Joint Distribution of the Number of Vertices with Given Different Outdegrees
in Galton--Watson Forest

T. Myll\"ari
Department of Mathematics
\AA bo Akademy University
\AA bo, Finland


A Galton--Watson forest consisting of N roots (or trees) and n non-root
vertices is considered.  _Outdegree of a vertex_ is the number of branches
emanating from this vertex.  We study limit distributions of the number of
vertices of a given outdegree in such a forest.  Earlier, the local limit
theorems for this characteristic were proved.  It was shown that the limiting
distribution is normal with parameters depending on  how N and n approach
infinity.  Now, more general case, namely joint distribution of the number of
vertices with k given different outdegrees, is considered.  Local limit theorem
for this characteristic is proved.  In this case the normal limiting
distribution is obtained too.