Compounding binomial with arbitrary ([0,1]) valued random variable.
Let $Z$ be a random variable which takes values in the interval [0,1] with continuous density function $f_Z$. Now we have a coin with probability for heads equal to $Z$. Let $X_n$ be the number of heads when flipping the coin $n$ times. Now consider the function $$f_n(x)=\begin{cases} 0, & x <0 \\ 0, & x\ge 1 \\ (n+1)\operatorname{P}(X_n=d),& x\in \Big[\frac{d}{n+1}, \frac{d+1}{n+1}\Big)\ \text{for some }d\in \{0,1,\ldots,n\} \end{cases}$$ It turns out that $f_n$ converges to $f_Z$ in some sense. We will prove now pointwise convergence. Proof. Fix $x\in(0,1)$ and we should prove that $f_n(x)\to f_Z(x)$ as $n$ tends to infinity (from now on all our considerations will be for the intervarl $[0,1]$). For a fixed $n$ $x \in \Big[\frac{\left \lfloor{(n+1)x}\right \rfloor }{n+1}, \frac{\left \lfloor{(n+1)x}\right \rfloor+1}{n+1}\B