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The moments of first entrance time distributions 57 Notes. Theorem 1 is elementary; Theorem 2 is Theorem 96 III HARDY [1J; the other Tauberian theorem cited is a consequence of Theorem 108 there. 1) in the case 1ni i < 00. Similar methods have been used by FELLER [2J in obtaining further results mentioned in the notes on § 6. The new proof of the ratio limit theorem is the original one announced in CHUNG [1 J. For the individual ratio limit theorem in a chain with independent increments, stated in a more specific form but equivalent to the general result, see CHUNG and ERDOS [1J; there is a complement due to KEMENY [1].

If either j E H or i"'" H then HP;} < 00. Proof. If jEH then HP\j)-::;'JP\j)=j;'j). 1. Next, let kEH, k *i and i"",k so that kP;'k)=/j'r»O for some m~ 1. _ j(n+tn) kijkjk=kik -ik' I t follows that o We proceed to derive two general formulas. : HP\1 1'=1 respectively and applying § 5, Lemma A, we 46 ,1. Discrete parameter obtain (9) (10) provided that kf£H and the limit on the right side exists. These two formulas are completely general, from which simpler ones will emerge under certain mild assumptions, as we shall noVl' see.

We have we have lim n ...... oo Since L ni; = i lim L.. "a(~) '1 "a(n) = 1 L.. '1 n .... OO iE] i~] 1, given any 1'> 0 1- = 1- we can choose J " n·· 'I' L.. iE] = J(e) so that L: ni; < e. j~] There exists no =no(e) such that if n>no. L a}j)