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A note on some concentration inequalities under a non-standard assumption
Christophe Chesneau, Jan Bulla, André Sesboüé
To cite this version:
Christophe Chesneau, Jan Bulla, André Sesboüé. A note on some concentration inequalities under a non-standard assumption. 2010. �hal-00419741v2�
A note on some concentration inequalities under a non-standard assumption
∗Jan Bulla, Christophe Chesneau & Andr´e Sesbo¨u´e†
19 June 2010
Abstract
We determine two bounds for the tail probability for a sum ofnindependent random variables. Our assumption on these variables is non-standard: we suppose that they have moments of order δ for some δ ∈ [1,2). Numerical examples illustrate the theoretical results.
1 Introduction
Letn be a positive integer and (Yi)i∈{1,...,n} benindependent random variables. For any t >0, we wish to determine the smallestpn(t) satisfying
P
n
X
i=1
Yi≥t
!
≤pn(t). (1)
To reach this aim, numerous inequalities exist: Markov’s inequality, Tchebychev’s in- equality, Chernoff’s inequality, Berry-Esseen’s inequality, Bernstein’s inequality, Mac- Diarmid’s inequality, Fuk-Nagaev’s inequality, . . . See, e.g., [1, 2, 3, 4] and the references therein for details.
In this note, we investigate pn(t) in a non-standard case, as we merely suppose that supi∈{1,...,n}E |Yi|δ
exists for someδ∈[1,2). That is, we have no information on the existence of the variance and thus most of the common inequalities cannot be applied.
We determine two bounds: the first one is a direct consequence of Markov’s inequality and von Bahr-Esseen’s inequality (see Lemma 1 below), and the second one, which is more technical and original, offers a suitable alternative. Considering the Pareto distribution, we compare the quality of these bounds via a numerical study.
The note is organized as follows. Section 2 presents the result and the proof. Section 3 provides an application.
∗Mathematics Subject Classifications: 60E15.
†Laboratoire de Math´ematiques Nicolas Oresme, Universit´e de Caen Basse-Normandie, Cam- pus II, Science 3, 14032 Caen, France, chesneau@math.unicaen.fr, andre.sesboue@math.unicaen.fr, bulla@math.unicaen.fr.
2 Results
Theorem 1. Let n be a positive integer and (Yi)i∈{1,...,n} be n independent random variables such that, for any i∈ {1, . . . , n},
− E(Yi) = 0,
− E |Yi|δ
exists for someδ∈[1,2) (we have no a priori information on the exis- tence of a moment of order2 or it does not exist).
Then, for anyt >0, we have the two following bounds.
Bound 1:
P
n
X
i=1
Yi≥t
!
≤(2−n−1)t−δ
n
X
i=1
E |Yi|δ .
Bound 2:
P
n
X
i=1
Yi≥t
!
≤min
y>0gn(t, y), where
gn(t, y) = exp − t2
8 Pn
i=1E Yi21{|Yi|<y}
+ty/3
!
+ 22δ(2−n−1)t−δ
n
X
i=1
E
|Yi|δ1{|Yi|≥y}
.
The proof of Bound 1 uses Markov’s inequality and von Bahr-Esseen’s inequality, whereas the proof of Bound 2 is more technical (truncation techniques, Markov’s in- equality, Bernstein’s inequality, von Bahr-Esseen’s inequality,. . . ).
Proof of Theorem 1. We prove Bounds 1 and 2 in turns.
Proof of Bound 1. We need the following version of the von Bahr-Esseen inequality (see [5]).
Lemma 1. (von Bahr-Esseen’s inequality) Let n be a positive integer, p ∈ [1,2) and (Xi)i∈{1,...,n} be n independent random variables such that, for any i∈ {1, . . . , n},E(Xi) = 0andE(|Xi|p)<∞. Then
E
n
X
i=1
Xi
p!
≤(2−n−1)
n
X
i=1
E(|Xi|p).
Using Markov’s inequality and von Bahr-Esseen’s inequality (see Lemma 1), we obtain
P
n
X
i=1
Yi≥t
!
≤t−δE
n
X
i=1
Yi
δ
≤(2−n−1)t−δ
n
X
i=1
E |Yi|δ .
Bound 1 is proved.
Proof of Bound 2. For any random eventA, let 1A be the indicator function onA.
