Stochastic Processes and Calculus by Uwe Hassler

Author:Uwe Hassler
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

From Grimmett and Stirzaker (2001, p. 310) or Pötscher and Prucha (2001, Theorem 5 and 9) we adopt the following results.

Lemma 8.3 (Implications of Convergence)

The following implications hold, . (a) Convergence in mean square implies convergence in probability:

(b) Convergence in probability implies convergence in distribution:

In general, the converse of Lemma 8.3(a) or (b) does not hold. However, if X = c is a constant, then convergence in probability and convergence in distribution are equivalent, see Grimmett and Stirzaker (2001, p. 310) or Pötscher and Prucha (2001, Theorem 10).

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