Doob's Second Martingale Convergence Theorem
It is important to note that the convergence in Doob's first martingale convergence theorem is pointwise, not uniform, and is unrelated to convergence in mean square, or indeed in any Lp space. In order to obtain convergence in L1 (i.e., convergence in mean), one requires uniform integrability of the random variables Nt. By Chebyshev's inequality, convergence in L1 implies convergence in probability and convergence in distribution.
The following are equivalent:
- (Nt)t>0 is uniformly integrable, i.e.
- there exists an integrable random variable N ∈ L1(Ω, P; R) such that Nt → N as t → +∞ both P-almost surely and in L1(Ω, P; R), i.e.
Read more about this topic: Doob's Martingale Convergence Theorems, Statement of The Theorems
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