  Extreme events appear in many physics phenomena, whenever the probability
distribution has a ''heavy tail'', differing very much from the equilibrium
one. Most unusual are the cases of power-law (Pareto) probability
distributions. Among their many manifestations in physics, from ''rogue waves''
in the ocean to L\'evy flights in random walks, Pareto dependences can follow
very different power laws. For some outstanding cases the power exponents are
less than 2, leading to indefinite mean values, let alone higher moments. Here
we present the first evidence of indefinite-mean Pareto distribution of photon
numbers at the output of nonlinear effects pumped by parametrically amplified
vacuum noise, known as bright squeezed vacuum (BSV). We observe a Pareto
distribution with power exponent 1.31 when BSV is used as a pump for
supercontinuum generation, and other heavy-tailed distributions (however with
definite moments) when it pumps optical harmonics generation. Unlike in other
fields, we can flexibly control the Pareto exponent by changing the
experimental parameters. This extremely fluctuating light is interesting for
ghost imaging and quantum thermodynamics as a resource to produce more
efficiently non-equilibrium states by single-photon subtraction, the latter we
demonstrate experimentally.
