Uniformly Integrable Martingales are L1 Convergent Martingales
Definition
Let’s assume we have a probability space $( \Omega , \mathcal{F} , P)$. A stochastic process $\left\{ X_{n} \right\}$ is said to converge to a random variable $X_{\infty}$ in the sense of $\mathcal{L}_{p}$, if it satisfies the following. $$ \lim_{n \to \infty} \| X_{n} - X_{\infty} \|_{p} = 0 $$ If a stochastic process $\left\{ X_{n} \right\}$ converges in the sense of $\mathcal{L}_{p}$, then the martingale $\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\}$ is said to converge in the sense of $\mathcal{L}_{p}$.
Theorem
If a martingale $\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\}$ is uniformly integrable, it converges in the sense of $\mathcal{L}_{1}$.
Explanation
From the viewpoint of measure theory, convergence in $p=1$ may not seem significant, but from the perspective of statistics, this level of convergence could be sufficient.
Proof
$$ \begin{align*} \int_{\Omega} |X_{n}| dP =& \int_{(|X_{n}| \le k)} |X_{n}| dP + \int_{(|X_{n}| > k)} |X_{n}| dP \\ \le & k P (|X_{n}| \le k) + \int_{(|X_{n}| > k)} |X_{n}| dP \\ \le & k+ \int_{(|X_{n}| > k)} |X_{n}| dP \end{align*} $$ That $\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\}$ is uniformly integrable means there exists some $k \in \mathbb{N}$ which satisfies the following for all $\varepsilon > 0$. $$ \sup_{ n \in \mathbb{N} } \int_{ \left( \left| X_{n} \right| \ge k \right) } \left| X_{n} \right| dP < \varepsilon $$ Therefore, applying $\displaystyle \sup_{n \in \mathbb{N}}$ to both sides of the formula obtained above, $$ \begin{align*} \sup_{n \in \mathbb{N}} \int_{\Omega} |X_{n}| dP \le & k + \sup_{n \in \mathbb{N}} \int_{ \left( \left| X_{n} \right| \ge k \right) } \left| X_{n} \right| dP \\ <& k + \varepsilon \\ <& \infty \end{align*} $$ summarizes to obtaining $\displaystyle \sup_{n \in \mathbb{N}} E | X_{n} | < \infty$.
Submartingale Convergence Theorem: Let’s assume we have a probability space $( \Omega , \mathcal{F} , P)$ and a submartingale $\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\}$.
If we set $\displaystyle \sup_{n \in \mathbb{N}} E X_{n}^{+} < \infty$, then $X_{n}$ almost surely converges to some random variable $X_{\infty}: \Omega \to \mathbb{R}$. $$E X_{\infty} < E X_{\infty}^{+} < \infty$$
$\displaystyle \sup_{n \in \mathbb{N}} E X_{n}^{+} \le \sup_{n \in \mathbb{N}} E | X_{n} | < \infty$ and since a martingale is a submartingale, according to the submartingale convergence theorem, the stochastic process $\left\{ X_{n} \right\}$ almost surely converges to some random variable $X_{\infty}$. Furthermore, almost sure convergence implies probabilistic convergence, hence it can be written as $X_{n} \overset{P}{\to} X_{\infty}$.
Vitali Convergence Theorem: Let’s assume we have a measure space $( X , \mathcal{E} , \mu)$.
When $1 \le p < \infty$, for a sequence of functions $\left\{ f_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{L}^{p}$ to converge in the sense of $\mathcal{L}_{p}$ to $f$, it’s necessary and sufficient that all three of the following conditions are met.
- (i): $\left\{ f_{n} \right\}$ converges in measure to $f$.
- (ii): $\left\{ | f_{n} |^{p} \right\}$ is uniformly integrable.
- (iii): For all $\varepsilon > 0$, $$ F \in \mathcal{E} \land F \cap E = \emptyset \implies \int_{F} | f_{n} |^{p} d \mu < \varepsilon^{p} \qquad \forall n \in \mathbb{N} $$ and there exists $E \in \mathcal{E}$ such that $\mu (E) < \infty$.
Probability $P$ is a finite measure that trivially satisfies condition (iii). Furthermore, since it’s assumed that $\left\{ X_{n} \right\}$ is uniformly integrable for $p=1$, it satisfies condition (ii), and since probabilistic convergence implies measure convergence, $X_{n} \overset{P}{\to} X_{\infty}$ implies that $X_{n}$ measure converges to $X_{\infty}$, satisfying condition (i). According to the Vitali Convergence Theorem$(\Leftarrow)$, $\left\{ X_{n} \right\}$ converges in the sense of $\mathcal{L}_{1}$, and the uniformly integrable martingale $\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\}$ is a martingale that converges in the sense of $\mathcal{L}_{1}$.
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