  Let (X_n) be a sequence of random variables (with values in a separable
metric space) and (N_n) a sequence of random indices. Conditions for X_{N_n} to
converge stably (in particular, in distribution) are provided. Some examples,
where such conditions work but those already existing fail, are given as well.
  Key words and phrases: Anscombe theorem, Exchangeability, Random indices,
Random sums, Stable convergence
