It is now well-known that the size of the model is the bottleneck when using model-based approaches to diagnose complex systems. To answer this problem, decentralized/distributed approaches have been proposed. The global system model is described through its component models as a set of automata and the global diagnosis is computed from the component diagnoses (also called local diagnoses). Another problem, which is far less considered, is the size of the diagnosis itself. However, it can also be huge enough, especially when dealing with uncertain observations. It is why we recently proposed to slice the observation flow into temporal windows and to compute the diagnosis in an incremental way from these diagnosis slices. In this context, we define in this paper two independence properties (transition and state independence) and we show their relevance to get a tractable representation of diagnosis. To illustrate the impact on the diagnosis size, experimental results on a toy example are given.