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Limits of truncation experiments

Marohn F.

Probability and Mathematical Statistics

2001

Vol. 21, Fasc. 1

71--88

Given n i.i.d. copies X1,…., Xn of a random variable X with distribution Pѵ, ѵ θ∈Θ⊂Rk, we are only interested in those observations that fall into some set D = D(n) ⊂ Rd having but a small probability of occurrence. The truncation set D is assumed to be known and non-random. Denoting the distribution of the truncated random variable X1D(X) by Pnѵ we consider the triangular array of experiments (Rd, ßd, (Pnѵθ ), n∈N, and investigate the asymptotic behavior of the n-fold products ((Rd)n, (ßd)n, (Pnѵ)nѵ∈Θ ). 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 D is bounded. Finally, we investigate invariance properties of the occurring Poisson limits.