First off, your question is impossible to answer without more conditions on the initial distribution of momenta of the other particles (for instance they could be carefully arranged so that you can get any sequence of measurements that you wish for). A reasonable one is that the smaller particles are in equilibrium at some temperature T. With this assumption you can now apply the well-developed principles of statistical mechanics.

You can easily write down a Langevin equation for your ball-spring system. This can then be solved with the Fokker-Planck formalism to your system to figure out the probability distribution of the positions as a function of time as the ball-spring system equilibrates with the bath of particles. Of course here you'll have to put in initial conditions (if precisely well-defined, the initial distributions will be delta-function-like) for your ball and spring as well, as that will affect the approach to equilibrium.

Qualitatively, at very long times, you will get diffusion of the center of mass following the Einstein relation and from this, the positions of the two balls can be worked out as well.

I can try to add more details if you have specific questions. Or you can look for books on Fokker-Planck or "approach to equilibrium" in general (I learned the stuff from courses a while back, so I don't have a good reference at hand, sorry).

I can't really answer your question directly unless I know more about what "arbitrary sequences" you have in mind. Also, I've framed this answer from the point of view that you can look at a lot of such sequences and average over them to look at their distribution. If you're just given one such sequence, as far as I know you can't say very much.