Dependencies

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.

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.

The filtration \((\mathcal{F}_t)_{t\ge 0}\) is generated by \(X\): the \(\sigma \)-algebra \(\mathcal{F}_t\) is generated by \(X^t=(X_1,\ldots ,X_t)\), and \(\mathcal{F}_0\) is trivial. For a distribution \(P\) of \(X\), \(P_t\) is the distribution of \(X^t\).

Let \(\mathcal{X}=\mathbb {R}\), \(P_\theta =\mathrm{N}(\theta ,1)^{\otimes \infty }\), and \(\mathcal{P}=\{ P_\theta :\theta \in \mathbb {R}\} \). For \(s{\gt}0\) and \(\mu \in \mathbb {R}\), let \(Q_\mu =\mathrm{N}(\mu ,s^2)^{\otimes \infty }\), and, for \(\rho \ge 0\), let \(Q=\int Q_\mu \mathrm{N}(0,\rho ^2)(\mathrm{d}\mu )\), where \(\mathrm{N}(0,0)\) is the unit mass at zero.

For a probability measure \(\mu \) and a measurable \(f\) with values in \([0,\infty ]\), \(\mathrm{E}_\mu (\log f)\le \log \mathrm{E}_\mu (f)\).

Let \(s\in (0,1)\) and \(\rho {\gt}0\), fix integers \(r\ge 4\) and \(K\ge 1\), a horizon \(T\ge r^K\), and a constant \(c{\gt}0\). Let \(t_k=r^k\) for \(0\le k\le K\), let \(\xi \in \Delta _T\) be uniform on \(\{ t_0,\ldots ,t_K\} \), and let \(\gamma ^2=\rho ^2+s^2\). Let \(\theta \sim \mathrm{N}(0,\gamma ^2)\), let \(X\sim P_\theta \) given \(\theta \), and let \(Z_t=t^{1/2}(\bar X_t-\theta )\). Let \(\tau '\) be the first \(t_k\) with \(c\le |Z_{t_k}|\le c+1\), or \(\infty \) if there is none. With probability \(1/2\), let \(\tau =\tau '\), and otherwise let \(\tau \sim \xi \) be independent of \(X\); let \(\bar P\) be the distribution of \((\tau ,X^\tau )\). For \(j{\lt}k\) and \(t=t_k\), let \(W_{jk}(y)=t_j^{1/2}\sum _{i=t_j+1}^ty_i/\{ i(i-1)\} ^{1/2}\), let \(\mathcal{Z}_k(y)=\{ z:c\le |z|\le c+1\} \smallsetminus \bigcup _{j{\lt}k}\{ z:c\le |(t_j/t)^{1/2}z-W_{jk}(y)|\le c+1\} \), and let \(p_k(y)=\int _{\mathcal{Z}_k(y)}\varphi _1(z)\mathrm{d}z\). Let \(c'=c-(c+1)r^{-1/2}\), \(A_k=\{ y:\max _{j{\lt}k}|W_{jk}(y)|{\lt}c'\} \), \(\beta '=\frac12\log \{ \rho ^2/(\gamma ^2+1)\} -\log 2\), and \(\beta =\log (K+1)+\log \varphi _1(c+1)+\log (\rho /\gamma )\).

Let \(\ell =\varphi _{s^2}/\varphi _1\), where \(\varphi _v\) is the density of \(\mathrm{N}(0,v)\), let \(\kappa =(s^2-1-\log s^2)/2\), let \(E^{\star }_t=\prod _{i=2}^t\ell (Y_i)\), which is one for \(t\le 1\), and let \(L^\theta _t=\prod _{i=1}^t\ell (X_i-\theta )\), with \(L^\theta _0=1\).

Under the standing assumption, \(\Gamma _T=\max _{\xi \in \Delta _T}\inf _{(E_t)}\sum _{t=0}^T\xi _tR_t(E_t)\), for every \(\mathcal{X}\).

For \(j\ge 1\), let \(T_j=\lceil \exp (e^j)\rceil \), \(q_j=1+s^2/\{ (1-s^2)(j+1)\} \), and \(u_j=\{ (j+1)/(s^2j)\} (2+2\log T_j/\log q_j)^{1-s^2j/(j+1)}\). Let \(E_t=\sum _{j=1}^\infty E^{\star }_{t\wedge T_j}/\{ j(j+1)u_j\} \).

