  A multivariate fractional Poisson process was recently defined in Beghin and
Macci (2016) by considering a common independent random time change for a
finite dimensional vector of independent (non-fractional) Poisson processes;
moreover it was proved that, for each fixed $t\geq 0$, it has a suitable
multinomial conditional distribution of the components given their sum. In this
paper we consider another multivariate process
$\{\underline{M}^\nu(t)=(M_1^\nu(t),\ldots,M_m^\nu(t)):t\geq 0\}$ with the same
conditional distributions of the components given their sums, and different
marginal distributions of the sums; more precisely we assume that the
one-dimensional marginal distributions of the process
$\left\{\sum_{i=1}^mM_i^\nu(t):t\geq0\right\}$ coincide with the ones of the
alternative fractional (univariate) Poisson process in Beghin and Macci (2013).
We present large deviation results for
$\{\underline{M}^\nu(t)=(M_1^\nu(t),\ldots,M_m^\nu(t)):t\geq 0\}$, and this
generalizes the result in Beghin and Macci (2013) concerning the univariate
case. We also study moderate deviations and we present some statistical
applications concerning the estimation of the fractional parameter $\nu$.
