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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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  • $\begingroup$ In the formula you probably meant $F_k$ instead of $F_n$. Also, replacing $n-k$ with $k$ seems more natural. $\endgroup$ Oct 11, 2010 at 17:41
  • $\begingroup$ 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 $n$. $\endgroup$
    – Tom Smith
    Oct 12, 2010 at 5:46
  • $\begingroup$ I only mentioned that because then you can take $n=\infty$. $\endgroup$ Oct 13, 2010 at 5:09

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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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  • $\begingroup$ The URL above doesn't work for some reason. $\endgroup$
    – Peter Shor
    Oct 12, 2010 at 1:49
  • $\begingroup$ I also get an error. I added a link to the paper on Boris Solomyak's homepage. $\endgroup$ Oct 12, 2010 at 3:53
  • $\begingroup$ Thanks - I'll take a look at that later when I have time. $\endgroup$
    – Tom Smith
    Oct 12, 2010 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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