= Lp martingale convergence theorem
{c}
{title2=$L^p,\quad1<p<\infty$}
A <martingale> satisfying $\sup_n\mathbb E|M_n|^p<\infty$, for $1<p<\infty$, has <almost sure convergence> and <convergence in Lp> to a limit $M_\infty\in L^p$. Also $M_n=\mathbb E[M_\infty\mid\mathcal F_n]$. The <Doob Lp maximal inequality> makes $\sup_n|M_n|$ integrable to the power $p$; the <Martingale convergence theorem> gives the almost-sure limit, and the <dominated convergence theorem> gives convergence in the <Lp norm>. Boundedness in $L^1$ alone does not imply <convergence in L1>.
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