logistic strategy {#sec:diff-hindsight}

In the last subsection we clarified the behavior of the capital process for the hindsight strategy. Now we employ the standard Laplace method to evaluate the difference of the hindsight strategy and the logistic strategy (Section 5 of [@cover-universal-portfolios], Chapter 3.1 of [@jensen]).

\[lem:laplace\] Let $\pi$ be a prior density supporting a neighborhood of the origin and let ${{\cal K}}_n^\pi$ denote its capital process. For $\xi \in E_1$ such that $\lim_n{{\cal V}}_n^{-1}{{\cal S}}_n=0$, $$\label{eq:laplace}
\lim_n \frac{ \log {{\cal K}}_n^{{\hat\theta_n^*}} - \log  {{\cal K}}_n^\pi }{(1/2)\log \det {{\cal V}}_n} = 0.$$

For $\theta$ close to the origin, expanding $\log{{\cal K}}_n^\theta$ around ${\hat\theta_n^*}$ we have $$\log{{\cal K}}_n^\theta = \log {{\cal K}}_n^{{\hat\theta_n^*}} - \frac{1}{2} (\theta - {\hat\theta_n^*}){'}H_\psi(\tilde \theta_n)(\theta - {\hat\theta_n^*}),$$ where $\tilde \theta_n$ is on the line segment joining $\theta$ and ${\hat\theta_n^*}$. Hence $${{\cal K}}_n^\theta = {{\cal K}}_n^{{\hat\theta_n^*}} \times \exp(-\frac{1}{2} (\theta - {\hat\theta_n^*}){'}H_\psi(\tilde \theta_n)(\theta - {\hat\theta_n^*})).$$ Now by the standard Laplace method we obtain .

Finally we give a proof of Theorem \[thm:main\].

By Corollary \[cor:hindsight\] and Lemma \[lem:laplace\] $$\log {{\cal K}}_n^\pi = \frac{1}{2} \log\det {{\cal V}}_n \; 
\big( \frac{ {{\cal S}}_n{'}{{\cal V}}_n^{-1}{{\cal S}}_n} {\log \det {{\cal V}}_n} - 1 + o(1)\big).$$ Hence if $\limsup_n {{\cal S}}_n{'}{{\cal V}}_n^{-1}{{\cal S}}_n/ \log \det 