Mh. Math. 115, 191-213 (1993) MonaL~efte f/it 9 Springer-Verlag 1993 Printed in Austria Semistable Measures and Limit Theorems on Real and p-adic Groups By Riddhi Shah, Bombay (Received 4 May 1992; infinal form 1 September 1992) Abstract. For any locally compact group G, we show that any locally tight homomorphism from a real directed semigroup into MI(G) (semigroup of probability measures on G) has a 'shift' which extends to a continuous one-parameter semigroup. If G is a p-adic algebraic group then the above holds even iff is not locally tight. These results are applied to give sufficient conditions for embeddability of some translate of limits of sequences of the form {v kn} and # E M 1 (G) such that z(#) = #k, for some k > 1 and zeAut G (cf. Theorems 2.1, 2.4, 3.7). Introduction Let G be a locally compact (Hausdorff, second countable) group and let MI(G) denote the topological semigroup consisting of all probability measures on G with weak topology and the convolution as the semigroup operation. Let Aut G be the group of all bicontinu- ous automorphisms of G. A continuous one-parameter semigroup {#t}t>~0 in MI(G) is said to be (z, c)-semistable, for some z~Aut G and for some c e ]0, 1[ w ] 1, o0 [, if r(#t) = Pct for all t i> 0. Given a # e M 1(G) and a zeAut G such that z(#) = #k for some k~N\{1} (where N is the set of all natural numbers), we are interested to know if # can be embedded in a (~, k)-semistable continuous one-parameter semigroup {#t}t>~o as #1 = #. We prove this to be so in the case when G is a connected aperiodic (real) Lie group (cf. Corollary 2.2). This general- ises a result of NOBEL who has proved it for every aperiodic strongly root compact group G (cf. [19], Remark 4). We also show that if G is a connected real Lie group or a p-adic algebraic group (for a prime p) with -c, # and k as above (z is an algebraic morphism in the p-adic case) then there exists an element x in G, such that x# can be embedded in a (z,k)-semistable continuous one-parameter semigroup {#t}t~>o as 1991 Mathematics Subject Classification: 60B15, 22E35, 60B10.