Set
V =
n
X
i=1
Yi1{|Yi|≥y}−E Yi1{|Yi|≥y}
and
W =
n
X
i=1
Yi1{|Yi|<y}−E Yi1{|Yi|<y}
.
SinceE Yi1{|Yi|≥y}
+E Yi1{|Yi|<y}
=E(Yi) = 0, we haveV +W =Pn i=1Yi. Using{V +W ≥t} ⊆ {V ≥t/2} ∪ {W ≥t/2}, we obtain
P
n
X
i=1
Yi≥t
!
=P(V +W ≥t)≤P
V ≥ t 2
∪
W ≥ t 2
≤A+B, (2)
where
A=P
V ≥ t 2
=P
n
X
i=1
Yi1{|Yi|≥y}−E Yi1{|Yi|≥y}
≥ t 2
!
and
B=P
W ≥ t 2
=P
n
X
i=1
Yi1{|Yi|<y}−E Yi1{|Yi|<y}
≥ t 2
! .
We treat boundAandB in turn.
Upper bound forA. For anyi∈ {1, . . . , n}, set
Xi=Yi1{|Yi|≥y}−E Yi1{|Yi|≥y}
.
We haveE(Xi) = 0. It follows from Markov’s inequality and von Bahr-Esseen’s inequality (see Lemma 1) applied with the independent variables (Xi)i∈{1,...,n}
that
A≤2δt−δE
n
X
i=1
Xi
δ
≤2δ(2−n−1)t−δ
n
X
i=1
E |Xi|δ
. (3)
Using the elementary inequality|x+y|a≤2a−1(|x|a+|y|a), (x, y)∈R2, a≥1, and Jensen’s inequality with the convex functionϕ(x) =|x|δ,x∈R, we obtain
E |Xi|δ
≤ 2δ−1 E
|Yi|δ1{|Yi|≥y}
+
E Yi1{|Yi|≥y}
δ
≤ 2δ−1 E
|Yi|δ1{|Yi|≥y}
+E
|Yi|δ1{|Yi|≥y}
= 2δE
|Yi|δ1{|Yi|≥y}
. (4)
Thus, from (3) and (4) follows A≤22δ(2−n−1)t−δ
n
X
i=1
E
|Yi|δ1{|Yi|≥y}
. (5)
The upper bound for B. We will utilize one of Bernstein’s inequalities (see, for instance, [3]), presented in the following.
Lemma 2. (Bernstein’s inequality) Let nbe a positive integer and(Xi)i∈{1,...,n}
ben independent random variables such that, for any i∈ {1, . . . , n},E(Xi) = 0 and|Xi| ≤M <∞. Then we have
P
n
X
i=1
Xi≥λ
!
≤exp
− λ2
2 (Pn
i=1E(Xi2) +λM /3)
,
for anyλ >0.
For anyi∈ {1, . . . , n}, set
Xi=Yi1{|Yi|<y}−E Yi1{|Yi|<y}
. We haveE(Xi) = 0 and
|Xi| ≤ |Yi|1{|Yi|<y}+E |Yi|1{|Yi|<y}
≤2y.
Therefore, Bernstein’s inequality (see Lemma 2) applied with the independent variables (Xi)i∈{1,...,n} and the parametersλ=t/2 andM = 2y gives
B≤exp
− t2
8 (Pn
i=1E(Xi2) +ty/3)
. SinceE Xi2
=V Yi1{|Yi|<y}
≤E Yi21{|Yi|<y}
for anyi∈ {1, . . . , n}, we have
B≤exp − t2
8 Pn
i=1E Yi21{|Yi|<y}
+ty/3
!
. (6)
Combining (2), (5) and (6), we obtain the inequality P
n
X
i=1
Yi≥t
!
≤ exp − t2
8 Pn
i=1E Yi21{|Yi|<y}
+ty/3
!
+ 22δ(2−n−1)t−δ
n
X
i=1
E
|Yi|δ1{|Yi|≥y}
.
Sincey >0 is arbitrary, we obtain the desired inequality.
2 Remark. For anyt >0, contrary to Bound 1, Bound 2 is always inferior to 1. Indeed, due to the dominated convergence theorem, we have limy→∞E
|Yi|δ1{|Yi|≥y}
= 0 and, since limy→∞Pn
i=1E Yi21{|Yi|<y}
+ty/3 =∞, P
n
X
i=1
Yi≥t
!
≤min
y>0gn(t, y)≤ lim
y→∞gn(t, y) = 1.
