2
votes
3answers
203 views

Speed of convergence for Weyl's Equidistribution theorem

If $f$ is a continuous periodic fonction on $[0,1]$ and $a\not\in\mathbb{Q}$, the Weyl's equidistribution theorem states that $$\frac{1}{n}\sum_{k=0}^{n-1}f(ak)\rightarrow \int_0^1 f(x)dx.$$ Can we ...
5
votes
3answers
733 views

On the continuity of $\sum_{n=1}^{\infty} sin(nx) / n^\alpha$

I know that the series $\sum_{n=1}^\infty \sin(nx) / n^\alpha$, with $0 < \alpha < 1$, converges for $x \in [0,2\pi]$. I'm trying to understand if the function $f(x) = \sum_{n=1}^\infty ...
0
votes
1answer
418 views

Is an L_1 bounded sequence of random variables with uniformly converging CDFs uniformly integrable?

Changing my question in light of Dan's answer. Thanks, Dan. Consider a sequence of real random variables $X_i$ bounded in $L_1$, that is $\mathbb E\left|X_i\right|\leq M$ for all $i$. Suppose that ...