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:


 * $S_n$ is $\map \sigma {X_0, \ldots, X_n}$-measurable.

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:

almost surely.

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.