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Probabilistic proofs of large deviation results for sums of semiexponential random variables and explicit rate
function at the transition
Fabien Brosset, Thierry Klein, Agnès Lagnoux, Pierre Petit
To cite this version:
Fabien Brosset, Thierry Klein, Agnès Lagnoux, Pierre Petit. Probabilistic proofs of large deviation results for sums of semiexponential random variables and explicit rate function at the transition. 2021.
�hal-02895684v2�
Large deviations at the transition for sums of Weibull-like random variables
Fabien Brosset
1, Thierry Klein
2, Agnès Lagnoux
3, and Pierre Petit
11
Institut de Mathématiques de Toulouse; UMR5219. Université de Toulouse;
CNRS. UT3, F-31062 Toulouse, France.
2
Institut de Mathématiques de Toulouse; UMR5219. Université de Toulouse;
ENAC - Ecole Nationale de l’Aviation Civile , Université de Toulouse, France.
3
Institut de Mathématiques de Toulouse; UMR5219. Université de Toulouse;
CNRS. UT2J, F-31058 Toulouse, France.
January 19, 2021
Abstract
Deviation probabilities of the sumSn=X1+· · ·+Xnof independent and identically distributed real-valued random variables have been extensively investigated, in particular whenX1 is Weibull-like distributed, i.e. logP(X>x)∼ −qx1−ǫasx→ ∞. For instance, A.V. Nagaev formulated exact asymptotic results for P(Sn > xn) when xn> n1/2 (see, [16, 17]). In this paper, we derive rough asymptotic results (at logarithmic scale) with shorter proofs relying on classical tools of large deviation theory and giving an explicit formula for the rate function at the transition xn= Θ(n1/(1+ǫ)).
Key words: large deviations, sums of independent and identically distributed random vari- ables, Weibull-like, semiexponential, stretched exponential, Gärtner-Ellis theorem, contraction principle.
AMS subject classification: 60F10, 60G50.
1 Introduction
Moderate and large deviations of the sum of independent and identically distributed (i.i.d.) real-valued random variables have been investigated since the beginning of the 20th century.
Kinchin [11] in 1929 was the first to give a result on large deviations of i.i.d. Bernoulli dis- tributed random variables. In 1933, Smirnov [22] improved this result and in 1938 Cramér [4]
gave a generalization to i.i.d. random variables satisfying the eponymous Cramér’s condition which requires the Laplace transform of the common distribution of the random variables to be finite in a neighborhood of zero. Cramér’s result was extended by Feller [9] to sequences of not necessarily identically distributed random variables under restrictive conditions (Feller con- sidered only random variables taking values in bounded intervals), thus Cramér’s result does not follow from Feller’s result. A strengthening of Cramér’s theorem was given by Petrov in [19] together with a generalization to the case of non-identically distributed random variables.
Improvements of Petrov’s result can be found in [20]. Deviations for sums of heavy-tailed i.i.d.
random variables were studied by several authors: an early result appears in [13] and more recent references are [2, 3, 6, 14].
In [16,17], A.V. Nagaev studied the case where the commom distribution of the i.i.d. random variables is absolutely continuous with respect to the Lebesgue measure with density p(t) ∼ e−|t|1−ǫ as |t| tends to infinity, with ǫ ∈(0,1). He distinguished five exact-asymptotics results corresponding to five types of deviation speeds. In [18], S.V. Nagaev generalizes to the case where the tail writes as e−t1−ǫL(t), where ǫ∈(0,1) andL is a suitably slowly varying function at infinity. Such results can also be found in [2, 3].
