A note on almost sure convergence and convergence in measure. (English) Zbl 1313.28003

Kříž, Pavel; Štěpán, Josef

Commentat. Math. Univ. Carol. 55, No. 1, 29-40 (2014).

The purpose of the paper is to study the conditions under which the almost everywhere convergence coincides with the convergence in measure. The authors give some conditions under which the Cauchy property in \(\nu \) measure implies the convergence \(\nu \)-almost surely. As an application to the statistical estimation, the authors show that under such conditions a weakly consistent estimator is automatically a strong one. The results are closely connected with the results by A. Ionescu-Tulcea [Z. Wahrscheinlichkeitstheor. Verw. Geb. 26, 197–205 (1973; Zbl 0289.46030); Adv. Math. 12, 171–177 (1974; Zbl 0301.46032)].

Reviewer: Masafumi Akahira (Ibaraki)

MSC:

28A20	Measurable and nonmeasurable functions, sequences of measurable functions, modes of convergence
62F12	Asymptotic properties of parametric estimators
62C10	Bayesian problems; characterization of Bayes procedures

Keywords:

convergence in measure; almost sure convergence; pointwise compactness; Lusin property; strongly consistent estimators

Citations:

Zbl 0289.46030; Zbl 0301.46032