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Nemirovski's Inequalities Revisited

Lutz Dümbgen, Sara A. van de Geer, Mark C. Veraar and Jon A. Wellner
The American Mathematical Monthly
Vol. 117, No. 2 (February 2010), pp. 138-160
DOI: 10.4169/000298910x476059
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Page Count: 23
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Nemirovski's Inequalities Revisited
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Abstract We study generalizations of the well-known fact that the variance of a sum of real-valued independent random variables is the sum of the variances. When the random variables take values in a Banach space or some other vector space, we study generalizations which bound the square of the norm of the sum of independent elements in terms of a constant K (which depends on the space and norm) times the sum of the expected values of the square of the norms of the independent summands. Such moment inequalities for sums of independent random vectors are important tools for statistical research. Nemirovski and coworkers (1983, 2000) and Pinelis (1994) derived one particular type of such inequalities. We present and compare three different approaches to obtain such inequalities: The results of Nemirovski and Pinelis are based on deterministic inequalities for norms. A second method involves type and cotype inequalities, a tool from probability theory on Banach spaces. Finally, we use a truncation argument plus Bernstein’s inequality to obtain another version of the moment inequality above. Interestingly, all three approaches have their own merits.

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