1.35. Random variables defined on ® are also referred to as defined

"modulo I". The theory of addition of independent r.v.'s on ® is somewhat simpler than on ,921, as exemplified by the following theorem. Let {X j, j ~ I}

be independent and identically distributed r. v.' s on ® and let S k = L~= 1 X j.

Then there are only two possibilities for the asymptotic distributions of Sk.

Either there exist a constant c and an integer n > 1 such that Sk -kc converges in dist. to the equidistribution on {jn-I , 0 :s j :s n -I}; or Sk converges in dist. to the uniform distribution on @. [HINT: Consider the possible limits of

(f fJ.-(n»k as k ~ 00, for each n.]