Now, let us present the setting of this article. Let ǫ∈(0,1) and letX be a real-valued random variable verifying: there exists q >0 such that
logP(X >x)∼ −qx1−ǫ asx→ ∞. (1) Such a random variable X is often called a Weibull-like (or semiexponential, or stretched exponential) random variable. One particular example is that of [16, 17] where X has a density p(x) ∼ e−x1−ǫ. Moreover, unlike in [16, 17], this unilateral assumption is motivated by the fact that we focus on upper deviations of the sum. Observe that (1) implies that the Laplace transform of X is not defined on the right side of zero. Nevertheless, all moments of X+ := max(X,0) are finite. A weaker assumption on the left tail is required:
∃γ >0 ρ:=E[|X|2+γ]<∞. (2) We assume that X is centered (E[X] = 0) and denote by σ the standard deviation of X (Var(X) = σ2). For all n ∈ N∗, let X1, X2, ..., Xn be i.i.d. copies of X. We set Sn = X1+· · ·+Xn. In this paper, we are interested in the asymptotic behavior of logP(Sn>xn) for any positive sequence xn ≫n1/2. Not only does the logarithmic scale allow us to use the modern theory of large deviations and provide simpler proofs than in [2,3,18, 16,17], but we also obtain more explicit results. According to the asymptotics of xn, only three logarithmic asymptotic ranges appear. First, the Gaussian range: when xn ≪ n1/(1+ǫ), logP(Sn > xn) ∼ log(1−φ(σ−1n−1/2xn)),φ being the distribution function of the standard Gaussian law. Next, the domain of validity of the maximal jump principle: when xn ≫n1/(1+ǫ), logP(Sn >xn)∼ logP(max(X1, . . . , Xn) > xn). Finally, the transition (xn = Θ(n1/(1+ǫ))) appears to be an interpolation between the Gaussian range and the maximal jump one.
Logarithmic asymptotics were also considered in [12] for a wider class of distributions than in the present paper. Nevertheless, the setting was restricted to the particular sequence xn = n (that lies in the maximal jump range). In [8], the authors gave a necessary and sufficient condition on the logarithmic tails of the sum of i.i.d. real-valued random variables to satisfy a large deviation principle which covers the Gaussian range. In [1], Arcones proceeded analogously and covered the maximal jump range for symmetric random variables. In [10], the author studied a more general case of Weibull-like upper tails with a slowly varying function L, at a particular speed of the maximal jump range: xn =n1/(1−ǫ).
The transition at xn = Θ(n1/(1+ǫ)) is not considered in [2, 3]. It is treated in [18] and in [16, 17, Theorems 2 and 4]. Nevertheless, the rate function is given through non explicit formulae and hence is difficult to interpret. The main contribution of this work is to provide an explicit formula for the rate function at the transition. Moreover, we provide probabilistic proofs which apply both to the Gaussian range and to the transition.
The paper is organized as follows. In Section2, we recall two known results (Theorems 1and 2) and state the main theorem (Theorem 3). Section 3 is devoted to preliminary results. In particular, we recall a unilateral version of Gärtner-Ellis theorem inspired from [21] (Theorem 5) and establish a unilateral version of the contraction principle for a sum (Proposition 6), which has its own interest and which we did not find in the literature. The proof of Theorem 3 can be found in Section 4. On the way, we prove Theorem 1. And, to be self-contained, we give in Section 5 a short proof of Theorem 2which is new, up to our knowledge.
2 Main result
In this section, we summarize all regimes of deviations for the sum Sn defined in Section 1.
The two following results are known (see, e.g., [8] and [2]).
Theorem 1 (Gaussian range). Forn1/2 ≪xn≪n1/(1+ǫ), we have:
n→∞lim n
x2nlogP(Sn >xn) =− 1 2σ2.
Theorem 2 (Maximal jump range). For xn≫n1/(1+ǫ), setting Mn := max(X1, . . . , Xn),
n→∞lim 1
x1−ǫn logP(Sn>xn) = lim
n→∞
1
x1−ǫn logP(Mn >xn) = lim
n→∞
1
x1−ǫn logP(X >xn) = −q.
The Gaussian range occurs when all summands contribute to the deviations of Sn in the sense that logP(Sn > xn) ∼ logP(Sn > xn, ∀i ∈ J1, nK Xi < xǫn). In the maximal jump range, the main contribution of the deviations of Sn is due to one summand, meaning that logP(Sn>xn)∼logP(X>xn).
Now we turn to the main contribution of this paper: we estimate the deviations of Sn at the transition xn = Θ(n1/(1+ǫ)) and provide an explicit formula for the rate function. Notice that the sequence n1/(1+ǫ) is the solution (up to a scalar factor) of the following equation in xn: x2n/n=x1−ǫn , equalizing the speeds of of the deviation results obtained in the Gaussian range and in the maximal jump range. It appears that the behavior at the transition is a trade-off between the Gaussian range and the maximal jump range driven by the contraction principle for the distributions L(Sn−1 | ∀i Xi < xǫn)⊗ L(Xn | Xn>xǫn) and the function sum.
