  Invariant ensembles of random matrices are characterized by the distribution
of their eigenvalues $\{\lambda_1,\cdots,\lambda_N\}$. We study the
distribution of truncated linear statistics of the form $\tilde{L}=\sum_{i=1}^p
f(\lambda_i)$ with $p<N$. This problem has been considered by us in a previous
paper when the $p$ eigenvalues are further constrained to be the largest ones
(or the smallest). In this second paper we consider the same problem without
this restriction which leads to a rather different analysis. We introduce a new
ensemble which is related, but not equivalent, to the "thinned ensembles"
introduced by Bohigas and Pato. This question is motivated by the study of
partial sums of proper time delays in chaotic quantum dots, which are
characteristic times of the scattering process. Using the Coulomb gas
technique, we derive the large deviation function for $\tilde{L}$. Large
deviations of linear statistics $L=\sum_{i=1}^N f(\lambda_i)$ are usually
dominated by the energy of the Coulomb gas, which scales as $\sim N^2$,
implying that the relative fluctuations are of order $1/N$. For the truncated
linear statistics considered here, there is a whole region (including the
typical fluctuations region), where the energy of the Coulomb gas is frozen and
the large deviation function is purely controlled by an entropic effect.
Because the entropy scales as $\sim N$, the relative fluctuations are of order
$1/\sqrt{N}$. Our analysis relies on the mapping on a problem of $p$ fictitious
non-interacting fermions in $N$ energy levels, which can exhibit both positive
and negative effective (absolute) temperatures. We determine the large
deviation function characterizing the distribution of the truncated linear
statistics, and show that, for the case considered here
($f(\lambda)=1/\lambda$), the corresponding phase diagram is separated in three
different phases.
