63.8. LEVY’S THEOREM 2177

By continuity, this equation continues to hold for λ m = 0. Then iterate this, using a similarargument on the first factor of the right side to eventually obtain

E(

ei∑mj=1 λ j(X(t j)−X(t j−1))

)=

m

∏j=1

e−λ2

j2 (t j−t j−1).

Then letting all but one λ j equal zero, this shows the increment, X (t j)− X(t j−1

)is a

random variable which is normally distributed having variance t j− t j−1 and mean 0. Theabove formula also shows from Proposition 59.11.1 on Page 1891 that the increments areindependent. This proves the theorem.