About the constants in the Fuk-Nagaev inequalities

In this paper we give efﬁcient constants in the Fuk-Nagaev inequalities. Next we derive new upper bounds on the weak norms of martingales from our Fuk-Nagaev type inequality.


Introduction and previous results
In this paper, we are interested in the deviation on the right of sums of unbounded independent random variables with finite variances. So, let X 1 , X 2 , . . . , X n be a finite sequence of independent random variables with finite variances. Set S n = X 1 + X 2 + · · · + X n and σ 2 = IE(X 2 1 ) + IE(X 2 2 ) + · · · + IE(X 2 n ). (1.1) Assume that IE(S n ) = 0. Then the so-called Tchebichef-Cantelli inequality states that IP(S n ≥ x) ≤ σ 2 /(x 2 + σ 2 ) for any x > 0.
However (1.4) is too restrictive. A less restrictive condition is the existence of moments of order q > 2 for the positive parts of the random variables X 1 , X 2 , . . . , X n . Set X i+ = max(0, X i ) and C q (X) = IE(X q 1+ ) + IE(X q 2+ ) + · · · + IE(X q n+ ) for any q ≥ 1. If IE(X i ) = 0 for any i in [1, n], then an adequate version of the Fuk-Nagaev inequalities (see Fuk (1973) or Nagaev (1979), Corollary 1.8) yields IP(S n ≥ x ) ≤ (q + 2)C q (X) qx q + exp − 2x 2 (q + 2) 2 e q σ 2 for any x > 0 (1.7) and any q > 2 such that C q (X) < ∞. Therefrom IP(S n ≥ a q σ 2 log z + b q C q (X)z 1/q ) ≤ 2/z for any z > 1, (1.8) with a q = (1 + q/2)e q/2 and b q = 1 + (2/q). Fan, Grama and Liu (2017) obtain an extension of (1.7) to the case of martingales (see their Corollary 2.5), with the same constants a q and b q . In Section 3 of this paper, we will prove the following maximal version of (1.8) with the optimal constant a q . Let S * n = max(0, S 1 , . . . , S n ): under the assumptions of (1.7), IP S * n > σ 2 log z + 1 + (2/q) + (q/3)1 q>3 C q (X)z 1/q ≤ 1/z for any z > 1. (1.9) In the iid case, one can derive immediately the bounded law of the iterated logarithm (with the exact constant) from (1.9), which shows (in spirit) that the constant a q = 1 appearing here cannot be further improved. In Section 4 we give similar inequalities under weak moments conditions. In Section 5 we apply the results of Sections 3 and 4 to get constants in the weak Rosenthal inequalities of Carothers and Dilworth (1988). The results of Sections 3, 4 and 5 are given in the more general setting of martingale differences sequences. Section 2 deals with preliminary results, which are the starting point of this paper.

Preliminary results
In this section, we introduce some definitions and technical tools which will be used all along the paper. We start with the definition of the tail function, the quantile function and the integrated quantile function.
Definition 2.1. Let X be a real-valued random variable. Then the tail function H X of X is defined by H X (t) = IP(X > t). The quantile function Q X of X is the cadlag inverse of H X (note that Q X is nonincreasing).
The basic property of Q X is: x < Q X (u) if and only if H X (x) > u. This property ensures that Q X (U ) has the same distribution as X for any random variable U with the uniform distribution over [0, 1].
Definition 2.2. The integrated quantile functionQ X of the real-valued and integrable random variable X is defined byQ X (u) = u −1 u 0 Q X (s)ds (since Q X is nonincreasing, Q X is a nonincreasing function).
We start by a byproduct of Doob's inequality, which is a reformulation of Lemma 1 in Dubins and Gilat (1978).
The assumption M 0 ≥ 0, which seems to be necessary (this assumption ensures that Q M * n ≥ 0 ), is omitted in Dubins and Gilat (1978). Therefore I give a proof below. From the Doob inequality and Lemma 2.1(a) in Rio (2000), for any x ≥ 0, (2.1) Consider now a real-valued random variable X with a finite Laplace transform on a right neighborhood of 0. Define X by X (t) = log IE exp(tX) for any t ≥ 0.
and the function Q * X by Q * X (u) = T X (log(1/u)). for any u ∈]0, 1]. (2.4) As noted by Rio (2000, p. 159), T X is the inverse function of the Legendre transform of X . Furthermore the following properties are valid.
Proposition 2.5. (i) For any real-valued and integrable random variable X with a finite Laplace transform on a right neighborhood of 0,Q X ≤ Q * X . (ii) T is subadditive on Ψ.

