Sum of discrete uniform random variables

[avidaware]

Homework Statement

Let $X_k$ be iid uniform discrete on $\{0,...,9\}$. Find the distribution of $\sum\limits_{k=1}^{\infty} \frac{X_k}{10^k}$

Homework Equations

The Attempt at a Solution

I've tried a lot of things, I've tried decomposing $X_k$ into 10 bernoulli trials, I've tried using some form of central limit theorem. I've tried calculating the characteristic functions, then taking the limit and I get something really ugly. Any hints? I feel like there is some limit theorem I don't know.

 

[Ray Vickson]

Homework Statement

Let $X_k$ be iid uniform discrete on $\{0,...,9\}$. Find the distribution of $\sum\limits_{k=1}^{\infty} \frac{X_k}{10^k}$

Homework Equations

The Attempt at a Solution

I've tried a lot of things, I've tried decomposing $X_k$ into 10 bernoulli trials, I've tried using some form of central limit theorem. I've tried calculating the characteristic functions, then taking the limit and I get something really ugly. Any hints? I feel like there is some limit theorem I don't know.

If you observe $Y = \sum_{k=1}^{\infty} X_k / 10^k$ you will see a random number between 0 and 1 and whose decimal digits are iid uniformly distributed in 0--9. How do you think such a number will be distributed?