  The linear combination of Student's $t$ random variables (RVs) appears in
many statistical applications. Unfortunately, the Student's $t$ distribution is
not closed under convolution, thus, deriving an exact and general distribution
for the linear combination of $K$ Student's $t$ RVs is infeasible, which
motivates a fitting/approximation approach. Here, we focus on the scenario
where the only constraint is that the number of degrees of freedom of each
$t-$RV is greater than two. Notice that since the odd moments/cumulants of the
Student's $t$ distribution are zero, and the even moments/cumulants do not
exist when their order is greater than the number of degrees of freedom, it
becomes impossible to use conventional approaches based on moments/cumulants of
order one or higher than two. To circumvent this issue, herein we propose
fitting such a distribution to that of a scaled Student's $t$ RV by exploiting
the second moment together with either the first absolute moment or the
characteristic function (CF). For the fitting based on the absolute moment, we
depart from the case of the linear combination of $K= 2$ Student's $t$ RVs and
then generalize to $K\ge 2$ through a simple iterative procedure. Meanwhile,
the CF-based fitting is direct, but its accuracy (measured in terms of the
Bhattacharyya distance metric) depends on the CF parameter configuration, for
which we propose a simple but accurate approach. We numerically show that the
CF-based fitting usually outperforms the absolute moment -based fitting and
that both the scale and number of degrees of freedom of the fitting
distribution increase almost linearly with $K$.
