A generalization of Lévy's concentration-variance inequality
Sharp lower bounds are found for the concentration of a probability distribution as a function of the expectation of any given convex symmetric function \(\phi\). In the case \(\phi (x)=(x-c)^ 2\), where c is the expected value of the distribution, these bounds yield the classical concentration-variance inequality of Lévy. An analogous sharp inequality is obtained in a similar linear search setting, where a sharp lower bound for the concentration is found as a function of the maximum probability swept out from a fixed starting point by a path of given length.
scientific article; zbMATH DE number 4117544