Let \(s\in (0,1)\) and \(\rho ^2\ge 1-s^2\). Then \(\eta _{T_j}\le u_j\), the process \((E_t)\) is an e-process, adapted to \((\mathcal{G}_t)\), and \(\limsup _{t\to \infty }R_t(E_t)/\log \log t\le 1-s^2\). If, moreover, \(\Gamma _T/\log \log T\to 1-s^2\), then \(\limsup _{t\to \infty }R_t(E_t)/\log \log t\ge 1-s^2\).

For \(\xi \in \Delta _T\), \(Q^\xi \) is the distribution of \((\tau ,X^\tau )\) for \(X\sim Q\) and an independent time \(\tau \) with distribution \(\xi \). The set \(\mathcal{B}\) contains the mixtures \(\bar P=\int P_\lambda ^{\tau _\lambda }\pi (\mathrm{d}\lambda )\), where \(\pi \) is a probability measure on a measurable space, \(\lambda \mapsto P_\lambda \in \mathcal{P}\) is a probability kernel, and \((\lambda ,x,u)\mapsto \tau _\lambda (x,u)\) is measurable, with \(\tau _\lambda \) a randomized stopping time for every \(\lambda \).

The set \(\Delta _T\) contains the probability vectors \((\xi _0,\ldots ,\xi _T)\).

Let \(s\in (0,1)\) and \(\rho {\gt}0\), and let \(\Gamma _T\) satisfy (S9) for every admissible \(r\), \(K\), \(T\), and \(c\). Then, for every \(a{\lt}1-s^2\), \(a\log \log T{\lt}\Gamma _T\) for all large \(T\).

Let \(s\in (0,1)\). For every \(b{\gt}1-s^2\), \(\log \eta _T{\lt}b\log \log T\) for all large \(T\).

Let \(s\le 1\) and \(t\ge 1\), and let \(E\in \mathcal{E}_t\) be measurable with respect to \(\mathcal{G}_t\). Then, for every \(\mu \) and every \(\rho \), \(R^\mu _t(E)=\kappa +R_t(E)\), with any numéraires of \(Q_\mu \), so that \(\sup _\mu R^\mu _t(E)=\kappa +R_t(E)\).

The null distributions and \(Q\) are probability measures, \(E^{\star }_t\) is a numéraire at every horizon \(t\), in versions fixed once and for all, and \(\sup _{E\in \mathcal{E}_t}\mathrm{E}_Q(\log E){\lt}\infty \) for every \(t\ge 0\).

For every \(s{\gt}0\) and every \(\rho \), \(\sup _{E\in \mathcal{E}_t}\mathrm{E}_Q(\log E)\le t\kappa \), so that the standing assumption holds for \(Q\) with any numéraires. For \(\rho ^2\ge 1-s^2\), it holds with the numéraires \(E^{\star }_t=\prod _{i=2}^t\ell (Y_i)\), including \(t=0\).

The pairs \((t,x^t)\) are pairs of a time \(t\) in \(\{ 0,1,\ldots \} \cup \{ \infty \} \) and a path \(x\), under the \(\sigma \)-algebra that observes only \(x^t\). A randomized stopping time is a measurable map \((x,u)\mapsto \tau (x,u)\), with \(u\in [0,1]\), such that \(\tau (\cdot ,u)\) is a stopping time of \((\mathcal{F}_t)\) for every \(u\). For a distribution \(P\) of \(X\), \(P^\tau \) is the distribution of \((\tau ,X^\tau )\), where \(\tau =\tau (X,U)\) with \(X\sim P\) and \(U\) uniform on \([0,1]\) and independent of \(X\).

A test supermartingale, or test martingale, with respect to a filtration is a nonnegative process adapted to this filtration, with initial value one, that is a supermartingale, or a martingale, with respect to this filtration under every null distribution.

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 \(P\).

The random walk \((S_t)_{t\ge 0}\) has \(S_0=0\) and independent \(\mathrm{N}(0,1)\) increments; it is realized on the space of its increments, with distribution \(\mathrm{N}(0,1)^{\otimes \infty }\). The reward is \(h_0=1\) and \(h_t=s\exp \{ (1-s^2)S_t^2/(2t)\} \) for \(t\ge 1\).

For \(\mu \in \mathbb {R}\), \(R^\mu _t\) is the regret defined by (2) and (3) with \(Q_\mu \) in place of \(Q\), with any numéraires of \(Q_\mu \), and the minimax regret in the worst case over \(\mu \) is \(\inf _{(E_t)}\max _{0\le t\le T}\sup _{\mu \in \mathbb {R}}R^\mu _t(E_t)\).

For \(t\ge 1\), \(\mathcal{F}_t=\mathcal{G}_t\vee \sigma (X_1)\) (Section S2.4).

