Lemma S3 with \(q\) chosen so that \((1-s^2)q{\lt}b\), which replaces letting \(q\to 1\); the term \(\log (2+2\log T/\log q)\) is bounded explicitly by \(\log \log T+\log (2+2/\log q)\) for \(T\ge 3\).

The choices of the paper: \(c'=s(2\log K+4\log \log K)^{1/2}\), \(c=(c'+r^{-1/2})/(1-r^{-1/2})\), and \(K=\lfloor \log T/\log r\rfloor \). The \(O\) and \(o\) terms of the paper are explicit: with \(\delta =1-r^{-1/2}\), the right-hand side of (S9) is at least \((1-s^2/\delta ^2)\log K-28(\log K)^{1/2}-C_0\), and \(\log \log T-\log (2\log r)\le \log K\le \log \log T\) when \(\log T/\log r\ge 6\). The ratio \(r\) is chosen with \(1-s^2/\delta ^2{\gt}\max (a,0)\), so that the case of a coefficient that is not positive, which the paper treats with \(\Gamma _T\ge 0\), does not arise.

As in Section S1.6. Lemma S3 with \(T=T_j\) and \(q=q_j\) gives \(\eta _{T_j}\le u_j\), so that \(\mathrm{E}_\theta (E_{\tau \wedge n})\le 1\). The paper shows that \(E^{\star }_t\to 0\) almost surely by the strong law of large numbers; here, the nonnegative martingale \((E^{\star }_t)\) with respect to \((\mathcal{G}_t)\) converges almost surely by Doob’s theorem, which suffices for \(E_t\) to converge almost surely, by dominated convergence for the series over \(j\), and Fatou’s lemma gives \(\mathrm{E}_\theta (E_\tau )\le 1\). For the upper limit, \(j=\lfloor \log \log t\rfloor +1\) and the explicit bound \(\log \{ j(j+1)u_j\} \le (1-s^2)j+4\log (j+2)+C\). For the lower limit, the paper takes a maximizer \(t_T\) of \(R_t(E_t)\) over \(t\le T\); here, the argument is by contradiction with \(\Gamma _T\le \max _{t\le T}R_t(E_t)\).

The upper bounds follow from \(\Gamma _T\le \log \eta _T\) (Proposition 1(iii)) and Lemma 80, and the lower bounds from Lemma S6 and Lemma 81, with the numéraires of Proposition 2. Part (ii) is Lemma 82.

As in Section S2.5. For \(t\ge t_\rho \), the proof of Proposition 2 applies at the horizon \(t\), so that \(\Pi _t=\prod _{i=2}^t\ell (Y_i)\) is the numéraire, and the differences \(d_t\) between the optimal e-power and \(\mathrm{E}_Q(\log \Pi _t)\) are bounded. The paper takes \(C=\max _{t{\lt}t_\rho }d_t\) and \(t_\rho =\lceil (1-s^2)/\rho ^2\rceil \); here, \(C=\sum _{t{\lt}t_\rho }|d_t|\) and \(t_\rho =\lceil (1-s^2)/\rho ^2\rceil +1\). The e-process \((\Pi _{t\wedge T}/\eta '_T)_{t\ge 0}\), with \(\eta '_T\) the value of Lemma 2(ii), comes from Proposition 1(iii) for the mixture with \(\rho =1\), since the e-process property does not involve the alternative, and gives \(\Gamma _T\le C+\log \eta '_T\). The lower bound is Lemma S6, which holds for every \(\rho {\gt}0\), with Lemma 81. For the limit of \(\log \eta _T\), Proposition 1(iii) gives \(\Gamma _T\le \log \eta _T\), and \(\eta _T\le t_\rho +\eta '_T\): for \(t\ge t_\rho \), the numéraire equals \(\Pi _t\) \(Q\)-almost surely, hence \(P_\theta \)-almost surely, since \(P_{\theta ,t}\ll Q_t\) through the distributions of \((\bar X_t,Y^t)\), and, for \(t{\lt}t_\rho \), its expectation under every null distribution is at most one.

Proposition S5, which Section 4.3 cites.

As in Section S1.7. Every stopping time \(\tau \) of \((\mathcal{F}_t)\) satisfies \(\tau =0\) everywhere or \(\tau \ge 1\) everywhere, since \(\mathcal{F}_0\) is trivial, so that \(E_0\) may be replaced by one. For the lower bound, the paper averages \(R^\mu _t(E)\) over \(\mu \sim \mathrm{N}(0,\rho ^2)\); here, \(R^\mu _t(E)\le r\) for every \(\mu \) implies \(\kappa +R_t(E)\le r\), by averaging the e-powers \(\mathrm{E}_{Q_\mu }(\log E)\ge t\kappa -r\), which is the same computation with a bound in place of the average. For the upper bound, the e-process \((E^{\star }_{t\wedge T}/\eta _T)_{t\ge 0}\) depends on the data only through the \(Y_i\), and Lemma 60 gives its regrets against every \(Q_\mu \); so does the e-process of Proposition 3(ii), which gives part (ii).

Theorem 1 with \(\rho =1\); the minimax regret in the worst case over \(\mu \) lies between zero and \(\kappa +\log \eta _T{\lt}\kappa +\log \log T\) for large \(T\), and \(\log x=o(x^\varepsilon )\). The optimal e-power is Proposition S4(i).

For \(s{\lt}1\), Lemma 87; for \(s\ge 1\), Lemma 71.

For \(Q_\mu \), Proposition S4. For the mixtures, Proposition S5 if \(s{\lt}1\) and \(\rho {\gt}0\), Proposition 4 if \(s\ge 1\), and Proposition S4 if \(\rho =0\), since the mixture with \(\rho =0\) is \(Q_\mu \) with \(\mu =0\). Propositions S4 and 4 are stated for specific numéraires and transfer to any numéraires by uniqueness, since the minimax regret depends on the numéraires only up to \(Q\)-null sets.