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Distribution of minimum values of stochastic functionals

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  • Some mathematical problems of mechanics and physics have a form of the following variational problem. There is a functional, $I$, which is a sum of some quadratic positive functional and a linear functional. The quadratic functional is deterministic. The linear functional is a sum of a large number, $N$, of statistically independent linear functionals. The minimum value of the functional, $I$, is random. One needs to know the probability distribution of the minimum values for large $N$. The probability distribution was found in [2] in terms of solution of some deterministic variational problem. It was clear from the derivation that the class of quadratic and linear functionals for which this probability distribution can be used is not empty. It was not clear though how wide this class is. This paper aims to give some sufficient conditions for validity of the results of [2].
    Mathematics Subject Classification: Primary 05C38, 15A15; Secondary 05A15, 15A18.


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