An e-variable at horizon \(t\) is an \(\mathcal{F}_t\)-measurable random variable \(E\) with values in \([0,\infty ]\) such that \(\sup _{P\in \mathcal{P}}\mathrm{E}_P(E)\le 1\), and \(\mathcal{E}_t\) is the set of these e-variables. An e-process is a process \((E_t)_{t\ge 0}\) with values in \([0,\infty ]\), adapted to \((\mathcal{F}_t)\), such that \(\sup _{P\in \mathcal{P}}\mathrm{E}_P(E_\tau )\le 1\) for every stopping time \(\tau \) of \((\mathcal{F}_t)\) with values in \(\{ 0,1,\ldots \} \cup \{ \infty \} \), where \(E_\infty =\limsup _{t\to \infty }E_t\).

A numéraire at horizon \(t\) is an e-variable \(E^{\star }_t\in \mathcal{E}_t\) such that \(\mathrm{E}_Q(E/E^{\star }_t)\le 1\) for every \(E\in \mathcal{E}_t\) (Larsson, Ramdas and Ruf, 2025, Definition 2.1). The e-power of \(E\) is \(\mathrm{E}_Q(\log E)\). The numéraire is log-optimal if \(\mathrm{E}_Q(\log E^{\star }_t)=\sup _{E\in \mathcal{E}_t}\mathrm{E}_Q(\log E)\), and \(E^{\star }_t=\mathrm{d}Q_t/\mathrm{d}P^{\star }_t\) if \(Q_t\) has the density \(e\) with respect to a sub-probability measure \(P^{\star }_t\) on \(\mathcal{X}^t\), with \(E^{\star }_t=e(X^t)\) \(Q\)-almost surely.

The expectation of a random variable with values in \([-\infty ,\infty ]\) is the integral of its positive part minus that of its negative part. The regret of \(E\in \mathcal{E}_t\) is \(R_t(E)=\mathrm{E}_Q\{ \log (E^{\star }_t/E)\} \), the regret of an e-process \((E_t)\) at \(t\) is \(R_t(E_t)\), and the minimax regret is \(\Gamma _T=\inf _{(E_t)}\max _{0\le t\le T}R_t(E_t)\), where \((E_t)\) ranges over the e-processes.

Let \(q{\gt}1\), and let \(B_k=\{ t\le T:q^k\le t{\lt}q^{k+1}\} \). For \(z\ge 0\), \(\mathrm{pr}(\max _{t\in B_k}S_t^2/t\ge z^2)\le 2\exp \{ -z^2/(2q)\} \), and the blocks \(B_k\) with \(k\ge N=1+\lfloor \log T/\log q\rfloor \) are empty.

For an integer \(u\ge 1\) and \(x\ge 0\), \(\mathrm{pr}(\max _{t\le u}S_t\ge x)\le \exp \{ -x^2/(2u)\} \), and \(\mathrm{pr}(\max _{t\le u}|S_t|\ge x)\le 2\exp \{ -x^2/(2u)\} \).

For every \(\xi \in \Delta _T\), \(\sup _{(E_t)}\sum _{t=0}^T\xi _t\mathrm{E}_Q(\log E_t)=\sum _{t=0}^T\xi _t\mathrm{E}_Q(\log E^{\star }_t)-\inf _{(E_t)}\sum _{t=0}^T\xi _tR_t(E_t)\).

For \(t\ge 1\), \(x\in \mathbb {R}^t\), \(m\in \mathbb {R}\), and \(v{\gt}0\), \(\prod _{i=1}^t\varphi _v(x_i-m)=t^{-1/2}\varphi _{v/t}(\bar x_t-m)\prod _{i=2}^t\varphi _v(y_i)\).

For an e-process \((E_t)\) and \(\bar P\in \mathcal{B}\), the function \(H(t,x^t)=E_t(x^t)\) on the pairs satisfies \(\int H\mathrm{d}\bar P=\int \mathrm{E}_{P_\lambda }\{ E_{\tau _\lambda (X,U)}\} \pi (\mathrm{d}\lambda )\le 1\).

With \(b=s^{-2}-1\), \(\log \ell (y)=-\log s-by^2/2\). For \(m\in \mathbb {R}\), \(v\ge 0\) with \(1+bv{\gt}0\), and \(Z\sim \mathrm{N}(0,1)\),

which is eq. (S1), and the right-hand side equals \(\exp \{ -(1-s^2)m^2/2\} \) for \(v=1\).

For probability measures \(\mu \) and \(\nu \) and a measurable \(H\) with values in \([0,\infty ]\) and \(\int H\mathrm{d}\nu \le 1\), \(\mathrm{E}_\mu (\log H)\le \mathrm{KL}(\mu \| \nu )\).

