c
Applied Mathematics E-Notes, 9(2009), 300-306 Available free at mirror sites of http://www.math.nthu.edu.tw/∼amen/
ISSN 1607-2510
A Tail Bound For Sums Of Independent Random
Variables And Application To The Pareto
Distribution∗
Christophe Chesneau†
Received 11 December 2008
Abstract
We prove a bound of the tail probability for a sum of n independent random
variables. It can be applied under mild assumptions; the variables are not assumed
to be almost surely absolutely bounded, or admit finite moments of all orders. In
some cases, it is better than the bound obtained via the Fuk-Nagaev inequality.
To illustrate this result, we investigate the bound of the tail probability for a sum
of n weighted i.i.d. random variables having the symmetric Pareto distribution.
1
Introduction
∗
Let (Yi )i∈N∗ be a sequence of independent random variables. For any
Pnn ∈ N , we wish to
determine the smallest sequence of functions pn (t) such that P ( i=1 Yi ≥ t) ≤ pn (t),
t ∈ [0, ∞[. This problem is well-known; numerous results exist. The most famous
of them is the Markov inequality. Under mild assumptions on the moments of the
Xi ’s, it gives a polynomial bound pn (t). Under the same assumptions, this bound
can be improved by the Fuk-Nagaev inequality (see [2]). If the Xi ’s are almost surely
absolutely bounded, or admit finite moments of all orders (and these moments satisfy
some inequalities), the Bernstein inequalities provide better results. See [6] and [7] for
further details.
In this note, we present a new bound pn (t). It can be applied under mild assumptions on the Xi ’s; only knowledge of the order of a finite moment is required. The
main interest of the proposed inequality is that it can be applied when the ‘Bernstein
conditions’ are not satisfied, and can give better results than the Fuk-Nagaev inequality (and the Markov inequality). The tail probability for a sum of n weighted i.i.d.
random variables having the symmetric Pareto distribution is studied. This is particularly interesting because the exact expression of the distribution of such a sum is
really difficult to identify (see [8]). Moreover, there are some applications in economics,
actuarial science, survival analysis and queuing networks.
∗ Mathematics
Subject Classifications: 60E15
Mathématiques Nicolas Oresme, Université de Caen Basse-Normandie, Campus II,
Science 3, 14032 Caen, France
† Laboratoire de
300
301
C. Chesneau
The note is organized as follows. Section 2 presents a general tail bound. An
application of this bound to the Pareto distribution can be found in Section 3. Section 4 provides a comparative study of the considered inequality and the Fuk-Nagaev
inequality.
2
A General Tail Bound
Theorem 1 below presents a bound of the tail probability for a sum of n independent
random variables. As for the Fuk-Nagaev inequality (and the Markov inequality), it
requires knowledge only of the order of a finite moment.
THEOREM 1. Let (Yi )i∈N∗ be a sequence of independent random variables. We
suppose that
− for any n ∈ N∗ and any i ∈ {1, . . . , n}, E(Yi ) = 0,
− there exists a real number p ≥ 2 such that, for any n ∈ N∗ and any i ∈ {1, . . . , n},
E(|Yi |p ) < ∞.
Then, for any t > 0 and any n ∈ N∗ , we have
!
n
X
t2
p/2
P
Yi ≥ t ≤ Cp t−p max rn,p(t), (rn,2 (t))
+ exp −
,
16bn
(1)
i=1
where, for any u ∈ {2, p}, rn,u (t) =
Pn
u
2
E
|Y
|
1
3b
n
i
i=1
i=1 E Yi ,
{|Yi |≥ } , bn =
Pn
t
Cp = 22p cp , and cp refers to the Rosenthal inequality (see Lemma 1 below).
PROOF. Let n ∈ N∗ . For any t > 0, we have
!
!
n
n
X
X
P
Yi ≥ t = P
(Yi − E (Yi )) ≥ t ≤ U + V,
i=1
where
U =P
i=1
n X
i=1
and
V =P
n X
i=1
t
Yi 1{|Yi |≥ 3bn } − E Yi 1{|Yi|≥ 3bn }
≥
t
t
2
t
Yi 1{|Yi|< 3bn } − E Yi 1{|Yi|< 3bn }
≥
t
t
2
!
!
.
Let us bound U and V , in turn.
The upper bound for U . The Markov inequality yields
!
n X
p
p −p
U ≤2 t E Yi 1{|Yi|≥ 3bn } − E Yi 1{|Yi |≥ 3bn } .
t
t
i=1
Now, let us introduce the Rosenthal inequality (see [9]).
(2)
302
Tail Bound for Sums of Independent Random Variables
LEMMA 1 (Rosenthal’s inequality). Let p ≥ 2 and (Xi )i∈N∗ be a sequence of
independent random variables such that, for any n ∈ N∗ and any i ∈ {1, . . . , n},
E(Xi ) = 0 and E(|Xi |p ) < ∞. Then we have

