In this paper, the envelope-constrained H-infinity filtering problem is investigated for a class of discrete time-varying stochastic systems over a finite horizon. The system under consideration involves fading measurements, randomly occurring nonlinearities (RONs) and mixed (multiplicative and additive) noises. A novel envelope-constrained performance criterion is proposed to better quantify the transient dynamics of the filtering error process over the finite horizon. The purpose of the problem addressed is to design a time-varying filter such that both the H-infinity performance and the desired envelope constraints are achieved at each time step. By utilizing the stochastic analysis techniques combined with the ellipsoid description on the estimation errors, sufficient conditions are established in the form of recursive matrix inequalities (RMIs) reflecting both the envelope information and the desired H-infinity performance index. The filter gain matrix is characterized by means of the solvability of the deduced RMIs. Finally, a simulation example is provided to show the effectiveness of the proposed filtering design scheme. (C) 2015 Elsevier Ltd. All rights reserved.