## Abstract

Pinsker’s widely used inequality upper-bounds the total variation distance kP − Qk1in terms of the Kullback-Leibler divergence D(P||Q). Although in general a bound in the reverse direction is impossible, in many applications the quantity of interest is actually D∗ (v, Q) — defined, for an arbitrary fixed Q, as the

infimum of D(P||Q) over all distributions P that are v-far away from Q in total variation. We show that D∗ (v, Q) ≤ Cv2 + O(v3), where C = C(Q) = 1/2 for “balanced” distributions, thereby providing a kind of reverse Pinsker inequality. Some of the structural results obtained in the course of the proof may be of

independent interest. An application to large deviations is given

infimum of D(P||Q) over all distributions P that are v-far away from Q in total variation. We show that D∗ (v, Q) ≤ Cv2 + O(v3), where C = C(Q) = 1/2 for “balanced” distributions, thereby providing a kind of reverse Pinsker inequality. Some of the structural results obtained in the course of the proof may be of

independent interest. An application to large deviations is given

Original language | English |
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Number of pages | 8 |

State | Published - 2012 |

### Publication series

Name | CoRR |
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