There are a number of experimental and theoretical papers testifying that the large-time behavior of the particle-size distribution function heavily depends on the initial conditions at the final stages of phase transformation processes. However, still now there is no theoretically derived distribution confirming these conclusions. The present paper is concerned with a new theoretical approach verifying the fact that the large-time distribution can actually be dependent on the details of the initial data. The concluding stage of evolution of a particulate ensemble as a result of coalescence and coagulation processes is considered. The transient kinetic and balance equations for the particle-size distribution function are modified into a single nonlinear equation for arbitrary collision frequency factors. This integro-differential equation is solved analytically in the limit of large times. The obtained particle-size distribution function depends on the initial condition and describes the concluding stages of transient phase transformation processes. The area of applicability of the present asymptotic solution for the large-time particle-size distribution is discussed. The obtained distribution function is in agreement with experiments.