In this paper we introduce a technique for obtaining exponential
inequalities, with particular emphasis placed on results involving ratios. Our
main applications consist of approximations to the tail probability of the
ratio of a martingale over its conditional variance (or its quadratic variation
for continuous martingales). We provide examples that strictly extend several
of the classical exponential inequalities for sums of independent random
variables and martingales. The spirit of this application is that, when going
from results for sums of independent random variables to martingales, one
should replace the variance by the conditional variance and the exponential of
a function of the variance by the expectation of the exponential of the same
function of the conditional variance. The decoupling inequalities used to
attain our goal are of independent interest. They include a new exponential
decoupling inequality with constraints and a sharp inequality for the
probability of the intersection of a fixed number of dependent sets. Finally,
we also present an exponential inequality that does not require any
integrability conditions involving the ratio of the sum of conditionally
symmetric variables to its sum of squares.
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NEW YORK, NEW YORK 10027 E-MAIL: vp@wald.stat.columbia.edu
