Problem: Let $X_1,X_2,\dots$ be i.i.d. random variables with distribution function $F$ and a continuous density function $f$. Let $c\in\mathbb R$ be a number with $f(c)>0$ and consider $$N_n(a,b)=\sum_{k=1}^n\mathbf1_{\{c+a/n<X_k<c+b/n\}}.$$ Show that $N_n(a,b)$ converges in distribution for any $a<b$ and find the limit.
My Attempt: Define the random variables $$Y_{n,k}=\frac{\mathbf1_{\{c+a/n<X_k<c+b/n\}}-\displaystyle\int_{c+a/n}^{c+b/n}f(x)\,dx}{\sqrt n\cdot\sqrt{\displaystyle\int_{c+a/n}^{c+b/n}f(x)\,dx-\left[\displaystyle\int_{c+a/n}^{c+b/n}f(x)\,dx\right]^2}}$$ for $1\leq k\leq n$. We will check the conditions of the Lindeberg-Feller theorem. First, by construction we have $E[Y_{n,k}]=0$. Next, note that $$Y_{n,k}^2=\frac{1}{n\cdot\left[\displaystyle\int_{c+a/n}^{c+b/n}f(x)\,dx-\left(\displaystyle\int_{c+a/n}^{c+b/n}f(x)\,dx\right)^2\right]}\left[\mathbf1_{\{c+a/n<X_k<c+b/n\}}-2\cdot\mathbf1_{\{c+a/n<X_k<c+b/n\}}\cdot\int_{c+a/n}^{c+b/n}f(x)\,dx+\left(\int_{c+a/n}^{c+b/n}f(x)\,dx\right)^2\right],$$ and taking the expectation of both sides we get that $$E\left[Y_{n,k}^2\right]=\frac{1}n.$$ Hence, $$\lim_{n\to\infty}\sum_{k=1}^nE\left[Y_{n,k}^2\right]=1.$$ For the third condition, we need to verify that for any $\varepsilon>0$, $$\lim_{n\to\infty}\sum_{k=1}^n E\left[\vert Y_{n,k}\vert^2\cdot\mathbf1_{\{\vert Y_{n,k}\vert>\varepsilon\}}\right]=0.$$
I am stuck in the verification of the third condition.
I have also tried the same problem with the random variables defined as
$$Y_{n,k}=\mathbf1_{\{c+a/n<X_k<c+b/n\}}-\int_{c+a/n}^{c+b/n}f(x)\,dx,$$
and also had problems with the third condition of Lindeberg-Feller.
Does anyone have any hints on how to get started on the verification of the third condition of the Lindeberg-Feller in this example?
Thank you for your time and appreciate any feedback.
So, it's about calculating the limit of a sequence of characteristic functions.
To be more precise, for any real number $\lambda $, we have: $$\mathbb{E}( e^{ i\lambda N_n(a,b)})\underbrace{=}_{(X_i) \text{ are iids}} \left( \mathbb{E}\left( e^{ i\lambda 1_{ c+a/n < X_1< c+b/n} } \right) \right)^{n} = \left[ 1+ (e^{i\lambda}-1)\left( \int_{c+a/n}^{c+b/n}f(t)dt \right) \right]^n$$
So you see, as $$ \lim_{n \rightarrow \infty} n \left( \int_{c+a/n}^{c+b/n}f(t)dt \right) = (b-a)f(c)$$ (Thanks to the continuity of $f$)
From which, we imly : $$ \lim_{n \rightarrow +\infty} \mathbb{E}( e^{ i\lambda N_n(a,b)})= e^{ (b-a)f(c)(e^{i\lambda}-1) }$$
By Levy' continuity theorem, we have the conclusion.