Stochastic invertible mappings between power law and Gaussian probability distributions

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We construct "stochastic mappings" between power law probability distributions (PD's) and Gaussian ones. To a given vector N, Gaussian distributed (respectively Z, exponentially distributed), one can associate a vector X, "power law distributed", by multiplying X by a random scalar variable a, N= a X. This mapping is "invertible": one can go via multiplication by another random variable b from X to N (resp. from X to Z), i.e., X=b N (resp. X=b Z). Note that all the above equalities mean "is distributed as". As an application of this stochastic mapping we revisit the so-called "zero-th law of thermodynamics problem" that bedevils the practitioners of nonextensive thermostatistics.