4 The Gaussian location model
These are the results of Section 4.1 of the paper, of Sections S1.1 and S1.4, and of Section S2.5 on the alternatives \(Q_\mu \). In a path \(x\colon \mathbb {N}\to \mathbb {R}\) of the Lean code, the coordinate \(x_i\) is the observation \(X_{i+1}\).
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.
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 \(\ell =\varphi _{s^2}/\varphi _1\), 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 \(L^\theta _0=1\).
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 \(Y_i\sim \mathrm{N}(0,v)\).
The rows of the Helmert matrix form an orthonormal basis \((\omega _1,\ldots ,\omega _t)\) of \(\mathbb {R}^t\), with \(\langle \omega _1,x\rangle =t^{1/2}\bar x_t\) and \(\langle \omega _i,x\rangle =-y_i\) for \(i\ge 2\) (Lancaster, 1965). Parseval’s identity gives (i), with invariance from the orthogonality of \(\omega _i\), \(i\ge 2\), to \((1,\ldots ,1)\). For (iii), \(\mathrm{N}(m,v)^{\otimes \infty }\) is the image of \(\mathrm{N}(0,1)^{\otimes \infty }\) under \(z\mapsto m+v^{1/2}z\), and the standard Gaussian measure is invariant under linear isometries.
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 \(v=1\).
Completing the square, the translation invariance of Lebesgue measure, and the Gaussian integral of Mathlib.
For \(t\ge 1\), \(x\in \mathbb {R}^t\), \(m\in \mathbb {R}\), and \(v{\gt}0\), \(\prod _{i=1}^t\varphi _v(x_i-m)=t^{-1/2}\varphi _{v/t}(\bar x_t-m)\prod _{i=2}^t\varphi _v(y_i)\).
Lemma S2(i).
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).
The formula for \(\log \ell \), and \(\int \varphi _{s^2}=1\).
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).
\(\mathrm{N}(\theta ,s^2)\) has the density \(y\mapsto \ell (y-\theta )\) with respect to \(\mathrm{N}(\theta ,1)\), and the factors of \(L^\theta _t\) are independent with expectation one under \(P_\theta \).
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).
The joint distribution of \((X_1,Y_2)\) is \(\mathrm{N}(\theta ,1)\) followed by the kernel \(a\mapsto \mathrm{N}\{ (\theta -a)/2^{1/2},1/2\} \), which characterizes the conditional distribution.
For \(t\ge 1\), \(\mathcal{F}_t=\mathcal{G}_t\vee \sigma (X_1)\) (Section S2.4).
Each \(Y_i\), \(i\le t\), is \(\mathcal{F}_t\)-measurable, and, by Lemma S2(ii) and induction, \(\bar X_j\) is a function of \((X_1,Y_2,\ldots ,Y_j)\), so that \(X_k=k\bar X_k-(k-1)\bar X_{k-1}\) is one too.
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 \((\mathcal{G}_t)\).
As in Section S1.4, with the density \(\mathrm{d}Q_t/\mathrm{d}P^{\star }_t\) obtained from the distributions of \((\bar X_t,Y^t)\) instead of integrating (S2): under the two mixtures they are \(\mathrm{N}(0,\rho ^2+s^2/t)\otimes \mathrm{N}(0,s^2)^{\otimes (t-1)}\) and \(\mathrm{N}(0,\rho ^2+s^2/t)\otimes \mathrm{N}(0,1)^{\otimes (t-1)}\), by Gaussian convolution, and the density is pulled back through the injective linear map \(x^t\mapsto (\bar x_t,y_2,\ldots ,y_t)\). That \(E^{\star }_t\in \mathcal{E}_t\) follows from \(\mathrm{E}_\theta (E^{\star }_t)=1\), and Lemma 27 gives the numéraire. Under \(P_\theta \), the \(Y_i\) are independent \(\mathrm{N}(0,1)\) variables by Lemma S2(iii), so that the products form a test martingale; under \(Q\), they are independent \(\mathrm{N}(0,s^2)\) variables, which gives the e-power, and \(\kappa \) is computed from \(\log \ell \).
