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Suppose we have a family $F_0,F_1,\dots$ of independent random variables which take the value $1$ with probability $p$ and $0$ otherwise; let $\delta$ be a number between $0$ and $1$. Let

$X_n = \sum_{k=0}^n \delta^{n-k} F_k$.

I'm interested in the distribution of $X_n$. It seems straightforward enough to be known and have a name - does anybody know what it is?

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In the formula you probably meant $F_k$ instead of $F_n$. Also, replacing $n-k$ with $k$ seems more natural. –  Ori Gurel-Gurevich Oct 11 '10 at 17:41
    
You are right about the $k$. Using $n-k$ rather than $k$ is more natural in the context I'm considering but of course it makes no practical difference for individual $n$. –  Tom Smith Oct 12 '10 at 5:46
    
I only mentioned that because then you can take $n=\infty$. –  Ori Gurel-Gurevich Oct 13 '10 at 5:09
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2 Answers

Unless I misunderstood your intention (see my comment above), if you take $n=\infty$ you get a Bernoulli convolution. See the paper Sixty Years Of Bernoulli Convolutions by Peres, Schlag and Solomyak which can also the last paper here.

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The URL above doesn't work for some reason. –  Peter Shor Oct 12 '10 at 1:49
    
I also get an error. I added a link to the paper on Boris Solomyak's homepage. –  Ori Gurel-Gurevich Oct 12 '10 at 3:53
    
Thanks - I'll take a look at that later when I have time. –  Tom Smith Oct 12 '10 at 5:47
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It is a finite distribution over numbers which aren't necessarily integers. I wouldn't expect it to have a name.

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