p !
!p/2 
n
n
n
X
X
X
,
E Xi ≤ cp max 
E (|Xi |p ) ,
E Xi2
i=1
i=1
i=1
where, for any τ > p/2, cp = 2 max τ p , pτ p/2 eτ
R∞
0
xp/2−1 (1 − x)−τ dx .
Under mild assumptions on the Xi ’s, the constant cp of Lemma 1 can be improved.
We refer to [1], [4] and [10].
For any i ∈ {1, . . . , n}, set Zi = Yi 1{|Yi |≥ 3bn } − E Yi 1{|Yi |≥ 3bn } . Since E(Zi ) = 0
t
t
and E (|Zi |p ) ≤ 2p E |Yi |p 1{|Yi |≥ 3bn }
t
independent variables (Zi )i∈N∗ gives
≤ 2p E (|Yi |p ) < ∞, Lemma 1 applied to the

p !
!p/2 
n
n
n
X
X
X
.
E ≤ cp max 
Zi E (|Zi |p ) ,
E Zi2
i=1
i=1
(3)
i=1
It follows from (2) and (3) that
U
≤
p −p
2 t
n
X
cp max
p
E (|Zi | ) ,
i=1
≤
=

22pt−p cp max 
n
X
E
i=1
n
X
i=1
Zi2
!!
n
X
E Yi2 1{|Yi |≥ 3bn }
E |Yi |p 1{|Yi|≥ 3bn } ,
t
t
i=1
p/2
Cpt−p max rn,p (t), (rn,2(t))
,
!p/2 

(4)
where Cp = 22p cp .
The upper bound for V . Let us present one of the Bernstein inequalities. See, for
instance, [6].
LEMMA 2 (Bernstein’s inequality). Let (Xi )i∈N∗ be a sequence of independent
random variables such that, for any n ∈ N∗ and any i ∈ {1, . . . , n}, E(Xi ) = 0 and
|Xi | ≤ M < ∞. Then, for any λ > 0 and any n ∈ N∗ , we have
n
X
P
i=1
where d2n =
Pn
i=1
!
Xi ≥ λ
λ2
≤ exp −
2(d2n + λM
3 )
!
,
E(Xi2 ).
For any i ∈ {1, . . . , n}, set Zi = Yi 1{|Yi |< 3bn } − E Yi 1{|Yi |< 3bn } . Since E(Zi ) = 0
t
t
6bn
and |Zi | ≤ |Yi |1{|Yi|< 3bn } + E |Yi |1{|Yi|< 3bn } ≤ t , Lemma 2 applied with the
t
t
303
C. Chesneau
independent variables (Zi )i∈N∗ and the parameters λ = 2t and M = 6btn , gives


