On the Equivalence and Condition of Different Consensus Over a Random Network Generated by i.i.d. Stochastic Matrices

Our objective is to find a necessary and sufficient condition for consensus over a random network generated by i.i.d. stochastic matrices. We show that the consensus problem in all different types of convergence (almost surely, in probability, and in <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$L^{p}$</tex> </formula> for every <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$p\geq 1$</tex></formula> ) are actually equivalent, thereby obtain the same necessary and sufficient condition for all of them. The main technique we used is based on the stability in a projected subspace of the concerned infinite sequences.