By Brewer M. J.

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**Example text**

31) 30 2 Exponential and Information Inequalities Proof. We essentially use the duality formula for entropy. 1 that ψ ∗−1 [K (P, Q)] = inf λ∈(0,b) ψ (λ) + K (Q, P ) . 32) that for any nonnegative random variable Y such that EP [Y ] = 1 and every λ ∈ (0, b) EP [Y (Z − EP [Z])] ≤ ψ (λ) + K (Q, P ) λ where Q = Y P . 28) holds. 25) yields EP [Y (λ (Z − EP [Z]) − ψ (λ))] ≤ EP [Y U ] ≤ EntP [Y ] and therefore ψ (λ) + K (Q, P ) . 29). 31) since then ψ ∗−1 (t) = 2vt. EP [Y (Z − EP [Z])] ≤ Comment. 31) is related to what is usually called a quadratic transportation cost inequality.

34) is given with constant 1, and [40] for a proof with the optimal constant 1/2). Proof. Let Q = Y P and A = {Y ≥ 1}. Then, setting Z = 1lA , P −Q TV = Q (A) − P (A) = EQ [Z] − EP [Z] . 35). We turn now to an information inequality due to Birg´e [17], which will play a crucial role to establish lower bounds for the minimax risk for various estimation problems. 4 Birg´ e’s Lemma Let us ﬁx some notations. 8) for any a ∈ [p, 1] a p hp (a) = sup (λa − ψp (λ)) = a ln λ>0 + (1 − a) ln 1−a 1−p . 25).

Since the transition from centered to noncentered Gaussian variables is via the addition of a constant, in this chapter, we shall deal exclusively with centered Gaussian processes. The parameter space T can be equipped with the intrinsic L2 -pseudodistance 2 d (s, t) = E (X (s) − X (t)) 1/2 . Note that d is a distance which does not necessarily separate points (d (s, t) = 0 does not always imply that s = t) and therefore (T, d) is only a pseudometric space. One of the major issues will be to derive tail bounds for supt∈T X (t).