I would argue for the opposite, at least for stable distributions the random variable is better described by its characteristic function, which has a simple closed-form expression, than by its cumulants (which do not exist). In this case the characteristic function is also more helpful than the probability density, which lacks a closed-form expression.

I would argue for the opposite, at least for stable distributions the random variable is better described by its characteristic function, which has a simple closed-form expression, than by its cumulants (which do not exist). In this case the characteristic function is also more helpful than the probability density, which lacks a closed-form expression.

The OP also asks for the "probabilistic significance" of the characteristic function. In the context of stochastic processes of Lévy type, the characteristic function has the socalled Lévy–Khintchine representation, which describes the time dependence of the random variable in terms of a linear drift, a Brownian motion and a superposition of independent Poisson processes with different jump sizes.