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).
Lemma S2(iii) and \(\int \mathrm{N}(\theta ,b)\mathrm{N}(m,a)(\mathrm{d}\theta )=\mathrm{N}(m,a+b)\).
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).
The paper states the maximal invariance without proof: the difference \(x'^t-x^t\) is orthogonal to \(\omega _2,\ldots ,\omega _t\), so that its expansion in the Helmert basis is a constant vector. The distributions of \(Y^t\) are \(\mathrm{N}(0,s^2)^{\otimes (t-1)}\) under \(Q\) and \(\mathrm{N}(0,1)^{\otimes (t-1)}\) under \(P_\theta \).
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 proof of Proposition S5 bounds \(\sup _{E\in \mathcal{E}_t}\mathrm{E}_Q(\log E)\) by \(\mathrm{KL}(Q_t\| P_{\theta ,t})\) through (S3). Here, the Gibbs inequality is applied to each \(Q_\mu \) against \(P_\mu \), which gives \(\mathrm{E}_{Q_\mu }(\log E)\le t\kappa \), and the bound is averaged over \(\mu \), separately for the positive and negative parts of \(\log E\).
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.
(i) Under \(P_\theta \), \(\mathrm{E}_\theta \{ \ell (X_{t+1}-\mu )\mid \mathcal{F}_t\} =\exp \{ -(1-s^2)(\theta -\mu )^2/2\} \le 1\) by (S1), computed through the conditional distribution given \(\mathcal{F}_t\). The numéraire property follows from Lemma 27 with \(P^{\star }_t=P_{\mu ,t}\), and \(\Gamma _T=0\) from Proposition 1(iii) with \(\eta _T\le 1\), by optional stopping, instead of Remark 1. (ii) The numéraire is proved for every \(s\ge 1\), including \(t=0\), as in Proposition 2 with the mixing distribution \(\mathrm{N}\{ \mu ,(s^2-1)/t\} \), and Proposition 4 bounds \(\log \eta _T\).
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).
Proposition S4(i) and Lemma 50; the other claims are elementary.
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).
An integral of a function is at most its supremum; no integrability is needed.
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 variables \(Y^t\) have the same distribution \(\mathrm{N}(0,s^2)^{\otimes (t-1)}\) under every \(Q_\mu \) and under \(Q\), so that \(\mathrm{E}_{Q_\mu }(\log E)=\mathrm{E}_Q(\log E)\), and \(\mathrm{E}_{Q_\mu }(\log E^{\star }_t)=t\kappa =\kappa +(t-1)\kappa \). \(R_t(E)\) is computed directly, without the numéraire property for \(Q\), so that no condition on \(\rho \) is needed.
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).
The upper bound is Lemma 60 with \(E=E^{\star }_t\). For the lower bound, the paper averages the regrets over \(\mu \sim \mathrm{N}(0,\rho ^2)\); here, if \(\sup _\mu R^\mu _t(E)=r{\lt}\kappa \), then \(\mathrm{E}_{Q_\mu }(\log E)\ge t\kappa -r\) for every \(\mu \), and averaging under \(\mathrm{N}(0,1)\) gives \(\mathrm{E}_Q(\log E){\gt}(t-1)\kappa =\mathrm{E}_Q(\log E^{\star }_t)\), which contradicts log-optimality for \(\rho =1\).
As \(s\to 1\) with \(s\ne 1\), \(\kappa /\{ (1-s^2)^2/4\} \to 1\) (Section 5).
With \(u=s^2-1\), \(\kappa =\{ u-\log (1+u)\} /2\), and the Taylor bound \(|u-\log (1+u)-u^2/2|\le |u|^3/(1-|u|)\) of Mathlib gives \(2\{ u-\log (1+u)\} /u^2\to 1\).