the maximum and the minimum eigenvalues of ${{\cal V}}_n$. In this paper we consider the following regularity conditions:

-   $\lim_n \lambda_{\min,n}=\infty$.

-   $\limsup_n \lambda_{\max,n}/\lambda_{\min,n} < \infty$.

-   $\{ c_1,c_2,\dots\}$ is a bounded set.

Namely we take $E_1$ as $$\label{eq:our-regularity}
E_1 = \{\xi \mid \lim_n \lambda_{\min,n}=\infty, \limsup_n \lambda_{\max,n}/\lambda_{\min,n} < \infty \ \text{and}\ 
c_1,c_2,\dots \text{are bounded}\} .$$ The condition i) makes the meaning of “${{\cal V}}_n \rightarrow \infty$” precise. The condition ii) means that ${{\cal V}}_n$ stays away from being singular. For $d=1$ ii) is trivial and not needed.

Logistic betting strategy {#sec:logistic-betting}
=========================

In this section we introduce a betting strategy based on logistic modeling of Reality’s moves.

As in the previous section Skeptic models $x_n$ as a Bernoulli variable with the success probability $\hat p_n$. Furthermore we specify that Skeptic uses the following logistic regression model for the logarithm of the odds ratio: $$\label{eq:logistic-simple}
\log \frac{\hat p_n}{1-\hat p_n}= \log\frac{p_n}{1-p_n} + {\theta{'}}c_n,$$ where $\theta \in{\mathbb R}^d$ is a parameter vector.

In previous studies in game-theoretic probability, many strategies of Skeptic depend only on $x_i-p_i$, $i\le n-1$, and do not depend on $p_n$. However obviously it is more reasonable to consider Skeptic’s strategies which depend on $p_n$. Strategies explicitly depending on $p_n$ 