Theorem 3 (Transition). For all C >0 and xn=Cn1/(1+ǫ),
n→∞lim n
x2nlogP(Sn >xn) =− inf
06t61
(q(1−t)1−ǫ C1+ǫ + t2
2σ2
)
=:−J(C).
Let us give a somewhat more explicit expression for the rate function J. Let f(t) = q(1− t)1−ǫ/C1+ǫ+t2/(2σ2). An easy computation shows that, ifC 6Cǫ′ := (1+ǫ)((1−ǫ)qσ2ǫ−ǫ)1/(1+ǫ), then f is decreasing and its minimum 1/(2σ2) is attained att = 1. IfC > Cǫ′, then f has two local minima, at 1 and att(C): the latter corresponds to the smallest of the two roots in [0,1]
of f′(t) = 0, equation equivalent to
t(1−t)ǫ = (1−ǫ)qσ2 C1+ǫ .
If Cǫ′ < C 6 Cǫ := (1 +ǫ)(qσ2(2ǫ)−ǫ)1/(1+ǫ), then f(t(C)) > f(1). And, if C > Cǫ, then f(t(C))< f(1). As a consequence, for allC >0,
J(C) =
1
2σ2 if C 6Cǫ,
q(1−t(C))1−ǫ
C1+ǫ +t(C)2σ22 if C > Cǫ.
As a consequence, we see that the transition interpolates between the Gaussian range and the maximal jump one. First, when xn = Cn1/(1+ǫ), the asymptotics of the Gaussian range coincide with the one of the transition for C 6Cǫ. Moreover, t(Cǫ) = (1−ǫ)/(1 +ǫ) and one can check that −1/(2σ2) =−q(1−t(Cǫ))1−ǫ/Cǫ1+ǫ−t(Cǫ)2/(2σ2). Finally, for C > Cǫ, by the definition of t(C), we deduce that, as C → ∞, t(C) →0 leading to t(C)∼(1−ǫ)qσ2C−(1+ǫ). Consequently, C1+ǫJ(C)→1 asC → ∞, and we recover the asymptotic of the maximal jump range (recall that, when xn =Cn1/(1+ǫ),x2n/n=C1+ǫx1−ǫn ).
In Section4, we give a proof of Theorem3which also encompasses Theorem 1. Before turning to this proof, we establish several intermediate results useful in the sequel.
3 Preliminary results
First, we present a classical result, known as the principle of the largest term, that will allow us to consider the maximum of several quantities rather than their sum. The proof is standard (see, e.g., [5, Lemma 1.2.15]).
Lemma 4 (Principle of the largest term). Let (vn)n>0 be a positive sequence diverging to ∞, N be a positive integer, and, fori= 1, . . . , N, (an,i)n>0 be a sequence of non-negative numbers.
Then,
n→∞lim 1 vn
log
N
X
i=1
an,i
!
= max
i=1,...,N
n→∞lim 1 vn
logan,i
.
The next theorem is a unilateral version of Gärtner-Ellis theorem, which was proved in [21].
Its proof is omitted to lighten the present paper.
Theorem 5 (Unilateral Gärtner-Ellis theorem). Let (Yn)n>0 be a sequence of real random variables and a positive sequence (vn)n>0 diverging to ∞. Suppose that there exists a differen- tiable function Λ defined on R+ such that Λ′ is a (increasing) bijective function from R+ to R+ and, for all λ>0:
1
vnlogEhevnλYni−−−→
n→∞ Λ(λ).
Then, for all c>0,
−inf
t>cΛ∗(t)6 lim
n→∞
1 vn
logP(Yn> c)6 lim
n→∞
1 vn
logP(Yn>c)6−inf
t>cΛ∗(t), where, for all t>0, Λ∗(t) := sup{λt−Λ(λ) ; λ>0}.
Now, we present a unilateral version of the contraction principle for a sequence of random variables in R2 with independent coordinates where the function considered is the sum of the coordinates. Observe that only unilateral assumptions are required. The proof of the upper bound uses the same kind of decomposition as in the proof of [7, Lemma 4.3].