Fuk-Nagaev inequalities under strong moments assumptions
Throughout this section, (M j ) 0≤j≤n is a martingale in L 2 with respect to a nondecreasing filtration (F j ) j , such that M 0 = 0. We set X j = M j − M j−1 for any positive j. We assume that, for some constant q > 2, The increments X 1 , X 2 , . . . , X n are said to be conditionally symmetric if, for any j in [1, n] the conditional law of X j given F j−1 is symmetric. The main result of this section is Theorem 3.1 below.
From Theorem 3.1 and Lemma 2.3, we immediately get the corollary below.
Corollary 3.2. Under the assumptions of Theorem 3.1(a), for any z > 1, Note that β q ≤ 1 + (2/q). Hence Corollary 3.2 improves (1.8) for any value of z in the case q ≤ 3. If q > 3, the elementary inequality e log z ≤ qz 1/q can be used to replace (e/3) log z by (q/3)z 1/q in the above inequalities, which proves that Corollary 3.2 implies (1.9).
Proof of Theorem 3.1(a). We prove Theorem 3.1(a) in the case C q (M ) = 1. The general case follows by dividing the random variables by C q (M ).
Lemma 3.4. Let Z 1 , Z 2 , . . . , Z n be a finite sequence of random variables with finite variances, adapted to a nondecreasing filtration (F j ) j . Suppose furthermore that, for any j in [1, n], IE(Z j | F j−1 ) ≤ 0 almost surely. Let T n = Z 1 + Z 2 + · · · + Z n . Then, for any positive t, log IE e tTn ≤ (t), where Proof of Lemma 3.4. From the elementary inequality e x ≤ 1 + x + (x 2 /2) + k≥3 (x k + /k!), we infer that, for any positive t, We now apply Lemma 3.4 to the random variablesX 1 ,X 2 , . . . ,X n . Noticing that X j ≤ X j , which ensures that IE(X j | F j−1 ) ≤ 0 and thatX 2 j ≤ X 2 j , which implies that n j=1 IE(X 2 j | F j−1 ) ∞ ≤ σ 2 , we thus get that Usually the coefficients γ k are bounded up by σ 2 y k−2 . However this upper bound does not take into accounts the assumption on the moments of order q. Here we need the more precise upper bound below.
Proposition 3.5. Let Z 1 , Z 2 , . . . , Z n be a finite sequence of random variables, adapted to a nondecreasing filtration (F j ) j . Suppose furthermore that max(Z 1 , Z 2 , . . . , Z n ) ≤ c a.s. for some positive c and that n j=1 Then, first n j=1 IE(Z k j+ | F j−1 ) ≤ V (q−k)/(q−2) almost surely, for any real k ∈ [2, q] and second n j=1 IE(Z k j+ | F j−1 ) ≤ c k−q almost surely, for any real k ≥ q.
Proof of Proposition 3.5. Noting that Z k j+ ≤ Z q j+ c k−q for any k ≥ q, one immediately gets the second assertion. We now prove the first assertion. From the convexity of the exponential function, (q − 2)(Z j+ /a) k−2 ≤ (k − 2)(Z j+ /a) q−2 + (q − k) for any k in [2, q] and any positive a. Multiplying this inequality by a k−2 Z 2 j+ , we get that Taking the conditional expectation with respect to F j−1 and summing then on j, we infer Choosing a = V −1/(q−2) in this inequality, we then get the first part of Proposition 3.5.

Proof of Theorem 3.1(b).
It is enough to prove Theorem 3.1(b) in the case C q (M ) = 1. Set y = z 1/q and define the random variablesX j byX j = max(−y, min(X j , y)) for j in [1, n]. SetM n =X 1 +X 2 + · · · +X n . Then

Fuk-Nagaev inequalities under weak moments assumptions
Throughout this section, (M j ) 0≤j≤n is a martingale in L 2 with respect to a nondecreasing filtration (F j ) j , such that M 0 = 0. We set X j = M j − M j−1 for any positive j. We assume that, for some constant r > 2, (the letter w in C w r (M ) means weak). Let us now state our main result. Theorem 4.1. Let (M j ) 0≤j≤n be a martingale in L 2 satisfying (4.1), such that M 0 = 0. Then, for any z > 1,Q Mn (1/z) ≤ σ 2 log z + C w r (M )µ r z 1/r , where µ r = 2 + max(4/3, r/3) and (C w r (M ), σ) is defined by (4.2).
From Theorem 4.1 and Lemma 2.3, we immediately get the corollary below.  . The constant µ r appearing here can be improved. Nevertheless µ r ≤ 10/3 for any r in ]2, 4], which shows that Corollary 4.2 is suitable for numerical applications.

Upper bounds for weak norms of martingales
In this section, we apply the results of Sections 3 and 4 to weak norms of martingales. Let X be a real-valued and integrable random variable. For r ≥ 1, let Λ + r (X) = sup t>0 t (IP(X > t)) 1/r and Λ r (X) = max(Λ + r (X), Λ + r (−X) ).
(5.1) Then Λ r is a quasi-norm on the space weak-L r of real-valued random variables X such that Λ r (|X|) < ∞. From the properties of Q X given in Section 2, one can easily get the well-known equalities Λ + r (X) = sup u∈]0,1] u 1/r Q X (u) and Λ r (X) = max(Λ + r (X), Λ + r (−X)).
Hence Y q > (q/e). It follows that, for independent and identically distributed symmetric random variables in L q , (5.7) is more efficient for large values of n. Note however that 2 1/q < ζ q , so that one cannot compare (5.7) and (5.10) in the general case.