the usual strong law of large numbers in game-theoretic probability, we are interested in the convergence ${{\cal S}}_n/n \rightarrow 0$ as $n\rightarrow\infty$. Generalizing this case, in the presence of side information, we are interested in the convergence ${{\cal V}}_n^{-1} {{\cal S}}_n \rightarrow 0$, although the order of ${{\cal V}}_n$ may be different from $O(n)$. We call this convergence the usual form of the strong law of large numbers in BPFSI. See Theorem \[thm:usual-SLLN\] in Section \[subsec:usual-SLLN\]. However, as we prove in Theorem \[thm:main\] of Section \[subsec:main-result\], under mild regularity conditions, we can prove a stronger result $$\lim_n g({{\cal V}}_n)^{-1} {{\cal S}}_n =0,$$ where $g({{\cal V}})$ is close to ${{\cal V}}^{1/2}$ such as $g({{\cal V}})={{\cal V}}^{1/2+\epsilon}$, $\epsilon>0$.

Let $$\nu_n=\frac{M_n}{{{\cal K}}_{n-1}}$$ denote the fraction of the capital Skeptic bets on day $n$. Then the capital process ${{\cal K}}_n$ is written as $$\label{eq:ckn}
{{\cal K}}_n = \prod_{i=1}^n (1+\nu_i(x_i - p_i)).$$

Now suppose that Skeptic himself models Reality’s move as a Bernoulli variable with the success probability $\hat p_n \in (0,1)$. If Skeptic totally trusts Forecaster, then he sets $\hat p_n=p_n$. However if Skeptic does not totally trust Forecaster he may formulate $\hat p_n$ differently from $p_n$. Furthermore suppose that Skeptic uses the “Kelly 