I would like to know if one can find a probability distribution with finite mean and the following property: $F(2^{i+1})-F(2^i)\le p$ for given parameter $0<p<1$ all $i$.
That is, if we consider the buckets $[2^i,2^{i+1})$ for $i=0,1,\dots$ the probability mass in each bucket is bounded by a same parameter. Furthermore if possible, within each bucket the density function should look as uniform as possible.
The easiest way to construct one such distribution is of course to choose one of the first $\lceil 1/p\rceil$ buckets uniformly, then draw from an uniform distribution within that bucket. But I would like to know if one can be less aggressive and make the support to be $[0,\infty)$ while still have a finite mean.
I have considered Gamma distribution but it seems $F(2x)-F(x)$ only starts decreasing when $x$ is on the order of the mean.