Proposition 6 (Unilateral sum-contraction principle). Let ((Yn,1, Yn,2))n>0 be a sequence of R2-valued random variables such that, for each n, Yn,1 and Yn,2 are independent. Let (vn)n>0
be a positive sequence diverging to ∞. For all a∈R and i∈ {1,2}, let us define Ii(a) =−inf
u<a lim
n→∞
1 vn
logP(Yn,i> u) and Ii(a) = −inf
u<a lim
n→∞
1 vn
logP(Yn,i> u).
Assume that:
(H) for all M >0, there exists d >0 such that
n→∞lim 1 vn
logP(Yn,1 > d, Yn,2 <−d)<−M and lim
n→∞
1 vn
logP(Yn,1 <−d, Yn,2 > d)<−M.
Then, for all c∈R, one has
−inf
t>cI(t)6 lim
n→∞
1 vn
logP(Yn,1+Yn,2 > c)6 lim
n→∞
1 vn
logP(Yn,1+Yn,2 >c)6−inf
t>cI(t) where, for all t∈R,
I(t) := inf
a,b∈R a+b=t
I1(a) +I2(b) and I(t) := inf
a,b∈R a+b=t
I1(a) +I2(b).
Moreover I and I are nondecreasing functions.
Remark 7. A sufficient condition for assumption (H)is: fori∈ {1,2},
a→∞lim Ii(a) =∞.
Proof. Obviously, the functions I1, I1, I2, and I2 are nondecreasing. Let us prove that I is nondecreasing, the proof forI being similar. Let t1 < t2, letη >0, and let a∈Rbe such that I(t2)>I1(a) +I2(t2−a)−η. SinceI2 is nondecreasing, we have
I(t1)6I1(a) +I2(t1−a)6I1(a) +I2(t2−a)6I(t2) +η, which completes the proof of the monotony of I, letting η →0.
Lower bound. Letc∈R, lett > c, and letδ >0 be such that 0<2δ < t−c. For all (a, b)∈R2 such that a+b=t, we have
lim 1 vn
logP(Yn,1+Yn,2 > c)>lim 1 vn
logP(Yn,1 > a−δ) + lim 1 vn
logP(Yn,2 > b−δ)
>−I1(a)−I2(b).
Therefore,
lim 1 vn
logP(Yn,1+Yn,2 > c)>sup
t>c sup
(a,b)∈R2 a+b=t
(−I1(a)−I2(b)) =−inf
t>cI(t).
Upper bound. Letc∈Rand let M >0. Let d >0 be given by assumption (H). Define Z ={(a, b)∈[−d,∞)2 ; a+b> c} and K ={(a, b)∈[−d,∞)2 ; a+b=c}.
Write
P(Yn,1+Yn,2 >c)6P(Yn,1> d, Yn,2 <−d) +P(Yn,1 <−d, Yn,2 > d) +P(Yn,1, Yn,2)∈Z
=:Qn,1+Qn,2+Qn,3. By assumption,
lim 1 vn
log(Qn,1)<−M and lim 1 vn
log(Qn,2)<−M.
Let us estimate limvn−1logQn,3. For all (a, b)∈K,
−inf
u<a v<b
lim 1 vn
logPYn,1 > u, Yn,2 > v
>−inf
u<alim 1 vn
logP(Yn,1> u)−inf
v<blim 1 vn
logP(Yn,2 > v)
=I1(a) +I2(b).
Defining θ[δ] := min(θ−δ, δ−1) for all δ >0 and for all θ∈(−∞,∞], there exists ua< a and vb < b such that
−lim 1 vn
logPYn,1 > ua, Yn,2 > vb
>(I1(a) +I2(b))[δ]. (3) From the cover ((ua,∞)×(vb,∞))(a,b)∈K of the compact subset K, we can extract a finite subcover ((uai,∞)×(vbi,∞))16i6p. Since
Z ⊂
p
[
i=1
(uai,∞)×(vbi,∞),
we obtain, thanks to Lemma 4 and (3), lim 1
vn
logQ3 6lim 1 vn
log
p
X
i=1
PYn,1 > uai, Yn,2 > vbi
= max
16i6p
nlim 1 vn
logPYn,1 > uai, Yn,2 > vbi
o
6 max
16i6p
n−(I1(ai) +I2(bi))[δ]o
6− inf
(a,b)∈R2 a+b=c
n(I1(a) +I2(b))[δ]o.
