Sum of discrete uniform random variables

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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.
 

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Ray Vickson
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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?
 
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