  In Part I (G.Olshanski, math.RT/9804086) and Part II (A.Borodin,
math.RT/9804087) we developed an approach to certain probability distributions
on the Thoma simplex. The latter has infinite dimension and is a kind of dual
object for the infinite symmetric group. Our approach is based on studying the
correlation functions of certain related point stochastic processes.
  In the present paper we consider the so-called tail point processes which
describe the limit behavior of the Thoma parameters (coordinates on the Thoma
simplex) with large numbers. The tail processes turn out to be stationary
processes on the real line. Their correlation functions have determinantal form
with a kernel which generalizes the well-known sine kernel arising in random
matrix theory. Our second result is a law of large numbers for the Thoma
parameters. We also produce Sturm-Liouville operators commuting with the
Whittaker kernel introduced in Part II and with the generalized sine kernel.
