Limits of length functions of multi $p$-families of ideals

We show the asymptotic relationship between the limit of the normalized length function of a multi-$p-$family of ideals and that of its shifted family under linear growth conditions in a local domain of characteristic $p$. Examples of multi-$p-$families of ideals including products of Frobenius powers of different ideals. We apply our results to obtain a generalized version of a formula due to Wantanabe-Yoshida for certain $p-$families using results from Verma, and to provide an instance of the existence of a mixed multiplicity version of multi-$p$-families of ideals.