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Let /i be a regular Borel probability measure with support F on a compact semigroup S. Let XO, X+1, X+2-.. be a sequence of independent random variables with values in S, having identical distribution P(X E B)= 8t(B). The random walk W = X *-. X XXI *-XN n n -n 101 n is studied in this paper. Let D be the closed semigroup generated by F and let K be the kernel of D. An element x E D is called recurrent iff Px(Wn e Nx i.o.) = 1 for every open neighborhood Nx of x. We prove: x is essential for W if and only if x E K if and only if x is recurrent if n and only if ZP x(W e ) =UJn* [1z(x1 * )](N )oo for every N . Moreover all states in K are recurrent positive. These results extend results of the authors for the unilateral random walks (using different methods) and recent results of Larisse for the discrete case. 1. Let S be a compact semigroup, i.e., a compact Hausdorff space which is algebraically a semigroup with jointly continuous multiplication and let S have a countable basis. (See Concluding remarks on the blanket assumption of 2nd countability and how it can be removed.) Let yt be a regular probability measure defined on the Borel subsets-of S whose support is denoted by F. Let D = UF , the closed subsemigroup generated by F, and let K be the kernel of D. The measure ji induces various random walks on D among which the principal ones are the right, left and bilateral (symmetric) random walks induced by the probability transition functions (respectively) (prn(x, B)-n(x 1 B(,,)n(B), rP , B) it ~x1 B) n Pn(x, B) = Pn(x, B) jn(Bx1) (Ln) (B tPn(x, B) ,n x jnl(a, b); axb E B, a, b 6 Fn} On * X(Un)(B) for n = 1, 2,...,2 where It denotes the nth convolution (*) power of ji. For Received by the editors May 17, 1973 and, in revised form, November 5, 1973. AMS (MOS) subject classifications (1970). Primary 60G50, 60J15, 43A05.

Bilateral random walks on compact semigroups