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**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 U_{1},U_{2},..., urn U_{1} receiving each ball with probability p_{i}, i = 1,2,.... After n balls have been placed let L_{h} be the number of urns containing an odd number of balls. The event (L_{h} = 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 **{** p_{k} **}** 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 p_{i} > 0, i = 1,2,...,p_{1} \geq p_{2} \geq p_{3} \geq ..., and we set f_{n} = p_{n}+p_{n+1}+..., so that f_{1} = 1 and f_{n} decreases monotonically to zero. Theorem. A necessary and sufficient condition for recurrence is that (1) **sum**_{1}^{oo} {1 \over 2^{n}f_{n}} = oo.

**Classif.: ** * 60-99 Probability theory and stochastic processes

**Index Words: ** probability theory

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