2
t
V ≤ exp − P
 .
n
t 6bn
8
V
Y
1
3bn
+
i
i=1
6
t
{|Yi|< t }
Pn
Pn
Since i=1 V Yi 1{|Yi |< 3bn } ≤ i=1 E Yi2 = bn , it comes
t
t2
V ≤ exp −
.
16bn
(5)
Putting (4) and (5) together, we obtain the inequality
!
n
X
t2
p/2
−p
.
P
Yi ≥ t ≤ U + V ≤ Cp t max rn,p(t), (rn,2 (t))
+ exp −
16bn
i=1
Theorem 1 is proved.
Theorem 1 can be applied for a wide class of random variables. However, if the
variables are almost surely absolutely bounded, or have finite moments of all orders
satisfying some inequalities, the Bernstein inequalities can give more optimal results
than (1). When it is hard or not possible to prove that these conditions are satisfied,
Theorem 1 becomes of interest. This is illustrated in the example below and in Section
3 for the symmetric Pareto distribution. Other examples can be studied in a similar
fashion.
EXAMPLE. Let (Xi )i∈N∗ be i.i.d. random variables such that, for any x > 0,
γ
P(|X1 | ≥ x) ≤ ce−x , γ > 0, c > 0. Taking
t = tn = (κ16E(X12 )n log n)1/2
with κ > 0, the inequality (1) implies the existence of a constant Fp > 0 such that, for
n and p large enough,
!
1/2
n
X
3bn
t2n
)
+
exp
−
P
Xi ≥ tn
≤ Fp t−p
n
P(|X
|
≥
1
n
tn
16bn
i=1
0
2 ) 1/2
3 nE(X1
− 12 @
(16κ log n)1/2
(
)
≤ n−p/2+1 e
We used the inequality: rn,u (t) ≤ n E |X1 |2u
3
1/2 1γ
A
+ n−κ ≤ 2n−κ .
P(|X1 | ≥
1/2
3bn
tn )
, u ∈ {1, 2}.
Application to the Pareto Distribution
Proposition 1 below investigates the bound of the tail probability for a sum of n
weighted i.i.d. random variables having the symmetric Pareto distribution.
304
Tail Bound for Sums of Independent Random Variables
PROPOSITION 1. Let s > 2 and (Xi )i∈N∗ be i.i.d. random variables with the
probability density function
(
2−1 s|x|−s−1 , if |x| ≥ 1,
f(x) =
(6)
0
otherwise.
Pn
Let (ai )i∈N∗ be a sequence
ofnonzero real numbers such that i=1 |ai |s < ∞. Then,
P
1/s
n
for any n ∈ N∗ , any t ∈ 0, 3b
with ρn = ( ni=1 |ai |s )
and any p ∈ max( s2 , 2), s ,
ρn
we have
!
n
n
X
X
t2
−2p+s p−s
s
P
ai Xi ≥ t ≤ Kp t
bn
|ai | + exp −
,
(7)
16bn
i=1
i=1
p/2 Pn
s
s
s
2
p−s
where bn = s−2
a
,
K
=
3
max
,
22p cp ,
p
i=1 i
s−p
s−2


 1 + 2p/2 π −1/2 Γ p + 1 , 2 < p < 4,
2
cp =

 E (|θ − θ |p ) ,
p ≥ 4,
1
2
R∞
Γ(a) = 0 xa−1 e−x dx, a > 1, and θ1 , θ2 are independent Poisson random variables
with parameter 2−1 .
PROOF. Let n ∈ N∗ . Set, for any i ∈ {1, . . . , n}, Yi = ai Xi . Clearly, (Yi )i∈N are
independent random variablessuch that, for any i ∈ {1, . . . , n}, E(Yi ) = ai E(Xi ) = 0
and, for any p ∈ max( s2 , 2), s , E(|Yi |p ) ≤ supi=1,...,n |ai P
|pE(|Xi |p ) < ∞. WeP
are now
n
n
s
2
2
in the position to apply Theorem 1. We have bn =
E(Y
)
=
i
i=1
i=1 ai .
s−2
s
For any u ∈ {2, p} and any p ∈ max( 2 , 2), s , let us investigate the bound for
Pn
P
rn,u (t) = i=1 E |Yi |u 1{|Yi |≥ 3bn } = ni=1 |ai |u E |Xi |u 1n|X |≥ 3bn o . Recall that
t
i
|ai |t
Pn
1/s
3bn
s
n
ρn = ( i=1 |ai | )
and σn = supi=1,...,n |ai |. Since t ∈ 0, ρn ⊆ 0, 3b
σn , for any
u−s
R∞
3bn
s
i ∈ {1, . . . , n}, we have E |Xi |u 1n|X |≥ 3bn o = s 3bn xu−s−1dx = s−u
.
|ai |t
i
Therefore,
rn,u (t) =
s
s−u
3bn
t
and
p/2
max rn,p(t), (rn,2 (t))
where Rp = max
s
s−p ,
s
s−2
≤ Rp
p/2 |ai |t
|ai |t
3bn
t
p
u−s X
n
|ai |s
i=1

