Solving an integral equation

Hello everyone,

I am trying to solve a seemingly simple integral equation:

where is known to be zero for .

is a probability distribution function of a random process

where are iid random variables from a uniform distribution. I ran simulations of this process and got that is constant for and decays for . The decay does not seem to be exponential or 1/z. I did not manage to find an analytical solution or some transform (Laplace/Mellin) that can help.

Any suggestions would be welcome!