Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Suppose you are given a number $0 < c < 1$, and an i.i.d. sequence of random variables $x_i$, such that $P \left[ 0 \le x_i \le 1 - c \right] = 1$. The series $\sum_{k=0}^\infty x_k c^k$ converges a.s. to some random variable $\xi$ with $P \left[ 0 \le \xi \le 1 \right] = 1$. I'd like to find examples where we can identify the distirbution of $\xi$ explicitly (it is easy to compute all moments recursively, but that's not what I am talking about). One example is given by $c=\frac{1}{N}$, and $x_i$ having a discrete distribution on $ 0, 1, 2, \ldots , N-1 $ ($N$ is a natural number): the limit is uniform on $[0,1]$. Any other case?

share|improve this question
    
Do you want to scale your discrete example so that the support fits into $[0,1-c]$? –  Brendan McKay Sep 20 '13 at 1:36
    
Well, I wrote it wrong (not what I had in mind): the distribution of $x_i$ should have been written as uniform on $0,\frac{1}{N}, \frac{2}{N}, \ldots, \frac{N-1}{N}$. My bad. –  user40238 Oct 27 '13 at 23:13

3 Answers 3

One thing that naturally comes to mind is the Cantor staircase distribution (where $c=1/3$ and $x_i$ takes values $0$ and $2/3$ with probability $1/2$) and its immediate generalizations. Other than that I don't think much is known explicitly. You also certainly can write down the characteristic function as an infinite product. You also can view the distribution as an invariant distribution for process $Z_{n+1}=cZ_{n}+x_{n+1}$.

share|improve this answer

One particular (relatively) well studied case is that of Bernoulli convolution, where the RV are supported on just 2 points. Take a look here to get an idea about what is known.

share|improve this answer

The Fabius function is the cumulative distribution function of a sum of exponentially scaled uniform distributions

$$\sum_{i=1}^\infty \frac{U_i(0,1)} {2^i}.$$

This is an example of a somewhat natural smooth function which is not analytic at any point of $[0,1]$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.