A tail bound for sums of independent random variables : application to the symmetric Pareto distribution

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A tail bound for sums of independent random variables : application to the symmetric Pareto distribution

Christophe Chesneau

To cite this version:

Christophe Chesneau. A tail bound for sums of independent random variables : application to the

symmetric Pareto distribution. 2007. �hal-00192500�

A tail bound for sums of independent random variables : application to the

symmetric Pareto distribution

Christophe Chesneau

Laboratoire de Math´ematiques Nicolas Oresme, Universit´e de Caen Basse-Normandie,

Campus II, Science 3, 14032 Caen, France.

http://www.chesneau-stat.com chesneau@math.unicaen.fr

Abstract:In this note we prove a bound of the tail probability for a sum of nindependent 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. Moreover, in some cases, it is signifi- cantly better than the bound obtained via the standard Markov inequality.

To illustrate this result, we investigate the bound of the tail probability for a sum ofnweighted i.i.d. random variables having the symmetric Pareto distribution.

AMS 2000 subject classifications:60E15.

Keywords and phrases:Tail bound, symmetric Pareto distribution.

1. MOTIVATION

Let (Yi)i∈N^∗ be independent random variables. For any n ∈ N^∗, we wish to determine the smallest sequence of functionspn(t) such that

P Ã _n

X

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 theXi’s, it gives a polynomial boundpn(t). In many cases, this bound can be improved. For instance, if theXi’s are almost surely absolutely bounded, or admit finite mo- ments of all orders (and these moments satisfy some inequalities), the Bernstein inequalities provide better results. The obtained boundspn(t) are exponential.

SeePetrov(1995) and Pollard(1984) for further details and complete bibliog- raphy.

In this note, we present a new inequality which provides a boundpn(t) of the formpn(t) =vn(t) +wn(t), wherevn(t) is polynomial, andwn(t) is exponential.

It can be applied under mild assumptions on theXi’s; as for the Markov inequal- ity, only knowledge of the order of a finite moment is required. The main interest

Christophe Chesneau/A tail bound for sums of independent random variables 2

not satisfied, and can give better results than the Markov inequality. In order to illustrate this, we investigate the bound of the tail probability for a sum of n weighted i.i.d. random variables having the symmetric Pareto distribution.

This is particularly interesting because the exact expression of the distribution of such a sum is really difficult to identify. See, for instance,Ramsay (2006).

Moreover, there are some applications in economics, actuarial science, survival analysis and queuing networks.

The note is organized as follows. Section2 presents the main result. In Sec- tion3 we illustrate the use of this result by considering the symmetric Pareto distribution. The technical proofs are postponed to Section4.

2. MAIN RESULT

Theorem 2.1 below presents a bound of the tail probability for a sum of n independent random variables. As mentioned in Section1, it requires knowledge only of the order of a finite moment.

Theorem 2.1. Let(Yi)i∈N^∗ be independent random variables. We suppose that

• for any n∈N^∗, and any i∈ {1, ..., n}, we have, w.l.o.g.,E(Yi) = 0,

• there exists a real number p ≥ 2 such that, for any n ∈ N^∗, and any i∈ {1, ..., n}, we haveE(|Yi|^p)<∞.

Then, for anyt >0, and any n∈N^∗, we have

P Ã _n

X

i=1

Yi≥t

!

≤Cpt^−pmax³

rn,p(t),(rn,2(t))^p/2´ + exp

µ

− t² 16bn

¶

, (2.1)

where, for anyu∈ {2, p},rn,u(t) =Pn i=1E³

|Yi|^u1{^|Yi|≥3bnt }

´

,bn=Pn i=1E¡

Y_i²¢ andCp= 2^2p+1max¡

p^p, p^p/2+1e^pR∞

0 x^p/2−1(1−x)^−pdx¢ .

The proof of Theorem2.1uses truncation technics, the Rosenthal inequality and one of the Bernstein inequalities. SeeRosenthal(1970) andPetrov(1995).

