I try to show the following:
Suppose $(X_n),n\geq1$ is a sequence of random variables with uniform distribution on $\{1/n,\dots,n/n \}$. Show that $(X_n)$ converges in distribution to a random variable $X\sim U(0,1)$
So I try to show that the characteristic function $\Phi_{X_n}(t)\to \Phi_{X}(t)$ which implies convergence in distribution.
Since I have a discrete RV the characteristic function looks like this:
$$\Phi_{X_n}(t)=\sum\limits_{k=1}^n e^{itk} P(X_n=k)=\sum\limits_{k=1}^n e^{itk}\frac{1}{n}$$
And the characteristic function of the limit looks like this
$$\Phi_{X_n}(t)=\int\limits_{0}^1 e^{itX}\mathrm{d}P=\frac{e^{ibt}-e^{iat}}{i(b-a)t}=\frac{e^{it}-1}{it}$$
Now I have to show that
$$\lim\limits_{n\to\infty}\sum\limits_{k=1}^n e^{itk}\frac{1}{n}=\frac{e^{it}-1}{it}$$
But how can I do this and is this possible with convergence of a discrete measure to a contrinuous..
Best Answer
There is a mistake in the expression of the characteristic function. We have that $$ \varphi_{X_n}(t)=\operatorname Ee^{itX_n}=\frac1n\sum_{k=1}^ne^{itk/n}. $$ The sum $$ \frac1n\sum_{k=1}^ne^{itk/n} $$ is a right Riemann sum that converges to $$ \int_0^1e^{itx}\mathrm dx $$ as $n\to\infty$.