## The Annals of Probability

### Monotonicity of conditional distributions and growth models on trees

Thomas M. Liggett

#### Abstract

We consider a sequence of probability measures $\nu_n$ obtained by conditioning a random vector $X =(X_1,\ldots,X_d)$ with nonnegative integer valued components on

$$X_1 + \dots + X_d = n - 1$$

and give several sufficient conditions on the distribution of $X$ for $\nu_n$ 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 confirm her conjecture, by showing that certain $\nu_n$’s defined in terms of $d$-ary Catalan numbers are stochastically increasing in $n$. The proof uses certain combinatorial identities satisfied by the $d$-ary Catalan numbers.

#### Article information

Source
Ann. Probab., Volume 28, Number 4 (2000), 1645-1665.

Dates
First available in Project Euclid: 18 April 2002

https://projecteuclid.org/euclid.aop/1019160501

Digital Object Identifier
doi:10.1214/aop/1019160501

Mathematical Reviews number (MathSciNet)
MR1813837

Zentralblatt MATH identifier
1044.60094

#### Citation

Liggett, Thomas M. Monotonicity of conditional distributions and growth models on trees. Ann. Probab. 28 (2000), no. 4, 1645--1665. doi:10.1214/aop/1019160501. https://projecteuclid.org/euclid.aop/1019160501

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