Clearly, Theorem 2.1 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, the Bernstein inequalities can give more optimal results than (2.1). But, when these conditions are not satisfied, Theorem2.1 becomes of interest. This fact is illustrated in Section3 below for the symmetric Pareto distribution. Other examples can be studied in a similar fashion.

3. APPLICATION: SYMMETRIC PARETO DISTRIBUTION Proposition 3.1 below investigates the bound of the tail probability for a sum ofnweighted i.i.d. random variables having the symmetric Pareto distribution.

Proposition 3.1. Let s >2 and(Xi)i∈N^∗ be i.i.d. random variables with the probability density function f(x) = 2⁻¹s|x|^−s−11{|x|≥1}. Let (ai)i∈N^∗ be a se- quence of nonzeros real numbers such that Pn

i=1|ai|^s < ∞. Then, for any n ∈ N^∗, any t ∈ (0,^3b_ρnⁿ), where ρn = (Pn

i=1|ai|^s)^1/s, and any p ∈ (2, s), we have

P Ã _n

X

i=1

aiXi≥t

!

≤Kp

¡t^−2p+sb^p−s_n ¢

n

X

i=1

|ai|^s+ exp µ

− t² 16bn

¶

, (3.1)

wherebn =³

s s−2

´Pn

i=1a²_i,Kp= 3^p−smax µ³

s s−p

´ ,³

s s−2

´p/2¶

Cp, and Cp = 2^2p+1max¡

p^p, p^p/2+1e^pR∞

0 x^p/2−1(1−x)^−pdx¢ .

Notice that, since the distribution of the variables is symmetric, the constant Cp (associated to the Rosenthal inequality) can be improved. For its optimal form, we refer toIbragimov and Sharakhmetov (1997).

In the literature, there exist several results on the approximation of the tail probability of a sum ofni.i.d. random variables having the symmetric Pareto distribution. But, to our knowledge, contrary to Proposition3.1, these results are asymptotic (i.e.t→ ∞). See, for instance,Goovaerts, Kaas, Laeven, Tang, and Vernic (2005).

Illustration. Here, we consider a simple example to compare the precision between (3.1) and the bound obtained via the Markov inequality.

Lets >2 and (Xi)i∈N^∗ be i.i.d. random variables with the probability density functionf(x) = 2⁻¹s|x|^−s−11{|x|≥1}. For any integernsuch thatn^1/2−1/s(logn)^−1/2>

2^3/2

3 s^1/2, and anyp∈¡

max(^s₂,2), s¢

, if we take t=tn= 2^3/2(snlogn)^1/2, then we can balance the two terms of the bound in (3.1); there exist two constants, Q1>0 andQ2>0, such that

P Ã _n

X

i=1

Xi≥tn

!

≤Q1n^1−s/2(logn)^−p+s/2+n^1−s/2≤Q2n^1−s/2. (3.2)

Under the same framework, for any p < s, the Markov inequality combined with the Rosenthal inequality (see Lemma 4.1 below) implies the existence of two constants,R1>0 andR2>0, such that

P Ã _n

X

i=1

Xi≥tn

!

≤t^−p_n E ï

¯

¯

¯

¯

n

X

i=1

Xi

¯

¯

¯

¯

¯

p!

≤R1t^−p_n n^p/2≤R2(logn)^−p/2. (3.3)

Therefore, forn large enough, the rate of convergence in (3.2) is really faster than those in (3.3). In this case, (3.1) gives a better result than the Markov inequality.

Christophe Chesneau/A tail bound for sums of independent random variables 4

4. PROOFS

. Proof of Theorem2.1.Letn∈N^∗. For anyt >0, we have

P Ã _n

X

i=1

Yi≥t

!

=P Ã _n

X

i=1

(Yi−E(Yi))≥t

!

≤U+V,

where

U =P Ã _n

X

i=1

³

Yi1{^|Yi|≥3bnt } −E³

Yi1{^|Yi|≥3bnt }

´´≥ t 2

!

and

V =P Ã _n

X

i=1

³

Yi1{^|Yi|<3bnt } −E³

Yi1{^|Yi|<3bnt }

´´≥ t 2

! . Let us boundU and V, in turn.

