8 Additional results
These are the results of Sections S2.3, S2.4, and S2.6 of the paper, the threshold test of Section 5, and the adjuster of Section S3.
Let \(c_\theta (X_1)=\{ 2/(1+s^2)\} ^{1/2}\exp \{ -(1-s^2)(X_1-\theta )^2/(2+2s^2)\} \), the right-hand side of eq. (S15). Let \(\tilde E_t=\prod _{i=2}^t\ell (X_i-\bar X_{i-1})\), the likelihood ratio \(L^\theta _t\) with the running mean in place of \(\theta \). The mutual information \(I(f;g)\) of two random variables is the Kullback–Leibler divergence of their joint distribution from the product of their distributions.
For every \(\theta \in \mathbb {R}\) and every \(s\ne 1\),
which are eqs. (S15) and (S16), where the supremum runs over the stopping times of \((\mathcal{F}_t)\) bounded by \(2\). The rule \(\tau =1+\mathbf{1}\{ \mathrm{E}_\theta (E^{\star }_2\mid X_1){\gt}1\} \), which Section S2.3 calls the best stopping time up to \(t=2\), attains the supremum.
Eq. (S15) follows from the conditional distribution of \(Y_2\) given \(X_1\) and eq. (S1) with \(v=1/2\); it holds for every \(s{\gt}0\). Eq. (S16) follows from the decomposition over \(\{ \tau \le 1\} \in \mathcal{F}_1\). The supremum exceeds one because \(\{ \mathrm{E}_\theta (E^{\star }_2\mid X_1){\gt}1\} =\{ X_1\in U\} \) for an open set \(U\), which contains \(\theta \) if \(s{\lt}1\) and every point far enough from \(\theta \) if \(s{\gt}1\), so that it has positive probability under \(\mathrm{N}(\theta ,1)\).
Let \(s\ne 1\) and \(\rho ^2\ge 1-s^2\). The conditional expectation \(\mathrm{E}_\theta (E^{\star }_2\mid X_1)\) exceeds one when \((X_1-\theta )^2{\lt}\{ (1+s^2)/(1-s^2)\} \log \{ 2/(1+s^2)\} \) if \(s{\lt}1\), and when the reverse strict inequality holds if \(s{\gt}1\). The numéraires are not an e-process with respect to \((\mathcal{F}_t)\), the null distributions of \(X^2\) are absolutely continuous with respect to \(Q_2\), and \(\Gamma _T{\gt}0\) for every \(T\ge 2\), so that the minimax regret is positive also for \(s{\gt}1\) (Sections S2.3 and 4.1).
A \(Q_2\)-null set is \(Q_{\mu ,2}\)-null for some \(\mu \), since \(Q\) is a mixture, and Gaussian product measures on \(\mathbb {R}^2\) are equivalent to Lebesgue measure. If \(\Gamma _2=0\), Proposition S1(i) gives versions of the numéraires that form an e-process stopped at \(2\); they agree with the numéraires \(P_\theta \)-almost surely, since \(P_{\theta ,2}\ll Q_2\), and the rule of Proposition S2 contradicts (S16). Hence \(\Gamma _2{\gt}0\), by Lemma S1, and Proposition 1 gives \(\Gamma _T\ge \Gamma _2\).
(i) For every \(\theta \in \mathbb {R}\), \(X_1=\theta -\sum _{i\ge 2}Y_i/\{ i(i-1)\} ^{1/2}\) almost surely under \(P_\theta \), the series converging as the limit of its partial sums. (ii) Let \(\theta \in \mathbb {R}\) and \(t\ge 1\). Under \(P_\theta \), the conditional distribution of \(X_1\) given \(\mathcal{G}_t\) is \(\mathrm{N}[\theta -\sum _{i=2}^tY_i/\{ i(i-1)\} ^{1/2},1/t]\), and \(I(X_1;Y^t)=\frac12\log t\). (iii) Let \((E_t)\) be a process with values in \([0,\infty ]\) adapted to \((\mathcal{G}_t)\). Then \(\sup _\tau \mathrm{E}_\theta (E_\tau )\) is the same for every \(\theta \in \mathbb {R}\), where \(\tau \) ranges over the stopping times of \((\mathcal{F}_t)\), and \((E_t)\) is an e-process for \(\mathcal{P}\) if and only if it is an e-process for \(\{ P_\theta \} \) for some \(\theta \).
