The Poisson Boundary of Amenable Extensions

 We obtain two new criteria of amenability of a closed normal subgroup H in a locally compact group G. The first one is formulated in terms of convergence of convolutions of probability measures on G, and the second one in terms of the Poisson boundaries of random walks on G and on the quotient group G/H. Received 10 April 2001; in revised form 18 December 2001