For \(0\le k\le K\) and \(t=t_k\),

Let \(s\ge 1\) and \(\varepsilon {\gt}0\), and let \(\tau _\varepsilon \) be the first \(t\ge 1\) with \(S_t^2/t\le \varepsilon \). Then \(\mathrm{E}(h_{\tau _\varepsilon \wedge T})\ge s\exp \{ -(s^2-1)\varepsilon /2\} \mathrm{pr}(\tau _\varepsilon \le T)\).

Let

where \(\tau \) ranges over the stopping times of \((\mathcal{F}_t)\). For \(P\in \mathcal{P}\), the Snell envelope of \((E^{\star }_{t\wedge T})_{t\ge 0}\) under \(P\) is defined by \(\Lambda ^P_T=E^{\star }_T\) and

Let \(\bar X_t=t^{-1}\sum _{i=1}^tX_i\) and \(Y_i=\{ (i-1)/i\} ^{1/2}(X_i-\bar X_{i-1})\) for \(i\ge 2\), let \(Y^t=(Y_2,\ldots ,Y_t)\), and let \((\mathcal{G}_t)\) be the filtration generated by \((Y_t)_{t\ge 2}\), so that \(\mathcal{G}_0\) and \(\mathcal{G}_1\) are trivial.

Let \(T\ge 0\). For every e-process \((E_t)\), every \(\xi \in \Delta _T\), and every \(\bar P\in \mathcal{B}\),

which is eq. (5), with the convention \(0\times \infty =0\) in the sums. Consequently, for every \(\xi \in \Delta _T\), \(\Gamma _T\ge \inf _{(E_t)}\sum _t\xi _tR_t(E_t)\ge \sum _t\xi _t\mathrm{E}_Q(\log E^{\star }_t)-\inf _{\bar P\in \mathcal{B}}\mathrm{KL}(Q^\xi \| \bar P)\), which is eq. (6).

For every \(\rho {\gt}0\) and any numéraires of \(Q\), if \(c'{\gt}0\) and \(\beta \ge \beta '\), then

which is eq. (S9).

Let \(s{\gt}0\). (i) For every \(\theta \in \mathbb {R}\) and every \(t\ge 1\), \(E^{\star }_t=sL^\theta _t\exp \{ (1-s^2)t(\bar X_t-\theta )^2/(2s^2)\} \). (ii) For every \(T\ge 0\), \(\eta _T=\sup _\tau \mathrm{E}(h_{\tau \wedge T})\), where \(\tau \) ranges over the stopping times of the filtration generated by \((S_t)\).

Let \(\rho {\gt}0\), \(0\le k\le K\), and \(t=t_k\). (i) For every \(x\in \mathbb {R}^t\), \(J_k(x){\gt}0\) and \(\log \{ (K+1)J_k(x)\} \ge \beta '\). (ii) For every \(x\in \mathbb {R}^t\), \(J_k(x)\ge \{ \rho /(2\gamma )\} \exp [-\{ 2(c+1)t^{-1/2}|\bar x_t|+(c+1)^2t^{-1}\} /(2\gamma ^2)]p_k(y)\). (iii) If \(c'{\gt}0\), then \(p_k(y)\ge 2\varphi _1(c+1)\) for every \(y\in A_k\).

For every \(\rho {\gt}0\) and any numéraires of \(Q\), the distribution \(\bar P\) belongs to \(\mathcal{B}\), and

which is eq. (S8).

Let \(t\ge 2\) and \(x\in \mathbb {R}^t\). (i) The map \(x\mapsto (t^{1/2}\bar x_t,y_2,\ldots ,y_t)\) is linear and orthogonal, the values \(y_2,\ldots ,y_t\) do not change when the same constant is added to every \(x_i\), and \(\sum _{i=1}^t(x_i-m)^2=t(\bar x_t-m)^2+\sum _{i=2}^ty_i^2\) for every \(m\in \mathbb {R}\). (ii) \(\bar x_t=\bar x_{t-1}+y_t/\{ t(t-1)\} ^{1/2}\). (iii) If \(X^t\sim \mathrm{N}(m,v)^{\otimes t}\), then \(\bar X_t,Y_2,\ldots ,Y_t\) are independent, with \(\bar X_t\sim \mathrm{N}(m,v/t)\) and \(Y_i\sim \mathrm{N}(0,v)\).

