Questions: Brownian Motion and the Wiener Process

Question 1

A symmetric random walk S_n with step size 1/√n is rescaled as X_n(t) = S_{⌊nt⌋}/√n. As n → ∞, what does X_n converge to in distribution?

• A. A Poisson process with rate 1
• B. Standard Brownian motion W(t), by Donsker's invariance principle (the functional central limit theorem)
• C. A deterministic linear function of time
• D. A Cauchy process, since the limiting distribution has heavy tails

Answer: (B)

Donsker's theorem (the functional CLT) states that the rescaled random walk converges in distribution to standard Brownian motion in the space C([0,1]) with the sup-norm topology. Each increment S_{k+1} - S_k has mean 0 and variance 1, so after rescaling by √n, the CLT applies at each fixed time. Donsker's theorem strengthens this from finite-dimensional convergence to convergence of the entire path. This is the deep justification for Brownian motion as the universal limit of random walks.

Question 2

Brownian motion has independent increments: W(t₂) - W(t₁) and W(t₄) - W(t₃) are independent whenever the intervals [t₁,t₂] and [t₃,t₄] do not overlap.

• T. True
• F. False

Answer: True

Independent increments over non-overlapping intervals is one of the defining properties of Brownian motion. Combined with the Gaussian distribution of each increment and continuity of paths, it fully characterizes the Wiener process. Note that overlapping increments like W(3) - W(1) and W(4) - W(2) are not independent because they share the contribution from [2,3].

Question 3

The sample paths of Brownian motion are almost surely continuous but almost surely nowhere differentiable. Explain why continuity does not imply differentiability in this context.

Model answer: Continuity means W(t+h) → W(t) as h → 0, but differentiability requires (W(t+h) - W(t))/h to converge. Since W(t+h) - W(t) ~ N(0,h), the increment is of order √h, so the difference quotient is of order 1/√h → ∞. The path is continuous because the increments shrink, but those increments don't shrink fast enough (relative to h) for a derivative to exist. This reflects the fractal, infinitely jagged nature of Brownian paths — they have Hausdorff dimension 3/2.

The nowhere-differentiability of Brownian motion is not an edge case — it holds with probability 1. The heuristic |W(t+h) - W(t)| ~ √h captures it: continuity needs increments to vanish (√h → 0), while differentiability needs them to vanish faster than h (√h/h = 1/√h → ∞). This gap between continuity and differentiability is why classical calculus fails for Brownian motion and why Itô calculus is necessary.

Question 4

The covariance function of standard Brownian motion is Cov(W(s), W(t)) = ?

• A. s·t
• B. min(s, t)
• C. |s - t|
• D. e^{-|s-t|}

Answer: (B)

For s ≤ t, write W(t) = W(s) + (W(t) - W(s)). Then Cov(W(s), W(t)) = Cov(W(s), W(s)) + Cov(W(s), W(t) - W(s)) = Var(W(s)) + 0 = s = min(s,t). The zero cross-term follows from independent increments: W(s) depends only on the process up to time s, and W(t) - W(s) is independent of the process up to time s. The min(s,t) covariance uniquely identifies the Gaussian process that is Brownian motion.