Forecaster’s moves and Reality’s moves $$\xi=p_1 c_1 x_1 p_2 c_2 x_2 \dots$$ is called a [*path*]{}. The set $\Omega$ of all paths is called the [*sample space*]{}. A subset $E\subset \Omega$ is an [*event*]{}. A [*strategy*]{} ${{\cal P}}$ of Skeptic determines $\hat p_n$ based on a partial path $p_1 c_1 x_1\dots p_{n-1} c_{n-1} x_{n-1} p_n c_n$: $${{\cal P}}: p_1 c_1 x_1\dots p_{n-1} c_{n-1} x_{n-1} p_n c_n \mapsto \hat p_n, \qquad n=1,2,\dots.$$ ${{\cal K}}_n^{{{\cal P}}}={{\cal K}}_n^{{{\cal P}}}(\xi)$ denotes the capital process when Skeptic adopts the strategy ${{\cal P}}$. We say that Skeptic can [*weakly force*]{} an event $E$ by a strategy ${{\cal P}}$ if ${{\cal K}}_n^{{{\cal P}}}$ is never negative and $$\limsup_n {{\cal K}}_n^{{{\cal P}}}(\xi) = \infty   \quad \forall \xi\not\in E.$$ For two events $E_1, E_2 \subset \Omega$, $E_1^C \cup E_2$ is denoted as $E_1 \Rightarrow E_2$, where $E_1^C$ is the complement of $E_1$. We say that by a strategy ${{\cal P}}$ Skeptic can weakly force a conditional event $E_1\Rightarrow E_2$ if ${{\cal K}}_n^{{{\cal P}}}$ is never negative and $$\limsup_n {{\cal K}}_n^{{{\cal P}}}(\xi) = \infty   \quad \forall \xi \in  E_1\cap E_2^C.$$ $E_1$ is interpreted as a set of regularity conditions for the event $E_2$ to hold.

Let $\lambda_{\max,n}$ and $\lambda_{\min,n}$ denote 