Details

Autor(en) / Beteiligte

• Landers, Dieter
• Rogge, Lothar

Titel

Nonuniform Estimates in the Conditional Central Limit Theorem

Ist Teil von

• The Annals of probability, 1987-04, Vol.15 (2), p.776-782

Ort / Verlag

Hayward, CA: Institute of Mathematical Statistics

Erscheinungsjahr

1987

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• Let Xn, n ∈ N, be i.i.d. with mean 0, variance 1, and$E(|X_1|^r) < \infty$for some$r > 3$. Let B be a measurable set such that its distances from the σ fields σ (X1,..., Xn) are of order O(n-1/2(log n)-r/2). We prove that for such B the conditional probabilities P(n-1/2∑n i=1Xi≤ t∣ B) can be approximated by the standard normal distribution Φ (t) up to the classical nonuniform bound (1 + |t|r)-1n-1/2. An example shows that this is not true any more if the distances of B from σ (X1,..., Xn) are only of order O(n-1/2(log n)-r/2+ε) for some$\varepsilon > 0$. For the case r = 3 one can obtain the corresponding assertion only under a strengthened assumption.