Equidistribution and moments

Question

Equidistribution of a set sequence of numbers $\{s_1, s_2, s_3, \cdots\}$, loosely speaking, is the property that the proportion of terms falling in a subinterval is proportional to the length of that subinterval (a precise definition can be found in Wikipedia).

For simplicity let's consider $s_n$ being real numbers in the interval $[0,1]$; then an example of an equidistributed sequence is $s_n = n\alpha \mod 1$ where $\alpha$ is an irrational number.

Is equidistribution equivalent to showing that the moments of the set sequence of numbers match the moments of a uniform random variable $X$ on the interval, for all moments?

E.g. in the case of the interval $[0,1]$ can I say $\{s_n\}$ is equidistributed if $$ \lim_{N\to\infty} \frac{1}{N}\sum_{n=1}^N s_n^k = \mathbb{E}[X^k]=\int_0^1dx x^k=\frac{1}{k+1} $$ for all non-negative integer $k$?

Why or why not? Thanks.

Answer 1

Yet, It true, because the uniform distribution on $[0,1]$ is characterized by its moments. When a probability measure $\mu$ is characterized by its moments and $(\mu_n)$ is any sequence of probability measures having moments of every order, $\mu_n \to \mu$ as $n \to +\infty$ narrowly if and only if for every $k \in \mathbb{N}$, $$\int x^k \mathrm{d}\mu_n(x) \to \int x^k \mathrm{d}\mu(x) \text{ as } n \to +\infty.$$