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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