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This work is devoted to explore fundamental aspects of the spectralproperties of few-body general operators. We first consider the followingquestion: when we know the probability distributions of a set of observables,what can we way on the probability distribution of the summation of them? Inconsidering arbitrary operators, we could not obtain a useful information overthird order moment, while under the assumption of the few-body operators, wecan rigorously prove a much stronger bound on the moment generating functionfor arbitrary quantum states. Second, by the use of this bound, we generalizethe Chernoff inequality (or the Hoeffding inequality), which characterizes theasymptotic decay of the probability distribution for the product states by theGaussian decay. In the present form, the Chernoff inequality can be applied toa summation of independent local observables (e.g., single-site operators). Weextend the range of application of the Chernoff inequality to the genericfew-body observables.

Connecting probability distributions of different operators and generalization of the Chernoff-Hoeffding inequality