Expected proposal count under a finite rejection envelope

Let $A_i$ be the event that proposal $i$ is accepted. The algorithm uses the same measurable acceptance region for each IID proposal-uniform pair, so mutual independence and common distribution in [F1] imply that the events $A_i$ are IID. By [F2], $\mathbb{P}(A_i)=p$ for every $i$. Therefore $\mathbb{P}(N>m)=q^m$ for $m\ge 0$, and $\mathbb{P}(N=k)=\mathbb{P}(A_1^c\cap \cdots \cap A_{k-1}^c\cap A_k)=q^{k-1}p$ for each $k\ge 1$.

For each $m\ge 1$, the pointwise identity $N\wedge m={\sum }_{j=1}^m{\mathbf{1}}_{\{N\ge j\}}$ holds, including on the null event $N=\mathrm{\infty }$. By [F3], [F4], and step 1.1, $\mathbb{E}[N\wedge m]={\sum }_{j=1}^m\mathbb{P}(N\ge j)={\sum }_{j=0}^{m-1}{q}^j\mathrm{.}$

The truncations $N\wedge m$ increase pointwise to $N$. By [F5] and then [F6], $\mathbb{E}[N]={\lim }_{m\to \mathrm{\infty }}\mathbb{E}[N\wedge m]={\sum }_{j=0}^{\mathrm{\infty }}{q}^j=\frac{1}{1-q}=\frac{1}{p}\mathrm{.}$ This is finite because $p>0$.

Substituting $p=Z_f\mathrm{/}(c{Z}_g)$ from [F2] gives $\mathbb{E}[N]=c{Z}_g\mathrm{/}{Z}_f$; when $Z_f=Z_g=1$, this is $c$.