  This note is a companion article to the recent paper L\"ocherbach,
Loukianova, Marini (2024). We consider mean field systems of interacting
particles. Each particle jumps with a jump rate depending on its position. When
jumping, a macroscopic quantity is added to its own position. Moreover,
simultaneously, all other particles of the system receive a small random kick
which is distributed according to a positive $\alpha-$stable law and scaled in
$N^{-1/\alpha},$ where $0 < \alpha < 1.$ In between successive jumps of the
system, the particles follow a deterministic flow with drift depending on their
position and on the empirical measure of the total system. In a more general
framework where jumps and state space do not need to be positive, we have shown
in L\"ocherbach, Loukianova, Marini (2024) that the mean field limit of this
system is a McKean-Vlasov type process which is solution of a non-linear SDE,
driven by an $ \alpha-$stable process. Moreover we have obtained in
L\"ocherbach, Loukianova, Marini (2024) an upper bound for the strong rate of
convergence with respect to some specific distance disregarding big jumps of
the limit stable process. In the present note we consider the specific
situation where all jumps are positive and particles take values in $[ 0, +
\infty [ . $ We show that in this case it is possible to improve upon the error
bounds obtained in L\"ocherbach, Loukianova, Marini (2024) by using an adhoc
distance obtained after applying a concave space transform to the trajectories.
The distance we propose here takes into account the big jumps of the limit $
\alpha-$stable subordinator.
