Symmetric random walk, its probability distribution is binomial coefficient, in the continuous limit, is Gaussian distribution:

$\displaystyle e^{- x^{2}}$

What kind of random walk, its probability distribution, in the limit, is Gamma distribution:

$\displaystyle xe^{- x}$ for $x \geqslant 0$ ?

or simpler, an exponential distribution:

$\displaystyle e^{- x}$ for $x \geqslant 0$ ?

We are looking for something as simple as a random walk at discrete time, in the continuous limit, exponential factor $e^{-x}$ factor shows up. At each step, rules to guide random walk should be as simple as possible. If possible, we would like each step of the random walk to be a iid (independent and identically distributed) random variable. Is this possible ?

Thank you.