For every \(t\ge 0\), \(E^{\star }_t\) is positive and finite \(Q\)-almost surely, \(\log E^{\star }_t\) is \(Q\)-integrable, and \(E^{\star }_t\) is log-optimal. For every \(E\in \mathcal{E}_t\), the negative part of \(\log (E^{\star }_t/E)\) is \(Q\)-integrable, so that \(R_t(E)\) is well defined, and \(R_t(E)=\mathrm{E}_Q(\log E^{\star }_t)-\mathrm{E}_Q(\log E)\in [0,\infty ]\).

Let \(s\in (0,1)\), \(a=1-s^2\), \(T\ge 1\), and \(q\in (1,1/a)\). Then \(\log \eta _T\le aq\log (2+2\log T/\log q)+\log \{ 1/(1-aq)\} \).

Let \(A(e)=(e-1-\log e)/(\log e)^2\) for \(e{\gt}1\) and \(A(1)=1/2\), the adjuster of their eq. (12). If \((M_t)\) is a test supermartingale with respect to a filtration coarser than \((\mathcal{F}_t)\), then the process with terms \(A(\max _{i\le t}M_i)\) is an e-process with respect to \((\mathcal{F}_t)\).

Let \(\mathcal{X}\) be finite, let \(T\ge 0\), and let \(\xi \in \Delta _T\). Then \(\inf _{\bar P\in \mathcal{B}}\mathrm{KL}(Q^\xi \| \bar P)\le \sup _{(E_t)}\sum _{t=0}^T\xi _t\mathrm{E}_Q(\log E_t)\), where \((E_t)\) ranges over the e-processes. Their primal-dual equivalence, applied to a data-independent stopping time with distribution \(\xi \), gives the equality over their Bayesian mixtures of stopped null distributions, which belong to \(\mathcal{B}\).

For every nonempty null of probability measures and every alternative probability measure, a numéraire exists at every horizon.

For the random walk \((S_t)_{t\ge 0}\) with \(S_0=0\) and independent \(\mathrm{N}(0,1)\) increments, almost surely, \(\limsup _{t\to \infty }S_t/(2t\log \log t)^{1/2}=1\) and \(\liminf _{t\to \infty }S_t/(2t\log \log t)^{1/2}=-1\).

Let \((G_n)_{0\le n\le N}\) be a nonnegative process adapted to a filtration, with \(\mathrm{E}_P(G_n){\lt}\infty \) for every \(n\). Its Snell envelope \(S\), defined by backward induction, dominates \(G\), is a supermartingale under \(P\), and is dominated by every supermartingale that dominates \(G\). Moreover, \(\mathrm{E}_P(S_0)=\sup _\tau \mathrm{E}_P(G_{\tau \wedge N})\), where \(\tau \) ranges over the stopping times of the 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 set \(\mathcal{B}\) contains the Bayesian mixtures of stopped elements of the null of Koolen and Grünwald (2022, Section 3), the mixtures over a probability measure on pairs of a null distribution and a stopping time of \((\mathcal{F}_t)\) (Section 3).

Under \(P_\theta \) and given \(X_1\), the variable \(Y_2=(X_2-X_1)/2^{1/2}\) is normal with mean \((\theta -X_1)/2^{1/2}\) and variance \(1/2\) (Section 4.1).

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 every \(\mathcal{X}\), every \(\xi \in \Delta _T\), every e-process \((E_t)\), and every \(\bar P\in \mathcal{B}\), \(\sum _{t=0}^T\xi _t\mathrm{E}_Q(\log E_t)\le \mathrm{KL}(Q^\xi \| \bar P)\) (Section 3).

For every null and every choice of numéraires, \(\eta _T\) is nondecreasing in \(T\) (Example S1).

Let \(s\in (0,1)\) and \(\rho {\gt}0\). For any numéraires of \(Q\), \(\Gamma _T/\log \log T\to 1-s^2\) and \(\log \eta _T/\log \log T\to 1-s^2\) as \(T\to \infty \) (Section 4.3).

For a finite \(\mathcal{X}\), \(\Gamma _T\) is the infimum of \(\max _{t\le T}R_t(E_t)\) over the bounded e-processes \((E_t)\) stopped at \(T\) (Section S2.1).

The alternatives \(Q^\xi \) with \(\xi \) a unit mass have disjoint supports: for \(t\ne t'\), the unit masses at \(t\) and \(t'\) give mutually singular distributions on the pairs (Section S2.1).

For every \(s{\gt}0\), the minimax regret in the worst case over \(\mu \) is bounded uniformly in \(T\) if and only if \(s\ge 1\), that is, if and only if the alternatives have at least the variance of the null (Section 1).

