On the Maximum of a Branching Process Conditioned on the Total Progeny

The maximum M of a critical Bienaymé-Galton-Watson process conditioned on the total progeny N is studied. Imbedding of the process in a random walk is used. A limit theorem for the distribution of M as N → ∞ is proved. The result is trasferred to the non-critical processes. A corollary for the maximal strata of a random rooted labeled tree is obtained.

A Stochastic Control Approach to a Parabolic Equation, Reciprocal Processes

2000 Mathematics Subject Classi cation: 49L60, 60J60, 93E20.

On the Equality of Sharp and Germ - Fields for Gaussian Processes and Fields

2000 Mathematics Subject Classification: 60G15, 60G60; secondary 31B15, 31B25, 60H15