Abstract: Given i.i.d. copies of a random variable with distribution we are only interested in those observations that fall into some set having but a small probability of occurrence. The truncation set is assumed to be known and non-random. Denoting the distribution of the truncated random variable by we consider the triangular array of experiments and investigate the asymptotic behavior of the -fold products Under a suitable density expansion, Gaussian shifts as well as Poisson experiments occur in the limit, where the latter case typically occurs when the number of expected observations falling in is bounded. Finally, we investigate invariance properties of the occurring Poisson limits.

1991 AMS Mathematics Subject Classification: 62B15.

Key words and phrases: Statistical experiment, truncation model, Gaussian experiment, local asymptotic normality, Poisson experiment, Poisson process, translation invariance, scale invariance, independent increments.