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This paper studies the problem of distributed resilient state estimation (RSE) for linear measurement models in the presence of locally-bounded Byzantine nodes that may arbitrarily deviate from the prescribed update rule. Necessary and sufficient conditions for the existence of a distributed algorithm to solve the RSE problem in the almost sure sense are characterized in term of the topology-associated robustly collective observability. Under these conditions, a distributed projected stochastic resilient filtering algorithm is proposed. Compared with the existing results where asymptotic or probabilistic finite-time analysis is established, the exponential convergence (in sense of mean square) of the proposed algorithm is proved. To further improve computational performance of the algorithm, an adaptive event-triggered mechanism is constructed without compromising its correctness of the estimate.