For every \(s{\gt}0\) and \(\mu \), \(\Gamma _T\) does not tend to infinity for the alternative \(Q_\mu \). For the mixtures \(Q\) with \(\rho \ge 0\), \(\Gamma _T\to \infty \) exactly when \(s{\lt}1\) and \(\rho {\gt}0\), however small \(\rho \) is (Section S2.5).

For \(s\in (0,1)\), the minimax regret in the worst case over \(\mu \) tends to infinity, and, for every \(\varepsilon {\gt}0\), its ratio to \((\log T)^\varepsilon \) tends to zero, while the optimal e-power \(T\kappa \) of each \(Q_\mu \) grows linearly (Section 4.3).

Two e-processes with finite regrets differing by \(o(t)\) have e-powers differing by \(o(t)\), so that \(t^{-1}\mathrm{E}_Q(\log E_t)\) and \(t^{-1}\mathrm{E}_Q(\log E'_t)\) have the same limit when one of them has a limit (Section 1).

As \(s\to 1\) with \(s\ne 1\), \(\kappa /\{ (1-s^2)^2/4\} \to 1\) (Section 5).

For every \(\theta \) and \(0\le k\le K\), with \(t=t_k\), \(\tau '=t\) exactly when \(Z_t\in \mathcal{Z}_k(Y^t)\). Since \(Z_t\sim \mathrm{N}(0,1)\) is independent of \(Y^t\) under \(P_\theta \), \(p_k(Y^t)\) is the conditional probability that \(\tau '=t\), given \(Y^t\). If \(c'{\gt}0\), then, on the event \(\{ Y^t\in A_k\} \), the rule \(\tau '\) stops at \(t\) whenever \(c\le |Z_t|\le c+1\) (Section S1.6, after Lemma S5).

For \(s{\lt}1\), \(\ell (y)\) decreases in \(|y|\), and, for every \(s{\gt}0\), its expectation under \(\mathrm{N}(0,1)\) is one (Section 4.1).

For \(\theta \in \mathbb {R}\), \(L^\theta _t\) is the density of \(Q_{\theta ,t}\) with respect to \(P_{\theta ,t}\), and, under \(P_\theta \), \((L^\theta _t)_{t\ge 0}\) is a nonnegative martingale with respect to \((\mathcal{F}_t)\) with initial value one (Section 4.2).

Assume the law of the iterated logarithm. If \(s{\lt}1\), then \(\limsup _{t\to \infty }\log h_t/\log \log t=1-s^2\) almost surely (Section 4.2).

For an e-process \((E_t)\) and \(\alpha {\gt}0\), \(P(E_t\ge 1/\alpha \text{ for some }t)\le \alpha \) for every \(P\in \mathcal{P}\), so that the test that rejects the null as soon as the e-process reaches \(1/\alpha \) has level \(\alpha \) (Section 1).

The vector \(Y^t\) takes the same value at \(x\) and \(x'\) exactly when \(x'^t\) is a translation of \(x^t\), so that \(Y^t\) is a maximal invariant under the translations. For \(\rho ^2\ge 1-s^2\), every \(\theta \), and \(t\ge 1\), \(E^{\star }_t\) is the density of the distribution of \(Y^t\) under \(Q\) with respect to its distribution under \(P_\theta \) (Section 4.1).

Let \(P^{\star }_t=\int P_{\lambda ,t}\pi (\mathrm{d}\lambda )\) be a mixture of null distributions of \(X^t\), and let \(E\in \mathcal{E}_t\) satisfy \(E=\mathrm{d}Q_t/\mathrm{d}P^{\star }_t\). Then \(E\) is a numéraire at \(t\), and \(P^{\star }_t\) is the measure in (2) (Section S4, for the Bernoulli example, and the proof of Proposition 2).

Let \((E_t)\) be a process with finite values, adapted to \((\mathcal{F}_t)\) and stopped at \(T\). Then \((E_t)\) is an e-process if and only if, for each \(P\in \mathcal{P}\), some supermartingale \((\Psi _t)_{0\le t\le T}\) under \(P\) with \(\Psi _t\ge E_t\) has initial value \(\Psi _0\le 1\) (Section S4, which states this for the Bernoulli tree).

For \(\rho ^2\ge 1-s^2\) and \(t\ge 1\), the measure \(P^{\star }_t\) of (2) is a mixture of null distributions under which \(\bar X_t\) has the same distribution as under \(Q_t\), namely \(\mathrm{N}(0,\rho ^2+s^2/t)\) (Section 4.1).

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).

