Sum of Independent Random Variables with Mean Zero is Martingale

From ProofWiki
Jump to navigation Jump to search

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:

\(\ds \expect {S_{n + 1} \mid \FF_n^X}\) \(=\) \(\ds \expect {X_{n + 1} + S_n \mid \FF_n^X}\)
\(\ds \) \(=\) \(\ds \expect {X_{n + 1} \mid \FF_n^X} + \expect {S_n \mid \FF_n^X}\) Conditional Expectation is Linear
\(\ds \) \(=\) \(\ds \expect {X_{n + 1} \mid \FF_n^X} + S_n\) Conditional Expectation of Measurable Random Variable

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.

$\blacksquare$


Sources

Retrieved from "https://proofwiki.org/w/index.php?title=Sum_of_Independent_Random_Variables_with_Mean_Zero_is_Martingale&oldid=607807"