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Azuma bound for simple random walk

Statement

Assume AC. For a simple symmetric random walk $S_{n}$ with $S_{0} = 0$ and independent increments taking values $- 1$ and $1$ , $P (∣ S_{n} ∣ \geq t) \leq 2 e^{- t^{2} / (2 n)}$ for every integer $n \geq 1$ and every $t > 0$ .

Facts & Assumptions

Given: The hypotheses, objects, and conventions in the Statement.

[F1]

Conditioning a known variable and an independent variable verifies the martingale property from independent centered increments.

[F2]

Symmetric bounded-increment Azuma bound supplies the two-sided concentration estimate.

[F3]

The Axiom of Choice is inherited from conditional expectation and Azuma.

Proof

1.1F1

Let $ξ_{k} = S_{k} - S_{k - 1} \in {- 1, 1}$ and use the natural filtration. Symmetry gives $E ξ_{k} = 0$ , and independence from the past plus F1 yields $E [S_{k} ∣ F_{k - 1}] = S_{k - 1}$ . Thus $S$ is a martingale and $∣ S_{k} - S_{k - 1} ∣ = 1$ .

2.1F2F3step 1.1∎

For the stated $n \geq 1$ , apply F2 with $c_{k} = 1$ . Since $\sum_{k = 1}^{n} c_{k}^{2} = n > 0$ , it gives exactly $2 e^{- t^{2} / (2 n)}$ . At $t = x \sqrt{n}$ the exponent is $- x^{2} / 2$ ; the prefactor and lack of lattice correction show this is a robust bound, not the exact binomial tail. AC is used only through F3.

Depends on

Used by

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Sources

Roch, Notes 20: Azuma's Inequality, Theorem 20.8, pp. 3–4 (standard reference, not scraped)