The numéraire \(E^{\star }_t\) is positive \(Q\)-almost surely and unique up to \(Q\)-null sets, and \(E^{\star }_0=1\) since \(\mathcal{F}_0\) is trivial (Section 2).

For every e-process \((E_t)\), every \(P\in \mathcal{P}\), and every randomized stopping time \(\tau \), \(\mathrm{E}_P\{ E_{\tau (X,U)}\} \le 1\) (Section 3).

For \(s\ge 1\) and every \(T\), the e-process \((E^{\star }_{t\wedge T}/\eta _T)_{t\ge 0}\) of Proposition 1(iii) has regret at most \(\log s\) against every \(Q_\mu \) at every \(t\le T\), so that the minimax regret in the worst case over \(\mu \) is at most \(\log s\) (Section 4.3).

For \(s\le 1\), the numéraire of \(Q_\mu \) at \(t\) is \(L^\mu _t\), and the measure \(P^{\star }_t\) of (2) is the single null distribution \(P_{\mu ,t}\). The mixing variance \(\rho ^2-(1-s^2)/t\) is nonnegative at every horizon \(t\ge 1\) if and only if \(\rho ^2\ge 1-s^2\). With \(\rho =0\), the alternative of Section 4 is \(Q_\mu \) with \(\mu =0\), and the condition reduces to \(s\ge 1\) (Section S2.5).

Let \(\mathcal{P}=\{ P\} \) satisfy the standing assumption. Then \(Q_t\ll P_t\), the numéraires \(E^{\star }_t=\mathrm{d}Q_t/\mathrm{d}P_t\) form a test martingale with respect to \((\mathcal{F}_t)\), and \(\Gamma _T=0\) for every \(T\) and every choice of numéraires (Section 1).

For \(s{\lt}1\) and a single horizon \(t\ge 1\), \(\inf _{E\in \mathcal{E}_t}\sup _\mu R^\mu _t(E)=\kappa \), attained by the numéraire \(E^{\star }_t=\prod _{i=2}^t\ell (Y_i)\) (Section 4.3 and the proof of Theorem 1).

For \(P\in \mathcal{P}\), \((\Lambda ^P_t)_{0\le t\le T}\) is the smallest supermartingale under \(P\) with respect to \((\mathcal{F}_t)\) that dominates \((E^{\star }_{t\wedge T})_{0\le t\le T}\), \(\Lambda ^P_0=\sup _\tau \mathrm{E}_P(E^{\star }_{\tau \wedge T})\), and \(\eta _T=\sup _{P\in \mathcal{P}}\Lambda ^P_0\) (Section 2).

For \(0\le t\le T\) and \(B\in \mathcal{F}_t\), \(Q^\xi [\{ t\} \times B]=\xi _tQ_t(B)\), and \(Q^\xi \) is the image of \((\sum _t\xi _t\delta _t)\otimes Q\) under the map that stops the path at the time (Section 3).

Test supermartingales with respect to \((\mathcal{F}_t)\) are e-processes (Section 1).

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.

For every finite \(b\ge \eta _T\), the process \((E^{\star }_{t\wedge T}/b)_{t\ge 0}\) is an e-process with regret \(\log b\) at every \(t\le T\) (Section 5).

For every \(E\), \(\sup _\mu R^\mu _t(E)\) is at least the average of \(R^\mu _t(E)\) under \(\mu \sim \mathrm{N}(0,\rho ^2)\), in \([-\infty ,\infty ]\) (Section 4.1).

Let \(T\ge 0\). Then \(\Gamma _T\le \Gamma _{T+1}\). (i) For every \(t\ge 0\), the process with terms \(E_i=E^{\star }_t\mathbf{1}(i\ge t)\) is an e-process with \(R_t(E_t)=0\). (ii) For every \(\xi \in \Delta _T\), the process with terms \(\sum _{i=0}^{t\wedge T}\xi _iE^{\star }_i\) is an e-process with regret at most \(\log (1/\xi _t)\) at every \(t\le T\), so that \(\Gamma _T\le \log (T+1)\). (iii) If \(\eta _T{\lt}\infty \), then \((E^{\star }_{t\wedge T}/\eta _T)_{t\ge 0}\) is an e-process with regret \(\log \eta _T\) at every \(t\le T\), so that \(\Gamma _T\le \log \eta _T\).

(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 \(\theta \).

