Concentration Inequalities for Sums and Martingales by Bernard Bercu Bernard Delyon & Emmanuel Rio

Author:Bernard Bercu, Bernard Delyon & Emmanuel Rio
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

The function h r is convex with respect to x and concave with respect to r. Now, for any x ≠ 0, its maximum with respect to r is attained for

Therefore,

(2.126)

However, for any r in [0, 1], h r is the log-Laplace transform of a centered random variable with two-value distribution given by and . We immediately deduce from Lemma 2.19 that, for any r in [0, 1] and for any real x, . Consequently, (2.126) clearly leads to (2.125). Hereafter, it follows from the conjunction of (2.122), (2.123), and (2.125) that, for any positive t,

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