$\pi(\theta)$ of $\theta$, the capital process ${{\cal K}}_{n}^\pi$ is written as $${{\cal K}}_{n}^\pi = \int_{{\mathbb R}^d} {{\cal K}}_n^\theta \pi(\theta)d\theta=
\int_{{\mathbb R}^d} \frac{e^{{\theta{'}}\sum_{i=1}^n c_i x_i}}{\prod_{i=1}^n (1+p_i (e^{{\theta{'}}c_i}-1))}
\pi(\theta)d\theta.$$ ${{\cal K}}_{n}^\pi$ is of the form where $$\hat p(x_1,\dots,x_n)= \prod_{i=1}^n p_i^{x_i} (1-p_i)^{1-x_i} \int_{{\mathbb R}^d}
\frac{e^{{\theta{'}}\sum_{i=1}^n c_i x_i}}{\prod_{i=1}^n (1+ p_i (e^{{\theta{'}}c_i}-1))}\pi(\theta)d\theta.$$

In this paper we consider a prior density which is positive in a neighborhood of the origin. We call such $\pi$ “a prior supporting a neighborhood of the origin”.

Properties of logistic betting strategy from the viewpoint of game-theoretic probability {#sec:forcing}
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In this section we prove game-theoretic properties of our Bayesian logistic strategy.

Weak forcing of the usual form of the strong law of large numbers {#subsec:usual-SLLN}
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The first theoretical result on our logistic betting strategy is the following theorem.

\[thm:usual-SLLN\] In BPFSI, by a Bayesian logistic strategy with a prior supporting a neighborhood of the origin, Skeptic can weakly force $$E_1 \ \Rightarrow \ \lim_n {{\cal V}}_n^{-1}{{\cal S}}_n =0,$$ where $E_1$ is given in .

The rest of this subsection is devoted to a proof of this theorem. The basic logic of our proof is the same as in Section 3.2 of [@ShaVov01].

We first consider the logarithm of ${{\cal 