The upper bound forU.The Markov inequality yields

U ≤2^pt^−pE ï

¯

¯

¯

¯

n

X

i=1

³

Yi1{^|Yi|≥3bnt } −E³

Yi1{^|Yi|≥3bnt }

´´

¯

¯

¯

¯

¯

p!

. (4.1)

Now, let us introduce the Rosenthal inequality. SeeRosenthal(1970).

Lemma 4.1 (Rosenthal’s inequality). Let p≥2 and(Xi)i∈N∗ be independent random variables such that, for any n ∈ N^∗, and any i ∈ {1, ..., n}, we have E(Xi) = 0andE(|Xi|^p)<∞. Then we have

E ï

¯

¯

¯

¯

n

X

i=1

Xi

¯

¯

¯

¯

¯

p!

≤cpmax





n

X

i=1

E(|Xi|^p), Ã _n

X

i=1

E¡ X_i²¢

!p/2

,

wherecp= 2 max¡

p^p, p^p/2+1e^pR∞

0 x^p/2−1(1−x)^−pdx¢ .

For any i ∈ {1, ..., n}, set Zi = Yi1{^|Yi|≥3bnt } −E³

Yi1{^|Yi|≥3bnt }

´. Since E(Zi) = 0 and E(|Zi|^p) ≤ 2^pE³

|Yi|^p1{^|Yi|≥3bnt }

´ ≤ 2^pE(|Yi|^p) < ∞, Lemma 4.1applied to the independent variables (Zi)i∈N∗ gives

E ï

¯

¯

¯

¯

n

X

i=1

Zi

¯

¯

¯

¯

¯

p!

≤cpmax





n

X

i=1

E(|Zi|^p), Ã _n

X

i=1

E¡ Z_i²¢

!p/2

, (4.2)

wherecp= 2 max¡

p^p, p^p/2+1e^pR∞

0 x^p/2−1(1−x)^−pdx¢ .

It follows from (4.1) and (4.2) that

U ≤ 2^pt^−pcpmax à _n

X

i=1

E(|Zi|^p), Ã _n

X

i=1

E¡ Z_i²¢

!!

≤ 2^2pt^−pcpmax





n

X

i=1

E

³

|Yi|^p1{^|Yi|≥3bnt }

´ ,

à _n X

i=1

E

³

Y_i²1{^|Yi|≥3bnt }

´

!p/2



= Cpt^−pmax³

rn,p(t),(rn,2(t))^p/2´

, (4.3)

whereCp= 2^2pcp.

The upper bound forV.Let us present one of the Bernstein inequalities. See, for instance,Petrov(1995).

Lemma 4.2(Bernstein’s inequality). Let(Xi)i∈N∗ be independent random vari- ables such that, for anyn∈N^∗ and anyi∈ {1, ..., n}, we haveE(Xi) = 0 and

|Xi| ≤M <∞. Then, for anyλ >0, and any n∈N^∗, we have

P Ã _n

X

i=1

Xi≥λ

!

≤exp Ã

− λ² 2(d²_n+^λM₃ )

! ,

whered²_n=Pn

i=1E(X_i²).

For any i ∈ {1, ..., n}, set Zi = Yi1{^|Yi|<3bnt } −E³

Yi1{^|Yi|<3bnt }

´. Since E(Zi) = 0 and |Zi| ≤ |Yi|1{^|Yi|<3bnt }+E

³

|Yi|1{^|Yi|<3bnt }

´

≤ ^6b_tⁿ, Lemma 4.2 applied with the independent variables (Zi)i∈N^∗, and the parametersλ= ₂^t and M =^6b_tⁿ, gives

V ≤exp



− t²

8³ Pn

i=1V³

Yi1{^|Yi|<3bnt }

´+₆^t¡_6bn

t

¢´



.

SincePn i=1V³

Yi1{^|Yi|<3bnt }

´≤Pn i=1E¡

Y_i²¢

=bn, it comes

V ≤exp µ

− t² 16bn

¶

. (4.4)

Putting (4.3) and (4.4) together, we obtain the inequality

P Ã _n

X

i=1

Yi≥t

!

