Counter-example for converse of Scheffé's theorem.

Question

Disclaimer: This is exercise 3.2.1 in Durret's Probability Theory.

I have to find a sequence of random variables $X_n$ converging weakly to a uniform distribution on $(0,1)$ such that they densities $f_n(x)$ not converging to $1$ for any $x \in [0,1]$.

This reduces to find differentiable functions $F_n(x)$ converging pointwise to $F(x)=x\cdot \mathbb{1}_{[0,1]}$ such that their derivatives don't converge pointwise to 1.

However I am unable to find such a sequence of functions. Could someone give me a hint?

Answer 1

Let us denote by $\delta_x$ the Dirac measure (that is, $\delta_x(B)=1$ if $x$ belongs to $B$ and $0$ otherwise. It is known that $n^{-1}\sum_{j=1}^n\delta_{j/n}$ converges to a uniform distribution. The idea is to approximate this by a distribution which has a density.

Define $$f_n(x) := n\sum_{j=1}^n\mathbf 1_{\left[\frac jn-\frac 1{2n^2},\frac jn+\frac 1{2n^2} \right] } (x).$$ Then $f_n$ is a density and a computation gives $$\int_0^1e^{itx}f_n(x)\mathrm dx=n\frac 2t e^{it\frac jn}\sin\left(\frac{t} {2n^2}\right) $$ which shows the weak convergence to a uniform distribution.

Moreover, a sequence of real numbers $(c_n)_{n\geqslant 1}$ such that for any $n$, $c_n\in \{0,n\}$ cannot converge to $1$.