$\renewcommand{\P}{\mathbb{P}} \newcommand{\E}{\mathbb{E}}$

In the rejection sampling procedure, we wish to sample from some distribution with density $h$. To accomplish this, we first sample from some dominating distribution with density $g$. First find a value $M$ such that $M g(x) \geq h(x)$ for all $x$. Now, for each draw $Y$ from $g$, you also draw a standard uniform $U$. If $U \leq h(x)/(Mg(x))$, then set $X$ equal to $Y$. Otherwise draw from $g$ again and repeat the process.

It is easy to show that rejection sampling works. First, we’ll need a couple of lemmas, both of which are quite interesting on their own.

## Lemma 1

If $A$ is a random variable whose support is a subset of $[0,1]$, and $U$ is a standard uniform random variable independent of $A$, then

\begin{align*} \P \{ U \leq A \} = \E A \end{align*}

### Proof

Let $f_A$ and $f_U$ be the densities of $A$ and $U$. Because they are independent, their joint density is equal to the product of the two marginal densities.

\begin{align*} \P \{ U \leq A \} &= \int_0^1 \int_0^a f_U (u) f_A (a) du \; da\\ &= \int_0^1 \left[ \int_0^a f_U (u) du \right] f_A (a) da\\ &= \int_0^1 \P \{ U \leq a \} f_A (a) da\\ &= \int_0^1 a f_A (a) da\\ &= \E A \end{align*}

## Lemma 2

The overall probability of accepting a draw is $1/M$.

### Proof

\begin{align*} \P \{ Y \; \text{is accepted} \} &= \P \{ U \leq h(Y)/(Mg(Y)) \}\\ &= \E [h(Y)/(Mg(Y))] \qquad \qquad \qquad \text{(by Lemma 1)}\\ &= \frac{1}{M} \int_{-\infty}^\infty \frac{h(y)}{g(y)} g(y) dy\\ &= \frac{1}{M} \int_{-\infty}^\infty h(y) dy\\ &= 1/M \end{align*}

## Rejection Sampling Theorem

The rejection sampling procedure described above produces draws from $X$ with density $h$.

### Proof

\begin{align*} \P \{X \in (x, x+\epsilon) \} &= \P \{ Y \in (x, x+\epsilon) \vert Y \; \text{is accepted} \}\\ &= \frac{\P \{ Y \; \text{is accepted} \vert Y \in (x, x+\epsilon) \} \P \{ Y \in (x, x+\epsilon) \}}{\P \{ Y \; \text{is accepted} \}}\\ &\approx \frac{ \P \{ U \leq h(Y)/(Mg(Y)) \vert Y \in (x, x+\epsilon) \} (\epsilon g(x))}{1/M}\\ &\approx \epsilon M g(x) \P \{ U \leq h(x)/(Mg(x)) \}\\ &= \epsilon M g(x) h(x)/(Mg(x))\\ &= \epsilon h(x) \end{align*}

A routine limiting argument (letting $\epsilon \rightarrow 0$) shows that this implies that $X$ has density $h$.