# unimodal distribution as a product with a uniform variable

There is a theorem (Khinchin/Khintchine 1938) stating that a distribution is unimodal with mode at zero iff it is the distribution function of the product of two independent random variables one of which is uniform on (0,1). I am looking for a corresponding result for a unimodal distribution with mode at c > 0. This is not a homework. Thanks a lot in advance

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Consider $X= U V$, where $U$ and $V$ are iid $U \sim \mathcal{U}_{(0,1)}$. Clearly $X$ is of the form you described. Yet $X>0$ with probability one, thus $X$ can not have mode at zero. Did you mean uniform on $[-1,1]$ instead? – Sasha Oct 16 '12 at 18:34
[Second edit] Sasha, thank you for looking into my question. The definition of the mode allows for a discontinuity at the mode. Khinchin's (sometimes spelled as Khintchine's] theorem is for U(0,1). To clarify, I am wondering whether there exists a theorem involving a decomposition of a unimodal distribution with a mode at c>0 into a product of two independent r.v. And/or, conversely, whether a product of two independent r.v., one of which is uniform on (a,b) with a>0 produces a unimodal distribution. Many thanks :) – TAK Oct 18 '12 at 15:14

There is a beautiful paper on the subject; both results are true, one can write X=WU, W\in R and U\in [0,1] and X=W_1U_1, W_1\in R^+ and U_1 \in [-1,1]; counter example does not hold, because Uniform is not unimodal in the sense of Khintchine's theorem.

M. C Jones (2002) On Khintchine's Theorem and its Place in Random Variate Generation, The American Statistician, 56:4, 304-307, DOI: 10.1198/000313002588

Yogen Chaubey

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I think a result like this is unlikely to hold for $c\gt0$. The reason that it holds for $0$ is that the contribution to the resulting distribution from each value $x$ of the unspecified distribution is spread over $[0,x]$, so whatever it contributes at some point it also contributes at all points closer to $0$ on the same side. You can't achieve the same effect with some other interval, since bounds other than $0$ change under multiplication.

A counterexample for the product of any distribution with a uniform distribution on $(a,b)$ with $a\gt0$ yielding a unimodal distribution is afforded by two peaks sufficiently far apart at $x_1$ and $x_2$ that their contributions at $[ax_1,bx_1]$ and $[ax_2,bx_2]$ don't overlap and hence form two separate modes. This effect doesn't occur for $a=0$ because in this case the contributions always overlap and add up at $0$.

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Joriki, many thanks, very intuitive. Do you think whether for a product with U(a,b) there will be at most two modes? Or would that depend on the other variable? And another question is what would be the decomposition for a distribution with a single mode at c>0? Many thanks – TAK Oct 19 '12 at 22:34