Almost sure convergence of randomly truncated stochastic algorithms under verifiable conditions

In this paper, we are interested in the almost sure convergence of randomly truncated stochastic algorithms. In their pioneering work Chen and Zhu [Chen, H., Zhu, Y., 1986. Stochastic Approximation Procedure with Randomly Varying Truncations. In: Scientia Sinica Series.] required that the family of the noise terms is summable to ensure the convergence. In our paper, we present a new convergence theorem which extends the already known results by making vanish this condition on the noise terms -- a condition which is quite hard to check in practice. The aim of this work is to prove an almost sure convergence result for randomly truncated stochastic algorithms under assumptions expressed independently of the algorithm paths so that the conditions can easily be verified in practical applications.