A Limit Theorem for Random Matrices with a Multiparameter

Based on multiparameter subadditive ergodic theorems of Akcoglu and Krengel (1981) and Schürger (1988) we derive an almost sure limit theorem for families of random matrices with a multiparameter which satisfy a supermultiplicativity condition. This gives a multiparameter analogue of results of Fürstenberg and Kesten (1960) and Klingman (1973, 1976) (note, however, that our supermultiplicativity assumption is more restrictive since it involves products in an arbitrary order). It turns out that a Borel-Cantelli argument in Kingman (1973, 1976) has to be replaced by a projection argument involving subadditive processes with lower dimensional indices. Finally, we outline how our main convergence result applies to a certain stochastic model of a large economy.