# Talk:Cantor distribution

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## Untitled

Funny distribution this one, but nmot as funny as the word "eventuate" that was in the previous version of the article. --Lucas Gallindo 03:16, 5 October 2006 (UTC)

From the Oxford English Dictionary:
Michael Hardy 16:11, 5 October 2006 (UTC)

## Alternate definition?

Is this distribution also the limiting (as n goes to infinity) distribution of

$\frac{1}{2} + \sum_{i=1}^{n}\frac{R_{i}}{3^{i}}, \,\!$

where the Ri are iid Rademacher distributions? Seems to me that it is, but my proof skills are quite rusty... Baccyak4H (Yak!) 20:15, 12 December 2006 (UTC)

• Since any number in the Cantor set can be (uniquely) expressed, in base 3, as 0.abcd... where a,b,c,d,... are either 0 or 2, then a simple way to simulate this Cantor distribution would be to add iid Bernoullis, scaled to 2/3:
$\sum_{i=1}^{n}\frac{2}{3^i} B_i(\frac{1}{2})\,$

Since Rademacher and Bernoulli are essentially the same distribution, just scaled ($R_i = 2 B_i(\frac{1}{2}) - 1\,$), then a simple substitution could prove your expression. Albmont (talk) 20:21, 17 July 2008 (UTC)

• BTW, I think the characteristic function of the Cantor distribution is wrong. It seems like a series that diverges everywhere. Albmont (talk) 20:21, 17 July 2008 (UTC)
Thanks, I convinced myself in the interim time since asking, but it's good to know I haven't completely lost it.
About the series diverging, I don't see that... all the terms of the product are in [-1, 1], but the terms' limit is +/- 1, depending on the sign of t, so the product part cannot diverge, I would think (I cannot rule out it might be 0 in the limit, but that's OK). Could you elaborate? Baccyak4H (Yak!) 20:35, 17 July 2008 (UTC)
About the series diverging, I don't see that - neither do I now, but first I swear I saw a Σ instead of the Π! BTW, I tried to use the Bernoulli series to check the formula, but I only came as far as $\Pi (1/2 + 1/2 \ \exp(2it/3^n))\,$. Albmont (talk) 20:49, 17 July 2008 (UTC)
Ouch! The derivation using Rademacher is obvious! Albmont (talk) 20:50, 17 July 2008 (UTC)