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Classes of residual lifetime distributions conditioned on failure histories

We describe a form of convergence which implies (*) for more general functionals: this involves replacing original processes by their prediction processes, in the sense of [2]. We shall argue that this form of convergence reflects more closely our intuitive idea of 'similarity' of stochastic processes, if we want similar processes to have similar structural properties. And it turns out that this form of convergence is particularly suited to the 'martingale approach' to processes.