Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 164.47502
Autor: Darling, D.A.; Erdös, Pál
Title: On the recurrence of a certain chain (In English)
Source: Proc. Am. Math. Soc. 19, 336-338 (1968).
Review: Let balls be placed successively and independently in urns U1,U2,..., urn U1 receiving each ball with probability pi, i = 1,2,.... After n balls have been placed let Lh be the number of urns containing an odd number of balls. The event (Lh = 0 for infinitely many n) has probability one or zero, termed respectively the "recurrent" and the "transient" cases. In F. Spitzer, Principles of random walk (1964; Zbl 119.34304), p. 94, it was stated that "it seems impossible to obtain a general criterion in terms of { pk } to ensure the recurrent case", and by D. A. Darling [Proc. 5th Berkeley Sympos. Math. Stat. Probab., Univ. Calif. 1965/66, 2, No. 1, 345-350 (1967; Zbl 201.50801)] 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 is also necessary. Without loss of generality we assume pi > 0, i = 1,2,...,p1 \geq p2 \geq p3 \geq ..., and we set fn = pn+pn+1+..., so that f1 = 1 and fn decreases monotonically to zero. Theorem. A necessary and sufficient condition for recurrence is that (1) sum1oo {1 \over 2nfn} = oo.
Classif.: * 60-99 Probability theory and stochastic processes
Index Words: probability theory
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag