Sum of Independent Random Variables with Mean Zero is Martingale
Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $\sequence {X_n}_{n \mathop \ge 0}$ be a sequence of integrable independent random variables with:
- $\expect {X_n} = 0$ for each $n \in \N$
and:
- $X_0 = 0$
For $n \ge 0$ define:
- $\ds S_n = \sum_{i \mathop = 0}^n X_i$
Let $\sequence {\FF_n^X}_{n \mathop \ge 0}$ be the natural filtration for $\sequence {X_n}_{n \mathop \ge 0}$.
Then $\sequence {S_n}_{n \mathop \ge 0}$ is a $\sequence {\FF_n^X}_{n \mathop \ge 0}$-martingale.
Proof
We first show that $\sequence {S_n}_{n \mathop \ge 0}$ is $\sequence {\FF_n^X}_{n \mathop \ge 0}$-adapted.
From the definition of the $\sigma$-algebra generated by a collection of mappings, we have:
- $X_i$ is $\map \sigma {X_0, \ldots, X_n}$-measurable for $0 \le i \le n$.
So from Pointwise Sum of Measurable Functions is Measurable: General Result, we have:
By the definition of the natural filtration, we have:
- $\map \sigma {X_0, \ldots, X_n} = \FF_n^X$
and hence $\sequence {S_n}_{n \mathop \ge 0}$ is $\sequence {\FF_n^X}_{n \mathop \ge 0}$-adapted.
Now let $n \ge 0$.
We have:
\(\ds \expect {S_{n + 1} \mid \FF_n^X}\) | \(=\) | \(\ds \expect {X_{n + 1} + S_n \mid \FF_n^X}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \expect {X_{n + 1} \mid \FF_n^X} + \expect {S_n \mid \FF_n^X}\) | Conditional Expectation is Linear | |||||||||||
\(\ds \) | \(=\) | \(\ds \expect {X_{n + 1} \mid \FF_n^X} + S_n\) | Conditional Expectation of Measurable Random Variable |
From Random Variable Independent of Sigma-Algebra Generated by Independent Random Variables, we have:
- $\map \sigma {X_{n + 1} }$ is independent from $\FF_n^X$.
So, from Conditional Expectation Unchanged on Conditioning on Independent Sigma-Algebra: Corollary we have:
- $\expect {X_{n + 1} \mid \FF_n^X} = \expect {X_{n + 1} } = 0$ almost surely.
So we have:
- $\expect {S_{n + 1} \mid \FF_n^X} = S_n$ almost surely.
So $\sequence {S_n}_{n \mathop \ge 0}$ is a $\sequence {\FF_n^X}_{n \mathop \ge 0}$-martingale.
$\blacksquare$
Sources
- 1991: David Williams: Probability with Martingales ... (previous) ... (next): $10.4$: Some examples of martingales