Monotonicity of Conditional Distributions and Growth Models on Trees

We consider a sequence of probability measures n obtained by conditioning a random and give several suucient conditions on the distribution of X for n to be stochastically increasing in n. The problem is motivated by an interacting particle system on the homogeneous tree in which each vertex has d + 1 neighbors. This system is a variant of the contact process, and was studied recently by A. Puha. She showed that the critical value for this process is 1 4 if d = 2, and gave a conjectured expression for the critical value for all d. Our results connrm her conjecture, by showing that certain n 's deened in terms of d-ary Catall an numbers are stochastically increasing. The proof uses certain combinatorial identities satissed by the d-ary Catall an numbers.