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Erschienen in:
2014 | OriginalPaper | Buchkapitel
6. Convergence Theorems
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Abstract
In the strong and the weak laws of large numbers, we implicitly introduced the notions of almost sure convergence and convergence in probability of random variables. We saw that almost sure convergence implies convergence in measure/probability. This chapter is devoted to a systematic treatment of almost sure convergence, convergence in measure and convergence of integrals.
The key role for connecting convergence in measure and convergence of integrals is played by the concept of uniform integrability.
In particular, we study Lebesgue’s dominated convergence theorem and conditions for interchangeability of derivative and integral.