3 Application
Design of the study
Let (Yi)i∈{1,...,n}beni.i.d. random variables having the symmetric Pareto distribution with parametersi.e. Y1 has the probability density function
f(x) =
(((s−1)/2)|x|−s, if |x| ≥1,
0 otherwise.
Ifs∈(1 +δ,3) withδ∈[1,2), then E(Y1) = 0, E |Y1|δ
= s−1
s−δ−1, E |Y1|δ1{|Y1|≥y}
= s−1
s−δ−1min(y−s+δ+1,1), E Y121{|Y1|<y}
= s−1
3−s max(y3−s,1)−1 andE Y12
does not exist. For t >0 then holds by Theorem 1:
Bound 1:
P
n
X
i=1
Yi≥t
!
≤(2−n−1)t−δn s−1
s−δ−1. (7)
Bound 2:
P
n
X
i=1
Yi≥t
!
≤min
y>0gn(t, y), (8)
where
gn(t, y) = exp
− t2
8 (n(s−1) (max(y3−s,1)−1)/(3−s) +ty/3)
+ 22δ(2−n−1)t−δn s−1
s−δ−1min(y−s+δ+1,1).
Numerical results
In what follows, we present numerical results for the bounds (7) and (8). We consider two examples: first, a large value of n (5000), secondly a small value of n (50). For the sake of simplicity, we takes= 3−10−10. Following the philosophy of reproducible research, the programs are made available freely for download at the address
http://www.math.unicaen.fr/∼chesneau/concentration2final.r
This code contains the scripts to reproduce Figures 1 and 2, and it requires at least R [6] to run properly.
Figure 1: Empirical boundary values for largen
This figure displays the values of Bound 1 and 2 for varying values oftandδ. For all four panels,n equals 5000. The horizontal gray line represents bound value of 1.
delta = 1.0
t
p
2000 4000 6000 8000 10000 12000 14000
0.050.21 Bound 1
Bound 2
delta = 1.3
t
p
2000 4000 6000 8000 10000 12000 14000
0.050.21
delta = 1.6
t
p
2000 4000 6000 8000 10000 12000 14000
0.050.21
delta = 1.9
t
p
2000 4000 6000 8000 10000 12000 14000
0.050.21
Figure 1 displays the first case, in which n takes the value 5000. The three panels display the evolution of Bound 1 and 2 for different values of δ. More precisely, the values are 1.0, 1.3, 1.6 and 1.9 from top to bottom in the four panels. Note that, for
δ= 1.0 and the considered values oft, Bound 1 is greater than 1.
Figure 2: Empirical boundary values for smalln
This figure displays the values of Bound 1 and 2 for varying values oftandδ. For all four panels,n equals 50. The horizontal gray line represents a bound value of 1.
delta = 1.0
t
p
500 1000 1500 2000
0.050.21 Bound 1
Bound 2
delta = 1.3
t
p
500 1000 1500 2000
0.050.21
delta = 1.6
t
p
500 1000 1500 2000
0.050.21
delta = 1.9
t
p
500 1000 1500 2000
0.050.21
It is visible that Bound 2 is (clearly) lower than Bound 1 for values of δ lying closer
to 1, whereas Bound 1 should be preferred when δ approaches two. Subsequently, Figure 2 deals with the case of smalln, more precisely the value isn= 50. The results correspond to those of n= 5000, although the switch from Bound 2 to Bound 1 for increasing δ should be carried out earlier. However, for practical purposes, this case may only be of limited interest.
References
[1] F.Chung and L. Lu, Concentration inequalities and martingale inequalities — a survey, Internet Math., 3 (2006-2007), 79–127.
[2] D.H. Fuk and S.V. Nagaev, Probability inequalities for sums of independent random variables, Theor. Probab. Appl., 16(1971), 643–660.
[3] V.V. Petrov, Limit Theorems of Probability Theory, Clarendon Press, Oxford, 1995.
[4] D. Pollard, Convergence of Stochastic Processes, Springer, New York, 1984.
[5] B. von Bahr and C-.G. Esseen, Inequalities for therth Absolute Moment of a Sum of Random Variables, 1≤r≤2, The Annals of Mathematical Statistics, 36 (1965), 299-303.
[6] R Development Core Team, R: A Language and Environment for Statistical Computing, R Foundation for Statistical Computing, Vienna (Austria), 2010, http://www.R-project.org.
