1.*7. If there exists a 8 > 0 such that 10 dt sup < 00,

then the random walk is not recurrent. [HINT: Use Exercise 3 of Sec. 6.2 to show that there exists a constant C(E) such that 11 -cos E-IX C(E) lIfE jU .'/)(ISIlI < E) :::: C(E) 2 f.Ln (dx) :::: --du Jut dt, :,7i'i x 2 0 -u where f.L1l is the distribution of Sn, and use (13). [Exercises 6 and 7 give, respectively, a necessary and a sufficient condition for recurrence. If f.L is of the integer lattice type with span one, then j 7T 1 ---dt=oo -7T 1 -J(t)
is such a condition according to Kesten and Spitzer.]