Cooperative Models of Stochastic Growth - On a class of reinforced processes with graph-based interactions

COSTA, MARCELO ROCHA (2018) Cooperative Models of Stochastic Growth - On a class of reinforced processes with graph-based interactions. Doctoral thesis, Durham University.
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Consider a sequence of positive integer-valued random vectors denoted by ${\bf x}_n = (x_1(n),\ldots,x_N(n))$ for $n= 0,1,2,\ldots \>$. Fix ${\bf x}_0$, and given ${\bf x}_n$, choose a \emph{random} coordinate $i_{n+1} \in \{1,\ldots,N\}$. The probability that $\{i_{n+1} = i\}$ for a particular coordinate $i$ is proportional to a non-decreasing function $f_i$ of $\sum_{j =1}^N a_{ij}x_j(n)$, where $a_{ij} \geq 0$ measures how strongly $j$ cooperates with $i$. Now, on the event that $\{i_{n+1}=i\}$, update the sequence in such a way that ${\bf x}_{n+1}={\bf x}_n + {\bf e}_i$, where ${\bf e}_i$ is the vector whose $i$-th coordinate is 1 and whose other coordinates are 0. Finally, given $A=(a_{ij})_{i,j=1}^N$ and $f_i, \> i =1,\ldots,N,$ what can one say about $\lim_{n \to \infty} n^{-1} {\bf x}_n$?

Item Type Thesis (Doctoral)
Uncontrolled Keywords Stochastic Growth; Urn models; Branching Processes; Stochastic Approximation Algorithm.
Divisions Faculty of Science > Mathematical Sciences, Department of
Date Deposited 11 May 2018 12:38
Last Modified 16 Mar 2026 18:40

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