#### Tight bounds on the convergence rate of generalized ratio consensus algorithms

##### Balázs Gerencsér, László Gerencsér

The problems discussed in this paper are motivated by the ratio consensus problem formulated and solved in the context of the push-sum algorithm proposed in Kempe et al. (2003), and extended in B\'en\'ezit et al. (2010) under the name weighted gossip algorithm. We consider a strictly stationary, ergodic, sequentially primitive sequence of $p \times p$ random matrices with non-negative entries $(A_n), ~n \ge 1.$ Let $x, w \in \mathbb R^p$ denote a pair of initial vectors, such that $w \ge 0, w \neq 0$. Our objective is to study the asymptotic properties of the ratios $$e_i^\top A_n A_{n-1} \cdots A_1 x/ e_i^\top A_n A_{n-1} \cdots A_1 w, \qquad i=1,...,p,$$ where $e_i$ is the unit vector with a single $1$ in its $i$-th coordinate. The main results of the paper provide upper bounds for the almost sure exponential convergence rate in terms of the spectral gap associated with $(A_n).$ It will be shown that these upper bounds are sharp. In the final section of the paper we present a variety of connections between the spectral gap of $(A_n)$ and the Birkhoff contraction coefficient of the product $A_n \cdots A_1.$ Our results complement previous results of Picci and Taylor (2013), and Tahbaz-Salehi and Jadbabaie (2010).

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