE

In the proof of Lemma C.2 below we apply the following assertion due to [31] (Lemma A.1 in the Appendix A).

Lemma C.1. For $k\in\mathbb{N}$ and for each sign vector $\tau\in\{\pm\}^{k}$ let $J^{\tau}=(J_{j}^{\tau_{j}})_{j\in[\![k]\!]}\in\mathbb{R}^{k}$. Then,

$$\frac{1}{2^{k}}\sum_{\tau\in\{\pm\}^{k}}\prod_{j\in[\![k]\!]}J_{j}^{\tau_{j}}=\prod_{j\in[\![k]\!]}\frac{J_{j}^{-}+J_{j}^{+}}{2}.$$

Lemma C.2 ( $\boldsymbol{\chi}^{\mathbf{2}}$ -divergence for mixtures over hypercubes over several classes). Let $\mathcal{S}$ be an arbitrary index set of finite cardinality $|\mathcal{S}|\subset\mathbb{N}$ . For each $s\in\mathcal{S}$ assume $k^{s}\in\mathbb{N}$ and $\theta_{\bullet}^{s}\in\ell^{2}(\mathbb{N})\subseteq\mathbb{R}^{\mathbb{N}}$ . For $\tau\in\{\pm\}^{k^{s}}$ define coefficients $\theta_{\bullet}^{s,\tau}\in\ell^{2}(\mathbb{N})$ and functions $g^{s,\tau}\in\mathcal{L}^{2}$ by setting

$$\theta_{j}^{s,\tau_{j}}=\begin{cases}\tau_{j}\theta_{j}^{s}\quad j\in[\![{k^{s}}]\!]\\ 0\quad{otherwise}\end{cases}\quad{and}\quad g^{s,\tau}=\textrm{e}_{0}+\sum_{|j|\in[\![{k^{s}}]\!]}\theta_{|j|}^{s,\tau_{|j|}}\textrm{e}_{j}.$$

Assuming $g^{s,\tau}\in\mathcal{D}$ for each $s\in\mathcal{S}$ and $\tau\in\{\pm\}^{k^{s}}$ , we consider the mixture $\mathbb{P}_{1}$ with probability density $\frac{1}{|\mathcal{S}|}\sum_{s\in\mathcal{S}}(\frac{1}{2^{k^{s}}}\sum_{\tau\in\{\pm\}^{k^{s}}}\prod_{j\in[\![n]\!]}g^{s,\tau}(z_{j}))$ , $z_{j}\in[0,1),j\in[\![n]\!]$ and denote $\mathbb{P}_{0}:=\mathbb{P}_{{\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}_{[0,1)}}$ . Then, the $\chi^{2}$ -divergence satisfies

$$\chi^{2}(\mathbb{P}_{1},\mathbb{P}_{0})\leq\frac{1}{|\mathcal{S}|^{2}}\sum_{s,t\in\mathcal{S}}\exp\left(2n^{2}\sum_{j\in[\![{k^{s}\wedge k^{t}}]\!]}(\theta_{j}^{s}\theta_{j}^{t})^{2}\right)-1.$$

Proof of Lemma C.2. Recall that $\chi^{2}(\mathbb{P}_{1},\mathbb{P}_{0})=\mathbb{E}_{0}\left(\frac{\textrm{d}\mathbb{P}_{1}}{\textrm{d}\mathbb{P}_{0}}(Z_{j},_{j\in[\![n]\!]})\right)^{2}-1$ where $(Z_{j})_{j\in[\![n]\!]}$ are independent with identical marginal density $\textrm{e}_{0}={\mathchoice{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.0mu l}{\rm 1\mskip-4.5mu l}{\rm 1\mskip-5.0mu l}}_{[0,1)}$ under $\mathbb{P}_{0}$. Let $z_{j}\in[0,1)$, $j\in[\![n]\!]$, then

$$\frac{\textrm{d}\mathbb{P}_{1}}{\textrm{d}\mathbb{P}_{0}}(z_{j},_{j\in[\![n]\!]})=\frac{1}{|\mathcal{S}|}\sum_{s\in\mathcal{S}}\left(\frac{1}{2^{k^{s}}}\sum_{\tau\in\{\pm\}^{k^{s}}}\prod_{j\in[\![n]\!]}g^{s,\tau}(z_{j})\right).$$

Squaring, taking the expectation under $\mathbb{P}_{0}$ and exploiting the independence yields

$$\mathbb{E}_{0}\left(\frac{\textrm{d}\mathbb{P}_{1}}{\textrm{d}\mathbb{P}_{0}}(Z_{j},_{j\in[\![n]\!]})\right)^{2}=\frac{1}{|\mathcal{S}|^{2}}\sum_{s,t\in\mathcal{S}}\frac{1}{2^{k^{s}}}\frac{1}{2^{k^{t}}}\sum_{\tau\in\{\pm\}^{k^{s}}}\sum_{\eta\in\{\pm\}^{k^{t}}}\prod_{j\in[\![n]\!]}\mathbb{E}_{0}(g^{s,\tau}(Z_{j})g^{t,\eta}(Z_{j}))$$

$${}=\frac{1}{|\mathcal{S}|^{2}}\sum_{s,t\in\mathcal{S}}\frac{1}{2^{k^{s}}}\frac{1}{2^{k^{t}}}\sum_{\tau\in\{\pm\}^{k^{s}}}\sum_{\eta\in\{\pm\}^{k^{t}}}(\mathbb{E}_{0}(g^{s,\tau}(Z_{1})g^{t,\eta}(Z_{1})))^{n}.$$

