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The simple L\'evy Poisson process and scaled forms are explicitly constructedfrom partial sums of independent and identically distributed random variablesand from sums of non-stationary independent random variables. For the latter,the weak limits are scaled Poisson processes. The method proposed here preparesgeneralizations to dependent data, to associated data in the first place.

$\ell^{\infty}$ Poisson invariance principles from two classical Poisson limit theorems and extension to non-stationary independent sequences