Letting δ→0 and using the definition of I, we deduce that lim 1
vn
logQ3 6− inf
(a,b)∈R2 a+b=c
(I1(a) +I2(b)) =−I(c) =−inf
t>cI(t).
Letting M → ∞, we get the desired upper bound.
4 Proof of Theorems 1 and 3
From now on, all non explicitly mentioned asymptotics are taken as n→ ∞. Replacing X by q−1/(1−ǫ)X, we may suppose without loss of generality that
logP(X >x)∼ −x1−ǫ asx→ ∞. (4)
The conclusions of Theorem 1 and 3 follow from Lemmas8, 9, 12 below, and the principle of the largest term (Lemma 4).
4.1 Principal estimates
By (4), the Laplace transform ΛX of X is not defined at the right of zero. In order to use the standard exponential Chebyshev inequality anyway, we introduce the following decomposition:
P(Sn> xn) =
n
X
m=0
n m
!
Πn,m(xn) where, for all m∈J0, nK and for alla>0,
Πn,m(a) :=P(Sn>a, ∀i∈J1, mK Xi >xǫn, ∀i∈Jm+ 1, nK Xi < xǫn).
Note that the only relevant truncation is at xǫn(and not at xnas in [16,17]). The asymptotics we want to prove are given by Lemmas 8 and 9, the proofs of which rely on the unilateral version of Gärtner-Ellis theorem (Theorem 5) and on the unilateral sum-contraction principle (Proposition 6).
Lemma 8. Let C >0. If n1/2 ≪xn6Cn1/(1+ǫ) and t >0, then lim n
x2nlog Πn,0(txn) =− t2
2σ2. (5)
Proof. Let us introduce X with distribution L(X | X < xǫn). For all n ∈ N∗, let X1, X2, ..., Xn be i.i.d. copies ofX and let Sn =X1+· · ·+Xn, so that
Πn,0(txn) = P(Sn >txn, X1, . . . , Xn< xǫn) =P(Sn >txn)P(X < xǫn)n∼P(Sn>txn),
by (4). We want to apply Theorem 5 to the random variables Sn/xn with vn = x2n/n. For u >0,
n x2n logE
eu
x2 n n
Sn xn
= n2
x2n logEheuxn Xn 1X<xǫn
i− n2
x2nlogP(X < xǫn). (6) The second term in the right side of the above equation goes to 0 as n → ∞ since logP(X <
xǫn) ∼ −P(X > xǫn) = O(e−xǫ(1−n ǫ)/2), by (4). As for the first term, if y < xǫn, then xny/n 6 x1+ǫn /n 6C1+ǫ. Now, up to changingγ inγ∧1, (1) is true for someγ ∈(0,1] and there exists c >0 such that, for all s6C1+ǫ, |es−(1 +s+s2/2)|6c|s|2+γ. Hence,
EheuxnXn 1X<xǫn
i−e
u2x2 n σ2 2n2
6
EheuxnXn 1X<xǫn
i−E
"
1 + uxnX
n + u2x2nX2 2n2
!
1X<xǫn
#
+
E
"
1 + uxnX
n +u2x2nX2 2n2
!
1X<xǫn
#
− 1 + u2x2nσ2 2n2
!
+
1 + u2x2nσ2 2n2
!
−e
u2x2 n σ2 2n2
6cρ
uxn n
2+γ
+
E
"
1 + uxnX
n +u2x2nX2 2n2
!
1X>xǫn
#
+o x2n n2
!
. (7)
For n large enough, applying Hölder’s inequality,
E
"
1 + uxnX
n +u2x2nX2 2n2
!
1X>xǫn
#
6E[X21X>xǫn]
6E[X2+γ]2/(2+γ)P(X >xǫn)γ/(2+γ)
=o x2n n2
!
, (8)
by (4). Combining (6), (7), and (8), we get n
x2nlogE
eu
x2 n n
Sn xn
→ u2σ2
2 := Λ(u).
Since Λ∗(t) =t/2σ2, (5) stems from Theorem 5.
