Let {$Y_k$} be i.i.d. random variables on $\mathbb{R}$ with a common continuous density $f$. As we put more and more points $Y_k$ on the real line they tend to concentrate so let us spread them out by multiplying by $n$, and also look at them around the point $nc$, for some fixed $c \in R$.
In other words,for each $n ∈ \mathbb{N}$, define $$X_{n,k} =n(Y_k −c), 1 \leq k \leq n.$$
For each $n$ let $N_n(a,b)$ = $\sum^n_{k=1} I_{(a,b)}(X_{n,k})$ be the number of $X_{n,k}$ that fall into the interval $(a, b)$. How can I find the weak limit for $N_n(a, b)$ as $n \to \infty$?
Compute \begin{align} & P[X_{n, k} \in (a, b)] \\ = & P[a < X_{n, k} < b] \\ = & P[a < n(Y_k - c) < b] \\ = & P\left[c + \frac{a}{n} < Y_k < c + \frac{b}{n}\right] \\ = & \int_{c + \frac{a}{n}}^{c + \frac{b}{n}} f(x) dx \\ =: & p_n \end{align} Note that $p_n$ is independent of $k$ so that $I_{(a, b)}(X_{n, k})$ are also i.i.d. Bernoulli random variables. Therefore $N_n(a, b)$ is a binomial random variable, i.e., $$N_n(a, b) \sim \text{Bin}(n, p_n).$$ Also notice that (use that $f$ is continuous and integration mean value theorem) \begin{align} np_n = n\int_{c + \frac{a}{n}}^{c + \frac{b}{n}} f(x) dx \to f(c)(b - a). \end{align} Hence by the celebrated Poisson approximation theorem, $$N_n(a, b) \Rightarrow \text{Poisson}(f(c)(b - a)).$$