Law of small numbers as concentration inequalities for sums of independent random setsand random set valued mappings
In the paper we study concentration of sample averages (Minkowski's sums) of independent bounded random sets and set valued mappings around their expectations. Sets and mappings are considered in a Hilbert space. Concentration is formulated in the form of exponential bounds on probabilities of normalized large deviations. In a sense, concentration phenomenon reflects the law of small numbers, describing non-asymptotic behavior of the sample averages. We sequentially consider concentration inequalities for bounded random variables, functions, vectors, sets and mappings, deriving next inequalities from preceding cases. Thus we derive concentration inequalities with explicit constants for random sets and mappings from the sharpest available (Talagrand type) inequalities for random functions and vectors. The most explicit inequalities are obtained in case of discrete distributions. The obtained results contribute to substantiation of the Monte Carlo method in infinite dimensional spaces.