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We introduce the notion of a random matrix-valued multiplicative function,generalizing Rademacher random multiplicative functions to matrices. We providean asymptotic for the second moment based on a linear recurrence property forHilbert-Schmidt norms of sucessive products of random matrices. Moreover, weprovide upper bounds for the higher even moments related to the generalizedjoint spectral radius.

Random matrix-valued multiplicative functions and linear recurrences in Hilbert-Schmidt norms of random matrices