(i) Lemma S2(ii) gives \(\bar X_n=X_1+\sum _{i=2}^nY_i/\{ i(i-1)\} ^{1/2}\), and the strong law of large numbers gives \(\bar X_n\to \theta \); the absolute series diverges, so that the statement uses partial sums. (ii) The joint distribution of \((Y^t,X_1)\) is the distribution of \(Y^t\) followed by the kernel \(y\mapsto \mathrm{N}\{ \theta -g(y),1/t\} \). Mathlib has no differential entropy, so that the mutual information is computed as a Kullback–Leibler divergence: the density of the joint distribution against the product of the marginals is the ratio of the densities of \(\mathrm{N}\{ \theta -g(y),1/t\} \) and \(\mathrm{N}(\theta ,1)\), with expected logarithm \(\frac12\log t\). (iii) The paper uses that the shift of the data is a bijection of the stopping times; here, the supremum at \(\theta +c\) is bounded by that at \(\theta \) for every \(c\), and applied twice.
For every \(\theta \in \mathbb {R}\), \(\sup _\tau \mathrm{E}_\theta (E^{\star }_{\tau \wedge T})\) equals \(1\) when \(\tau \) ranges over the stopping times of \((\mathcal{G}_t)\), and equals \(\eta _T\) when \(\tau \) ranges over the stopping times of \((\mathcal{F}_t)\) (Section S2.4).
For the filtration \((\mathcal{G}_t)\), optional stopping at the bounded time \(\tau \wedge T\) for the test martingale of Proposition 2 gives both inequalities, in place of the citation of Proposition 2. For the filtration \((\mathcal{F}_t)\), Proposition S3(iii) shows that the supremum does not depend on \(\theta \), so that it equals \(\eta _T\).
Let \(s{\lt}1\) and \(\rho ^2\ge 1-s^2\). The process \((\tilde E_t)_{t\ge 0}\) is a test supermartingale with respect to \((\mathcal{F}_t)\) under every \(P_\theta \), and, for \(t\ge 1\) and \(T\ge 1\),
For every \(\mu \) and \(t\ge 1\), moreover, \(R^\mu _t(\tilde E_t)=\kappa +R_t(\tilde E_t)\).
As in Section S2.6: under \(P_\theta \), \(\mathrm{E}_\theta \{ \ell (X_i-\bar X_{i-1})\mid \mathcal{F}_{i-1}\} =\exp \{ -(1-s^2)(\theta -\bar X_{i-1})^2/2\} \le 1\) by (S1), computed through the conditional distribution given \(\mathcal{F}_{i-1}\). Since \(X_i-\bar X_{i-1}=\{ i/(i-1)\} ^{1/2}Y_i\), \(\log \ell (y)-\log \ell [\{ i/(i-1)\} ^{1/2}y]=by^2/(2i-2)\), and \(\mathrm{E}_Q(Y_i^2)=s^2\) gives the regret. The bound on \(\Gamma _T\) uses \(\sum _{i=1}^{t-1}i^{-1}\le 1+\log t\) from Mathlib, and the last claim is Lemma 60, since \(\tilde E_t\) depends on the data only through the \(Y_i\).
For a test supermartingale \((M_t)\) with respect to any filtration and \(c{\gt}0\), \(P(M_t\ge c\text{ for some }t)\le 1/c\) for every null distribution \(P\).
Optional stopping at the bounded hitting times of \([c,\infty )\), and monotone convergence.
For \(s{\gt}0\) and \(\alpha \in (0,1]\), the test that rejects the null as soon as the conditional e-variables \(E^{\star }_t\) reach \(1/\alpha \) has level \(\alpha \): \(P(E^{\star }_t\ge 1/\alpha \text{ for some }t)\le \alpha \) for every \(P\in \mathcal{P}\) (Section 5). The condition \(\alpha \le 1\) is not needed.
The numéraires form a test martingale with respect to \((\mathcal{G}_t)\) (Proposition 2), and Ville’s inequality applies in this filtration.
Assume the e-lifting theorem of Choe and Ramdas. For \(s{\gt}0\), the process with terms \(A(\max _{i\le t}E^{\star }_i)\), with the adjuster \(A\) of Choe and Ramdas (2026, Section 2.3), is an e-process with respect to \((\mathcal{F}_t)\) (Section S3).
The numéraires form a test martingale with respect to \((\mathcal{G}_t)\) (Proposition 2), and \(\mathcal{G}_t\subseteq \mathcal{F}_t\).
The Gaussian location model with the numéraires of Proposition 2 satisfies the standing assumption, and Proposition 1 applies to it. On \(\mathcal{X}=\{ 0,1\} \), the simple null of independent fair coins and the alternative of independent coins with probability \(1/4\) of one, with the likelihood ratios \(\mathrm{d}Q_t/\mathrm{d}P_t\) as numéraires, satisfy the standing assumption. For this model, \(\Gamma _T=0\) for every \(T\), and Proposition S1 and Lemma 41 apply.
An e-variable at horizon \(t\) is constant on the cylinders of the first \(t\) tosses, which have probability \(2^{-t}\) under the null, so that it is at most \(2^t\) and the optimal e-power is at most \(t\log 2\). The numéraires and \(\Gamma _T=0\) come from the claim of Section 1 on simple nulls.