Exploiting the orthonormality of $(\textrm{e}_{j})_{j\in\mathbb{Z}}$ we calculate

$$\mathbb{E}_{0}(g^{s,\tau}(Z_{1})g^{t,\eta}(Z_{1}))=\int g^{s,\tau}(z)g^{t,\eta}(z)\textrm{d}z=1+2\sum_{j\in[\![{k^{s}\wedge k^{t}}]\!]}\theta_{j}^{s,\tau_{j}}\theta_{j}^{t,\eta_{j}}.$$

Applying the inequality $1+x\leq\exp(x)$ for all $x\in\mathbb{R}$ we obtain

$$\mathbb{E}_{0}(g^{s,\tau}(Z_{1})g^{t,\eta}(Z_{1}))$$

$${}=1+2\sum_{j\in[\![{k^{s}\wedge k^{t}}]\!]}\theta_{j}^{s,\tau_{j}}\theta_{j}^{t,\eta_{j}}\leq\exp\left(2\sum_{j\in[\![{k^{s}\wedge k^{t}}]\!]}\theta_{j}^{s,\tau_{j}}\theta_{j}^{t,\eta_{j}}\right)$$

$$=\prod_{j\in[\![{k^{s}\wedge k^{t}}]\!]}\exp\left(2\theta_{j}^{s,\tau_{j}}\theta_{j}^{t,\eta_{j}}\right).$$

Hence,

$$\mathbb{E}_{0}\left(\frac{\textrm{d}\mathbb{P}_{1}}{\textrm{d}\mathbb{P}_{0}}(Z_{j},_{j\in[\![n]\!]})\right)^{2}\leq\frac{1}{|\mathcal{S}|^{2}}\sum_{s,t\in\mathcal{S}}\frac{1}{2^{k^{s}}}\frac{1}{2^{k^{t}}}\sum_{\tau\in\{\pm\}^{k^{s}}}\sum_{\eta\in\{\pm\}^{k^{t}}}\prod_{j\in[\![{k^{s}\wedge k^{t}}]\!]}\exp\left(2n\theta_{j}^{s,\tau_{j}}\theta_{j}^{t,\eta_{j}}\right),$$

where we apply Lemma C.1 to the $\eta$-summation with $J_{j}^{\eta_{j}}=\exp(2n\theta_{j}^{s,\tau_{j}}\theta_{j}^{t,\eta_{j}})$ and obtain

$$\mathbb{E}_{0}\left(\frac{\textrm{d}\mathbb{P}_{1}}{\textrm{d}\mathbb{P}_{0}}(Z_{j},_{j\in[\![n]\!]})\right)^{2}\leq\frac{1}{|\mathcal{S}|^{2}}\sum_{s,t\in\mathcal{S}}\frac{1}{2^{k^{s}}}\sum_{\tau\in\{\pm\}^{k^{s}}}\prod_{j\in[\![{k^{s}\wedge k^{t}}]\!]}\frac{\exp(-2n\theta_{j}^{s,\tau_{j}}\theta_{j}^{t})+\exp(2n\theta_{j}^{s,\tau_{j}}\theta_{j}^{t})}{2}.$$

Again applying Lemma C.1 to the $\tau$-summation with $J_{j}^{\tau_{j}}=\frac{\exp(-2n\theta_{j}^{s,\tau_{j}}\theta_{j}^{t})+\exp(2n\theta_{j}^{s,\tau_{j}}\theta_{j}^{t})}{2}$ yields

$$\mathbb{E}_{0}\left(\frac{\textrm{d}\mathbb{P}_{1}}{\textrm{d}\mathbb{P}_{0}}(Z_{j},_{j\in[\![n]\!]})\right)^{2}\leq\frac{1}{|\mathcal{S}|^{2}}\sum_{s,t\in\mathcal{S}}\prod_{j\in[\![{k^{s}\wedge k^{t}}]\!]}\frac{\exp(-2n\theta_{j}^{s}\theta_{j}^{t})+\exp(2n\theta_{j}^{s}\theta_{j}^{t})}{2}$$

$${}=\frac{1}{|\mathcal{S}|^{2}}\sum_{s,t\in\mathcal{S}}\prod_{j\in[\![{k^{s}\wedge k^{t}}]\!]}\cosh(2n\theta_{j}^{s}\theta_{j}^{t}).$$

Since $\cosh(x)\leq\exp(x^{2}/2)$, $x\in\mathbb{R}$, we obtain

$$\mathbb{E}_{0}\left(\frac{\textrm{d}\mathbb{P}_{1}}{\textrm{d}\mathbb{P}_{0}}(Z_{j},_{j\in[\![n]\!]})\right)^{2}\leq\frac{1}{|\mathcal{S}|^{2}}\sum_{s,t\in\mathcal{S}}\prod_{j\in[\![{k^{s}\wedge k^{t}}]\!]}\exp\left(2n^{2}(\theta_{j}^{s}\theta_{j}^{t})^{2}\right)=\frac{1}{|\mathcal{S}|^{2}}\sum_{s,t\in\mathcal{S}}\exp\left(2n^{2}\sum_{j\in[\![{k^{s}\wedge k^{t}}]\!]}\hskip-4.3pt(\theta_{j}^{s}\theta_{j}^{t})^{2}\right),$$

which completes the proof. $\Box$