3bn
max 
tρn
−s
,
3bn
tρn
−s !p/2
n
. Since t ∈ 0, 3b
and p > 2, we have
ρn


−s −s !p/2
−s
3bn
3bn
 = 3bn
,
.
max 
tρn
tρn
tρn

,
305
C. Chesneau
Hence,
p−s X
n
3bn
p/2
|ai |s .
max rn,p(t), (rn,2 (t))
≤ Rp
t
(8)
i=1
Theorem 1 and (8) imply that
P
n
X
ai Xi ≥ t
i=1
s
s−2
where bn =
!
≤ Kp t−2p+s bp−s
n
n
X
t2
|ai |s + exp −
,
16bn
p/2 i=1
Pn
2
p−s
max
i=1 ai , Kp = 3
s
s−p ,
s
s−2
Cp , and Cp = 22p cp . Using
the optimal form of the Rosenthal constant cp for the symmetric random variables (see
[1], [4] and [10]), we complete the proof of Proposition 1.
For recent asymptotic results on the approximation of the tail probability of a sum
of n i.i.d. random variables having the symmetric Pareto distribution, we refer to [3].
Section 4 below compares the precision between (7) and the bound obtained via
the Fuk-Nagaev inequality.
4
Comparison with the Fuk-Nagaev Inequality
For the sake of clarity, recall the Fuk-Nagaev inequality. The considered version can
be found in [5].
LEMMA 3. (Fuk-Nagaev inequality) Let p ≥ 2 and (Yi )i∈N∗ be a sequence of
independent random variables such that, for any n ∈ N∗ and any i ∈ {1, . . . , n},
E(Yi ) = 0 and E(Yi2 ) < ∞. Then, for any t > 0, we have
n
X
P
Yi ≥ t
i=1
≤
n
X
!
−p
P(Yi ≥ ηt) + (ηt)
i=1
where dn =
Pn
i=1
n
X
E
i=1
E(Yi2 ), η =
p
,
p+2
and c∗p =
Yip 1{0≤Yi≤ηt}
t2
+ exp − ∗
cp d n
,
(9)
ep (p+2)2
.
2
In some cases, (7) can give better results than the Fuk-Nagaev inequality. For
instance, consider the symmetric Pareto distribution (i.e. (Xi )i∈N∗ are i.i.d. with
the probability density
function (6) with s > 2). Suppose that n is large. For any
p ∈ max( 2s , 2), s , if we take t = tn = 23/2 (sn log n)1/2 (∈ 0, 3n1−1/s ), then we can
balance the two terms of the bound in (7); there exists a constant Qp > 0 such that
P
n
X
i=1
Xi ≥ tn
!
≤ Qp n1−s/2(log n)−p+s/2 + n1−s/2 ≤ 2n1−s/2.
(10)
306
Tail Bound for Sums of Independent Random Variables
For the same tn , the Fuk-Nagaev inequality (9) implies the existence of a constant
Rp > 0 such that, for any p > 0,
!
n
X
16
∗ (1−s/2)
P
Xi ≥ tn
≤ Rp n1−p/2 (log n)−p/2 + n cp
i=1
≤
p
16
∗ (1−s/2)
2 max n1−p/2 , n cp
.
(11)
2
Since c∗p = e (p+2)
> 8e2 > 16, for n large enough, the rate of convergence in (10)
2
is faster than the one in (11). Therefore, in this case, (7) gives a better result than
the Fuk-Nagaev inequality. This superiority can be extended to tn = κ(n log n)1/2 for
0 < κ < 23/2 s1/2 .
Acknowledgment. The author wishes to thank the reviewer for his/her valuable
comments and suggestions.
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