On the recurrence of a certain chain
Donald A. Darling, Paul L. Erdos · Proceedings of the American Mathematical Society · 1968
Let balls be placed successively and independently in urns Ui, Ui, ■ ■ ■ , urn Ui receiving each ball with probability pi, i=l,2, ■ ■ ■ .After n balls have been placed let LH be the number of urns containing an odd number of balls.The event [A" = 0 for infinitely many n] has probability one or zero, termed respectively the "recurrent" and the "transient" cases.In [l, p. 94] it was stated that "it seems impossible to obtain a general criterion in terms of {pk} to ensure the recurrent case," and in [2] it was stated "it would appear that the necessary and sufficient conditions are rather delicate and not to be exhibited in neat form."In this note we clarify matters, showing that the condition (1) given below, previously known to be sufficient for recurrence ([l] and [2]), is also necessary.Without loss of generality we assume pi>0, i=l, 2, • • • , pi^p2 z^p3^ • • • , and we set fn = pn+pn+i+ • • • , so that /i=l and /"