# Showing convergence to a uniform distribution

I've been told that X is a random variable with a discrete uniform distribution on the set (1/n,2/n,...,1) and have been asked to show that it converges in probability to U(0,1) which is a uniform distribution on the interval (0,1). However, I feel as if I'm missing information. I think I may need to find the pdf of X and us either the Delta Method or the Moment Generating Function Method to show convergence in distribution. Any help is greatly appreciated.

• X does not have a density, though its limit in distribution does. You could try the limit of its Cumulative Distribution Function, or of its moment generating function or characteristic function – Henry Sep 22 '20 at 0:35
• In this case, convergence of the CDF is especially straightforward – Thomas Lumley Sep 22 '20 at 1:36

## 1 Answer

As said in the comments, we can show that convergence of the CDF.

Let's call that discrete uniform variable $$X_n$$. For any $$0\leq t\leq1$$, we have:

$$P(X_n

Where $$[tn]$$ means the floor of $$tn$$. When $$n\rightarrow\infty$$, this expression converges to $$t$$. So, we have:

$$\lim_{n\rightarrow\infty}P(X_n

This means that the CDF of $$X_n$$ converges to the CDF of $$\text{Unif}(0,1)$$. This implies that $$X_n\overset{\mathcal D}\rightarrow U(0,1)$$ (convergence in distribution).