Lemma 9. Let C >0. If xn=Cn1/(1+ǫ), then lim n
x2nlog Πn,1(xn) =−J(C).
Proof. Recall that, for n∈N∗,
Πn,1(xn) =P(Sn>xn|X1 >xǫn, X2, . . . , Xn< xǫn)P(X >xǫn)P(X < xǫn)n−1. Using (4), it suffices to prove that
n
x2nlogP(Sn >xn |X1 >xǫn, X2, . . . , Xn < xǫn)→ −J(C).
To do so, we apply the contraction principle of Proposition 6 to (Yn,1, Yn,2) with
L(Yn,1) = L(x−1n X1 | X1 >xǫn) and L(Yn,2) =L(x−1n (X2+· · ·+Xn) | X2, . . . , Xn< xǫn),
and vn = x2n/n. First, one has obviously, P(X > uxn | X > xǫn) = 1 for u 6 0 and P(X >uxn | X >xǫn) = 0 for u>1. In addition, for u∈(0,1), using (4),
logP(X >uxn | X >xǫn)∼ −(uxn)1−ǫ.
Using the notation of Proposition 6, it follows that I1(a) =I1(a) =I1(a), where
I1(a) = sup
u<a
(
−lim n
x2nlogP(X >uxn | X >xǫn)
)
=
0 if a <0, C−(1+ǫ)a1−ǫ if a∈[0,1],
∞ if a >1.
(9)
Moreover, for all u >0, n
x2nlogP(X2+. . .+Xn >uxn | X2, . . . , Xn < xǫn) = n
x2nlog Πn−1,0(uxn)→ − u2 2σ2, by Lemma 8. Thus, we have I2(b) = b2/(2σ2) for all b > 0 and, since I2 is a nondecreasing and nonnegative function, we get I2(b) = 0 for allb 60. This, together with (9), leads to: for all t∈R,
I(t) = inf
a+b=t (a,b)∈R2
{I1(a) +I2(b)}= inf
t−16b6t{I1(t−b) +I2(b)}= inf
t−16b6t
((t−b)1−ǫ C1+ǫ + b2
2σ2
)
, since b < t −1 entails I1(t −b) = ∞ and b > t entails I1(t−b) +I2(b) > I1(0) + I2(t).
It is a standard result (see, e.g., [15, 4.c.]) that I is upper semicontinuous. Since I is also nondecreasing, I is right continuous and we get
inft>1I(t) = inf
t>1I(t) =I(1).
Applying Proposition 6, this completes the proof.
Notice that the very same argument shows that:
• ifxn =Cn1/(1+ǫ), then, for all m >1, lim n
x2nlog Πn,m(xn) = −J(C);
• ifxn ≪n1/(1+ǫ), then, for all m>1, lim n
x2nlog Πn,m(xn) =− 1 2σ2.
Our last step consist in proving that these estimates also hold for Pnm=2mnΠn,m(xn) instead of Πn,m(xn).
4.2 Two uniform bounds
Lemma 10. Fix a sequence xn → ∞. For all δ ∈(0,1) and M >0, there exists n(δ, M)>1 such that, for all n>n(δ, M), for all m∈J0, nK, for all u∈[0, Mnx−ǫn ],
logP(Sm >u, ∀i ∈J1, mK Xi < xǫn)6−(1−δ)u2 2nσ2 .
In particular, if xn6Cn1/(1+ǫ), taking M =C1+ǫ, the bound holds for u∈[0, xn].
Proof. Using the fact that1t>0 6et, for all λ >0,
P(Sm >u, ∀i∈J1, mK Xi < xǫn)6e−λuE[eλX1X<xǫn]m.
Up to changing γ inγ∧1, (1) is true for someγ ∈(0,1] and there existsc(M)>0 such that, for alls6Mσ−2, we havees 61+s+s2/2+c(M)|s|2+γ. Hence, forλ=u(nσ2)−1 6Mσ−2x−ǫn ,
E[eλX1X<xǫn] 61 + λ2σ2
2 +c(M)ρλ2+γ 61 + λ2σ2
2 (1 +δ), as soon as 2c(M)ρσ−2(Mσ−2x−ǫn )γ 6δ, i.e. for n>n(δ, M). Thus, since m 6n,
logP(Sm >u, ∀i∈J1, mK Xi < xǫn)6−λu+ nλ2σ2
2 (1 +δ) =−(1−δ)u2 2nσ2 .
