Suppose that Z is a random variable with a N(0, 1) distribution. Define a stochastic process via X(t) = (?t)Z, and note that for each t > 0, X(t) has a N(0, t) distribution. Is this Brownian motion?
SOLUTION: No. Increments are not independent; they all contain the same rv Z.
For example X(2) ? X(1) = Z, and X(4) ? X(3) = Z, they are the same hence not independent.