Suppose Z(t) is the standard Brownian process, show that the following processes defined by 

are also Brownian processes.

To show that X_i(t) is a Brownian process, i = 1, 2, 3, it suffices to show that

is normally distributed with zero mean, and 

Also, the increments over disjoint time intervals are independent and X_i(0) = 0.

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and X₁(t) = kZ(t/k²), k> 0, for t > 0 for t = 0, X₂(t) = {12(1) for t 0 X3 (t) = Z(t + h) – Z(h), - h> 0,