Lemma 11. Fix a sequence xn → ∞. For all δ ∈ (0,1), there exists n(δ)> 1 such that, for all n>n(δ), for all m>2, for all u∈[0, xn],
logP(Sm >u, ∀i∈J1, mK Xi >xǫn)6−(1−δ)u1−ǫ+m(1−2−ǫ)xǫ(1−ǫ)n .
Proof. The result is trivial foru < mxǫn. In the sequel, we supposeu>mxǫn. Letq′ = 1−2δ/3 and q′′ = 1−δ/3, so that 1−δ < q′ < q′′ < 1. Choose x(δ) >0 such that, for all x > x(δ), logP(X >x)6−q′′x1−ǫ. One has:
P(Sm >u, ∀i∈J1, mK Xi >xǫn)6P(Sm >u, ∀i∈J1, mK xǫn6Xi < u)
+P(∃i∈J1, mK Xi >u, ∀i∈J1, mK Xi >xǫn).
First,
P(∃i∈J1, mK Xi >u, ∀i∈J1, mK Xi >xǫn)6mP(X >u)P(X >xǫn)m−1
6me−q′(u1−ǫ+(m−1)xǫn(1−ǫ)) (10) as soon as xǫn > x(δ) (remember that u > mxǫn > xǫn), i.e. as soon as n > n1(δ). Secondly, denoting by ai integers,
P(Sm >u, ∀i∈J1, mK xǫn 6Xi < u) =Z
∀i xǫn6ui<u
1u1+···+um>u m
Y
i=1
P(X ∈dui)
6 X
∀i ⌈xǫn⌉6ai6⌈u⌉
1a1+···+am>u m
Y
i=1
P(ai−1< X 6ai)
6 X
∀i ⌈xǫn⌉6ai6⌈u⌉
1a1+···+am>u m
Y
i=1
e−q′′(ai−1)1−ǫ 6
Z
∀i xǫn6ui<u+2
1u1+···+um>u m
Y
i=1
e−q′′(ui−2)1−ǫdui
6
Z
∀i xǫn6ui<u+2 u1+···+um>u
e−q′(u1−1 ǫ+···+u1−mǫ)du1· · ·dum,
as soon as n is large enough (n>n2(δ)>n1(δ)) so that, for all v >xǫn, q′′(v−2)1−ǫ >q′v1−ǫ. Now, the functionf: (u1, . . . , um)7→ −q′(u1−ǫ1 +· · ·+u1−ǫm ) is convex, sof reaches its maximum on the domain of integration at the points where all the ui equal xǫn, except one equal to u−(m−1)xǫn. Therefore,
P(Sm >u,∀i∈J1, mK Xi >xǫn)6(u+ 2)mexph−q′(u−(m−1)xǫn)1−ǫ+ (m−1)xǫ(1−ǫ)n i.
Let
f(m, u) = (u−(m−1)xǫn)1−ǫ+ (m−1)xǫ(1−ǫ)n −u1−ǫ. Since
∂f
∂u(m, u) = (1−ǫ) 1
(u−(m−1)xǫn)ǫ − 1 uǫ
!
>0 and
f(m, mxǫn) =xǫ(1−ǫ)n m(1−m−ǫ)>xǫ(1−ǫ)n m(1−2−ǫ), we get
P(Sm >u, ∀i∈J1, mK xǫn6Xi < u)6(u+ 2)mexph−q′u1−ǫ+m(1−2−ǫ)xǫ(1−ǫ)n i. (11) Finally, putting together (10) and (11), and using the fact that, form>2,m−1>m(1−2−ǫ) and (u+ 2)m+m6(u+ 3)m,
P(Sm >u, ∀i∈J1, mK Xi >xǫn)6(u+ 3)mexph−q′u1−ǫ+m(1−2−ǫ)xǫ(1−ǫ)n i 6exph−(1−δ)u1−ǫ+m(1−2−ǫ)xǫ(1−ǫ)n i as soon as
log(u+ 3) 6log(xn+ 3)6 δ
3(1−2−ǫ)xǫ(1−ǫ)n , i.e. for n >n(δ)>n2(δ).