≤U+V ≤Cpt^−pmax³

rn,p(t),(rn,2(t))^p/2´ + exp

µ

− t² 16bn

¶ . Theorem2.1is proved.

Christophe Chesneau/A tail bound for sums of independent random variables 6

. Proof of Proposition 3.1. Let n ∈ N^∗. Set, for any i ∈ {1, ..., n}, Yi = aiXi. Clearly, (Yi)i∈N are independent random variables such that E(Yi) = aiE(Xi) = 0 and Pn

i=1E(Y_i²) = ³

s s−2

´Pn

i=1a²_i < ∞. In order to apply Theorem 2.1, let us bound the term rn,u(t) = Pn

i=1E³

|Yi|^u1{^|Yi|≥3bnt }

´ = Pn

i=1|ai|^uE µ

|Xi|^u1©

|Xi|≥_|^3bn

ai|t

ª

¶

for anyu∈ {2, p}, and anyp∈¡

max(^s₂,2), s¢ . Recall that ρn = (Pn

i=1|ai|^s)^1/s. Since t ∈ ³ 0,^3b_ρnⁿ´

⊆ ³ 0,^3b_σnⁿ´

, where σn = sup_i=1,...,n|ai|, we have E

µ

|Xi|^u1©

|Xi|≥_|^3bn

ai|t

ª

¶

= sR∞

3bn

|ai|tx^u−s−1dx =

³ s s−u

´ ³3bn

|ai|t

´u−s

.Hence,

rn,u(t) = µ s

s−u

¶ µ3bn

t

¶u−s n

X

i=1

|ai|^s.

Therefore,

max³

rn,p(t),(rn,2(t))^p/2´

≤Rp

µ3bn

t

¶p

max



 µ3bn

tρn

¶−s

, õ

3bn

tρn

¶−s!p/2

,

whereRp= max µ³

s s−p

´ ,³

s s−2

´p/2¶ .

Since t ∈ ³ 0,³_ρ^bn

n

´ and p > 2, we have max Ã

³3bn

tρn

´−s

, µ³

3bn

tρn

´−s¶p/2!

=

³3bn

tρn

´−s

.Hence,

max³

rn,p(t),(rn,2(t))^p/2´

≤Rp

µ3bn

t

¶p−s n

X

i=1

|ai|^s. (4.5)

Putting (4.5) in Theorem2.1, we obtain

P Ã _n

X

i=1

aiXi≥t

!

≤Kp¡

t^−2p+sb^p−s_n ¢

n

X

i=1

|ai|^s+ exp µ

− t² 16bn

¶ ,

wherebn =³

s s−2

´Pn

i=1a²_i,Kp = 3^p−smax µ³

s s−p

´ ,³

s s−2

´p/2¶

Cp, and Cp = 2^2p+1max¡

p^p, p^p/2+1e^pR∞

0 x^p/2−1(1−x)^−pdx¢

. Proposition3.1is proved.

References

Goovaerts, M., Kaas, R., Laeven, R., Tang, Q., and Vernic, R. (2005). The Tail Probability of Discounted Sums of Pareto-like Losses in Insurance.Scandina- vian Actuarial Journal, Issue 6, November 2005 , pp. 446 - 461

Ibragimov, R., and Sharakhmetov, Sh. (1997). On an exact constant for the Rosenthal inequality,Theory Probab. Appl., 42, pp. 294302.

Petrov, V. V. (1995). Limit Theorems of Probability Theory, Clarendon Press, Oxford.

Pollard, D. (1984). Convergence of Stochastic Processes, Springer, New York.

Ramsay, Colin M. (2006) The distribution of sums of certain i.i.d. Pareto vari- ates.Commun. Stat., Theory Methods. 35, No.1-3, pp. 395-405.

Rosenthal, H. P. (1970). On the subspaces ofL^p(p≥2) spanned by sequences of independent random variables,Israel Journal of Mathematics8: pp. 273-303.