  This paper delves on the versatility of the theory of Khinchin families for
asymptotic estimation. We show that in combination with Local Central Limit
theorems for lattice variables, Khinchin families furnish a convenient and
unified framework to deal with asymptotic results of the coefficients of large
powers of power series.
  We revisit in the present paper this classical theme from that point of view,
obtaining clean new proofs and a number of new results. Asymptotic results for
the coefficients of solutions of Lagrange's equation fall naturally into this
combined framework. We provide a direct proof of an extension of the Otter and
Meir-Moon asymptotic formula as well as asymptotic results for families of
Lagrangian probability distributions.