4.3 Upper bound for the sum of the Π
n,mUsing the uniform bounds of Lemmas 10 and 11, we are able to bound the remaining term
Pn m=2
n
m
Πn,m(xn) with an argument mimicing the proof of the upper bound in our unilateral sum-contraction principle.
Lemma 12. If n1/2 ≪xn≪n1/(1+ǫ), then lim sup
n→∞
n x2nlog
n
X
m=2
n m
!
Πn,m(xn)6− 1 2σ2. If xn=Cn1/(1+ǫ), then
lim sup
n→∞
n x2nlog
n
X
m=2
n m
!
Πn,m(xn)6−J(C).
Proof. Suppose n1/2 ≪xn 6Cn1/(1+ǫ). Fix some integer r>1. Noticing that
n(x, y)∈(R+)2 x+y>1o⊂
r
[
k=1
(
(x, y)∈(R+)2
x> k−1
r , y >1− k r
)
,
we have, for all m∈J2, nK,
Πn,m(xn) =P(Sn>xn, ∀i∈J1, mK Xi >xǫn, ∀i∈Jm+ 1, nK Xi < xǫn) 6
r
X
k=1
P
Sn−m > k−1
r xn, ∀i∈J1, n−mK Xi < xǫn
P
Sm >
1− k r
xn, ∀i∈J1, mK Xi >xǫn
6
r
X
k=1
exp
"
−(1−δ) ((k−1)/r)2x2n
2nσ2 +
1− k r
1−ǫ
x1−ǫn +m(1−2−ǫ)xǫ(1−ǫ)n
!#
,
for n large enough, applying Lemmas 10and 11. Hence, log
n
X
m=2
n m
!
Πn,m(xn)6log
r
X
k=1
exp
"
−(1−δ) ((k−1)/r)2x2n
2nσ2 +
1− k r
1−ǫ
x1−ǫn
!#
+ log
n
X
m=2
n m
!
em(1−2−ǫ)xǫn(1−ǫ)
where the latter sum is bounded.
• If xn ≪ n1/(1+ǫ), then x2n/n ≪ x1−ǫn and, applying the principle of the largest term (Lemma 4), we get
lim sup
n→∞
n x2nlog
n
X
m=2
n m
!
Πn,m(xn)6−(1−δ)r−1 r
2 1 2σ2, so, letting r → ∞and δ →0,
lim sup
n→∞
n x2nlog
n
X
m=2
n m
!
Πn,m(xn)6− 1 2σ2.
• If xn =Cn1/(1+ǫ), then x2n/n =C2n(1−ǫ)/(1+ǫ) =C1+ǫx1−ǫn and, applying the principle of the largest term (Lemma 4), we get
lim sup
n→∞
n x2nlog
n
X
m=2
n m
!
Πn,m(xn)6−(1−δ)minr
k=1
((k−1)/r)2
2σ2 + 1
C1+ǫ
1− k r
1−ǫ!
,
so, letting r → ∞and δ →0, lim sup
n→∞
n x2nlog
n
X
m=2
n m
!
Πn,m(xn)6− min
t∈[0,1]
t2
2σ2 + (1−t)1−ǫ C1+ǫ
!
=−J(C).
5 Proof of Theorem 2
To be complete, we mention a short proof of Theorem 2that we did not find in the literature.
Recall that we may assume that q= 1 without loss of generality (see the beginning of Section 4). First,
P(Mn>xn) = 1−(1−P(X >xn))n ∼nP(X >xn),
so x−1+εn logP(Mn>xn)→ −1 by (4). As for Sn, we introduce the following decomposition P(Sn>xn) = Pn+Rn
where
Pn:=P(Sn >xn, ∀i∈J1, nK Xi < xn) and Rn:=P(Sn>xn, ∃i∈J1, nK Xi >xn).
Theorem 2 is a direct consequence of Lemmas 4,13, and 14.
Lemma 13. If xn≫n1/(1+ǫ), then
lim 1
x1−ǫn logRn=−1. (12)
Proof. Notice that
P(Sn−1 >0)P(X >xn)6Rn6nP(X >xn).
The central limit theorem provides P(Sn−1 >0)→1/2 and the result follows.
