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Extreme events appear in many physics phenomena, whenever the probabilitydistribution has a ''heavy tail'', differing very much from the equilibriumone. Most unusual are the cases of power-law (Pareto) probabilitydistributions. Among their many manifestations in physics, from ''rogue waves''in the ocean to L\'evy flights in random walks, Pareto dependences can followvery different power laws. For some outstanding cases the power exponents areless than 2, leading to indefinite mean values, let alone higher moments. Herewe present the first evidence of indefinite-mean Pareto distribution of photonnumbers at the output of nonlinear effects pumped by parametrically amplifiedvacuum noise, known as bright squeezed vacuum (BSV). We observe a Paretodistribution with power exponent 1.31 when BSV is used as a pump forsupercontinuum generation, and other heavy-tailed distributions (however withdefinite moments) when it pumps optical harmonics generation. Unlike in otherfields, we can flexibly control the Pareto exponent by changing theexperimental parameters. This extremely fluctuating light is interesting forghost imaging and quantum thermodynamics as a resource to produce moreefficiently non-equilibrium states by single-photon subtraction, the latter wedemonstrate experimentally.

Indefinite-mean Pareto photon distribution from amplified quantum noise