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Uniformly Integrable Martingales are L1 Convergent Martingales 📂Probability Theory

Uniformly Integrable Martingales are L1 Convergent Martingales

Definition

Let’s assume we have a probability space (Ω,F,P)( \Omega , \mathcal{F} , P). A stochastic process {Xn}\left\{ X_{n} \right\} is said to converge to a random variable X∞X_{\infty} in the sense of Lp\mathcal{L}_{p}, if it satisfies the following. lim⁡n→∞∥Xn−X∞∥p=0 \lim_{n \to \infty} \| X_{n} - X_{\infty} \|_{p} = 0 If a stochastic process {Xn}\left\{ X_{n} \right\} converges in the sense of Lp\mathcal{L}_{p}, then the martingale {(Xn,Fn)}\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\} is said to converge in the sense of Lp\mathcal{L}_{p}.

Theorem

If a martingale {(Xn,Fn)}\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\} is uniformly integrable, it converges in the sense of L1\mathcal{L}_{1}.

Explanation

From the viewpoint of measure theory, convergence in p=1p=1 may not seem significant, but from the perspective of statistics, this level of convergence could be sufficient.

Proof

∫Ω∣Xn∣dP=∫(∣Xn∣≤k)∣Xn∣dP+∫(∣Xn∣>k)∣Xn∣dP≤kP(∣Xn∣≤k)+∫(∣Xn∣>k)∣Xn∣dP≤k+∫(∣Xn∣>k)∣Xn∣dP \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 {(Xn,Fn)}\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\} is uniformly integrable means there exists some k∈Nk \in \mathbb{N} which satisfies the following for all ε>0\varepsilon > 0. sup⁡n∈N∫(∣Xn∣≥k)∣Xn∣dP<ε \sup_{ n \in \mathbb{N} } \int_{ \left( \left| X_{n} \right| \ge k \right) } \left| X_{n} \right| dP < \varepsilon Therefore, applying sup⁡n∈N\displaystyle \sup_{n \in \mathbb{N}} to both sides of the formula obtained above, sup⁡n∈N∫Ω∣Xn∣dP≤k+sup⁡n∈N∫(∣Xn∣≥k)∣Xn∣dP<k+ε<∞ \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 sup⁡n∈NE∣Xn∣<∞\displaystyle \sup_{n \in \mathbb{N}} E | X_{n} | < \infty.

Submartingale Convergence Theorem: Let’s assume we have a probability space (Ω,F,P)( \Omega , \mathcal{F} , P) and a submartingale {(Xn,Fn)}\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\}.

If we set sup⁡n∈NEXn+<∞\displaystyle \sup_{n \in \mathbb{N}} E X_{n}^{+} < \infty, then XnX_{n} almost surely converges to some random variable X∞:Ω→RX_{\infty}: \Omega \to \mathbb{R}. EX∞<EX∞+<∞E X_{\infty} < E X_{\infty}^{+} < \infty

sup⁡n∈NEXn+≤sup⁡n∈NE∣Xn∣<∞\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 {Xn}\left\{ X_{n} \right\} almost surely converges to some random variable X∞X_{\infty}. Furthermore, almost sure convergence implies probabilistic convergence, hence it can be written as Xn→PX∞X_{n} \overset{P}{\to} X_{\infty}.

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Vitali Convergence Theorem: Let’s assume we have a measure space (X,E,μ)( X , \mathcal{E} , \mu).

When 1≤p<∞1 \le p < \infty, for a sequence of functions {fn}n∈N⊂Lp\left\{ f_{n} \right\}_{n \in \mathbb{N}} \subset \mathcal{L}^{p} to converge in the sense of Lp\mathcal{L}_{p} to ff, it’s necessary and sufficient that all three of the following conditions are met.

  • (i): {fn}\left\{ f_{n} \right\} converges in measure to ff.
  • (ii): {∣fn∣p}\left\{ | f_{n} |^{p} \right\} is uniformly integrable.
  • (iii): For all ε>0\varepsilon > 0, F∈E∧F∩E=∅  ⟹  ∫F∣fn∣pdμ<εp∀n∈N 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∈EE \in \mathcal{E} such that μ(E)<∞\mu (E) < \infty.

Probability PP is a finite measure that trivially satisfies condition (iii). Furthermore, since it’s assumed that {Xn}\left\{ X_{n} \right\} is uniformly integrable for p=1p=1, it satisfies condition (ii), and since probabilistic convergence implies measure convergence, Xn→PX∞X_{n} \overset{P}{\to} X_{\infty} implies that XnX_{n} measure converges to X∞X_{\infty}, satisfying condition (i). According to the Vitali Convergence Theorem(⇐)(\Leftarrow), {Xn}\left\{ X_{n} \right\} converges in the sense of L1\mathcal{L}_{1}, and the uniformly integrable martingale {(Xn,Fn)}\left\{ ( X_{n} , \mathcal{F}_{n} ) \right\} is a martingale that converges in the sense of L1\mathcal{L}_{1}.

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