* 22. X and Y are independent, identically distributed random variables with mean 0, variance 1, and characteristic function . If X + Y and

* 22. X and Y are independent, identically distributed random variables with mean 0, variance 1, and characteristic function φ. If X + Y and X − Y are independent, prove that

φ(2t ) = φ(t)3φ(−t ).

By making the substitution γ (t ) = φ(t )/φ(−t) or otherwise, show that, for any positive integer n,

φ(t ) =

(

1 −

1 2



t 2n

2

+ o



t/2n2

)4n

.

Hence, find the common distribution of X and Y. (Oxford 1976F)