Let \(T\ge 0\). (i) Suppose that \(\Gamma _T=0\), and that \(\mathcal{X}\) is finite or \(P_T\ll Q_T\) for every \(P\in \mathcal{P}\). Then \(E^{\star }_0,\ldots ,E^{\star }_T\) have versions such that \((E^{\star }_{t\wedge T})_{t\ge 0}\) is an e-process. (ii) On every sample space \(\mathcal{X}\), \(\Gamma _T=\max _{\xi \in \Delta _T}\inf _{(E_t)}\sum _{t=0}^T\xi _tR_t(E_t)\). (iii) If \(\mathcal{X}\) is finite, then, for every \(\xi \in \Delta _T\),

which is eq. (S11).

Let \(s\in (0,1)\) and \(\rho ^2\ge 1-s^2\). (i) As \(T\to \infty \), \(\Gamma _T/\log \log T\to 1-s^2\) and \(\log \eta _T/\log \log T\to 1-s^2\). (ii) The process \((E_t)_{t\ge 0}\) of Definition 78 is an e-process with \(\limsup _{t\to \infty }R_t(E_t)/\log \log t=1-s^2\).

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.

Let \(s{\gt}0\) and \(\rho ^2\ge 1-s^2\). For every \(t\ge 1\), \(E^{\star }_t=\prod _{i=2}^t\ell (Y_i)\) is the numéraire of \(Q\) at \(t\), with \(P^{\star }_t=\int P_{\theta ,t}\mathrm{N}\{ 0,\rho ^2-(1-s^2)/t\} (\mathrm{d}\theta )\), and \(\mathrm{E}_Q(\log E^{\star }_t)=(t-1)\kappa \), where \(\kappa =\mathrm{KL}\{ \mathrm{N}(0,s^2)\| \mathrm{N}(0,1)\} =(s^2-1-\log s^2)/2\). Under every \(P_\theta \), \((E^{\star }_t)_{t\ge 0}\) is a test martingale with respect to \((\mathcal{G}_t)\).

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, \(R^\mu _t(\tilde E_t)=\kappa +R_t(\tilde E_t)\).

If \(s\ge 1\), then \(\Gamma _T\le \log \eta _T\le \log s\) for every \(T\ge 0\), and, given the law of the iterated logarithm, \(\log \eta _T\to \log s\) as \(T\to \infty \).

Let \(s{\gt}0\), \(\mu \in \mathbb {R}\), and \(T\ge 0\), and let \(Q_\mu \) replace \(Q\) in (2) and (3). (i) If \(s\le 1\), then \(E^{\star }_t=\prod _{i=1}^t\ell (X_i-\mu )\) and \(\mathrm{E}_{Q_\mu }(\log E^{\star }_t)=t\kappa \) for \(t\ge 0\), the process \((E^{\star }_t)_{t\ge 0}\) is a test supermartingale with respect to \((\mathcal{F}_t)\), and \(\Gamma _T=0\). (ii) If \(s{\gt}1\), then \(E^{\star }_t=\prod _{i=2}^t\ell (Y_i)\) for \(t\ge 1\), as in Proposition 2, and \(\Gamma _T\le \log \eta _T\le \log s\), where \(\eta _T\) is the quantity (4) of Section 4.

Let \(s\in (0,1)\) and \(\rho {\gt}0\), and let \(Q=\int Q_\mu \mathrm{N}(0,\rho ^2)(\mathrm{d}\mu )\), without the condition \(\rho ^2\ge 1-s^2\). Numéraires of \(Q\) exist, by the theorem of Larsson, Ramdas and Ruf, and, for any numéraires of \(Q\), \(\Gamma _T/\log \log T\to 1-s^2\) and \(\log \eta _T/\log \log T\to 1-s^2\) as \(T\to \infty \).

If every \(\tau _\lambda \) is independent of \(X\), then

and the right-hand side of (5) is at most zero.

If \(E^{\star }_0,\ldots ,E^{\star }_T\) have versions such that \((E^{\star }_{t\wedge T})_{t\ge 0}\) is an e-process, then \(\Gamma _T=0\).

Let \(s\in (0,1)\) and \(\rho ^2\ge 1-s^2\). (i) For \(T\ge 1\), \(\kappa +\Gamma _T\le \inf _{(E_t)}\max _{0\le t\le T}\sup _{\mu \in \mathbb {R}}R^\mu _t(E_t)\le \kappa +\log \eta _T\), and the middle term divided by \(\log \log T\) tends to \(1-s^2\) as \(T\to \infty \). (ii) The e-process \((E_t)\) of Proposition 3(ii) satisfies \(\sup _{\mu \in \mathbb {R}}R^\mu _t(E_t)=\kappa +R_t(E_t)\) for \(t\ge 1\), so that \(\limsup _{t\to \infty }\sup _{\mu \in \mathbb {R}}R^\mu _t(E_t)/\log \log t=1-s^2\).