Velocity field provides a complementary avenue to constrain cosmological information, either through the peculiar velocity surveys or the kinetic Sunyaev Zel’dovich effect. One of the commonly used statistics is the mean radial pairwise velocity. Here, we consider the three-point mean relative velocity, i.e. the mean relative velocities between pairs in a triplet. Using halo catalogs from the Quijote suite of $N$-body simulations, we first showcase how the analytical prediction for the mean relative velocities between pairs in a triplet achieve better than 4-5% accuracy using standard perturbation theory at leading order for triangular configurations with a minimum separation of $r \geq 50\ h^{-1}\mathrm{Mpc}$. Furthermore, we present the mean relative velocity between pairs in a triplet as a novel probe of neutrino mass estimation. We explore the full cosmological information content of the halo mean pairwise velocities, and the mean relative velocities between halo pairs in a triplet. We undertake this through the Fisher-matrix formalism using 22,000 simulations from the Quijote suite, and considering all triangular configurations with a minimum and a maximum separation of $20\ h^{-1}\mathrm{Mpc}$ and $120\ h^{-1}\mathrm{Mpc}$, respectively. We find that the mean relative velocities in a triplet allows a 1$\sigma$ neutrino mass ($M_\nu$) constraint of 0.065 eV, that is roughly 13 times better than the mean pairwise velocity constraint (0.877 eV). This information gain is not limited only to neutrino mass, but extends to other cosmological parameters - $\Omega_{\mathrm{m}}$, $\Omega_{\mathrm{b}}$, $h$, $n_{\mathrm{s}}$ and $\sigma_{8}$ achieving an information gain of 8.9, 11.8, 15.5, 20.9 and 10.9 times respectively. These results illustrate the possibility of exploiting the mean three-point relative velocities for constraining the cosmological parameters accurately from future cosmic microwave background experiments and peculiar velocity surveys.

Starting from first principles, we derive the fundamental equations that relate the $n$-point correlation functions in real and redshift space. Our result generalises the so-called `streaming model' to higher-order statistics - the full $n$-point correlation in redshift-space is obtained as an integral of its real-space counterpart times the joint probability density of $n-1$ relative line-of-sight peculiar velocities. Equations for the connected $n$-point correlation functions are obtained by recursively applying the generalised streaming model for decreasing $n$. Our results are exact within the distant-observer approximation and completely independent of the nature of the tracers for which the correlations are evaluated. Focusing on 3-point statistics, we use an $N$-body simulation to study the joint probability density function of the relative line-of-sight velocities of pairs of particles in a triplet. On large scales, we find that this distribution is approximately Gaussian and that its moments can be accurately computed with standard perturbation theory. We use this information to formulate a phenomenological 3-point Gaussian streaming model. A practical implementation is obtained by using perturbation theory at leading order to approximate several statistics in real space. In spite of this simplification, the resulting predictions for the matter 3-point correlation function in redshift space are in rather good agreement with measurements performed in the simulation. We discuss the limitations of the simplified model and suggest a number of possible improvements. Our results find direct applications in the analysis of galaxy clustering but also set the basis for studying 3-point statistics with future peculiar-velocity surveys and experiments based on the kinetic Sunyaev-Zel’dovich effect.