## The Annals of Applied Probability

### Greedy Lattice Animals II: Linear Growth

#### Abstract

Let $\{X_\nu: \nu \in \mathbb{Z}^d\}$ be i.i.d. positive random variables and define $M_n = \max\big\{\sum_{\nu \in \pi}X_\nu: \pi \text{a self-avoiding path of length} n \text{starting at the origin}\big\}$, $N_n = \max\big\{\sum_{\nu \in \xi}X_\nu:\xi \text{a lattice animal of size} n \text{containing the origin}\big\}$. In a preceding paper it was shown that if $E\{X^d_0(\log^+ X_0)^{d+a}\} < \infty$ for some $a > 0$, then there exists some constant $C$ such that w.p.1, $0 \leq M_n \leq N_n \leq Cn$ for all large $n$. In this part we improve this result by showing that, in fact, there exist constants $M,N < \infty$ such that w.p.1, $M_n/n \rightarrow M$ and $N_n/n \rightarrow N$.

#### Article information

Source
Ann. Appl. Probab., Volume 4, Number 1 (1994), 76-107.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aoap/1177005201

Digital Object Identifier
doi:10.1214/aoap/1177005201

Mathematical Reviews number (MathSciNet)
MR1258174

Zentralblatt MATH identifier
0824.60100

JSTOR