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

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1 A tail boud for um of idepedet radom variable : applicatio to the ymmetric Pareto ditributio Chritophe Cheeau To cite thi verio: Chritophe Cheeau. A tail boud for um of idepedet radom variable : applicatio to the ymmetric Pareto ditributio. Short ote. 27. <hal-1925> HAL Id: hal Submitted o 28 Nov 27 HAL i a multi-dicipliary ope acce archive for the depoit ad diemiatio of cietific reearch documet whether they are publihed or ot. The documet may come from teachig ad reearch ititutio i Frace or abroad or from public or private reearch ceter. L archive ouverte pluridicipliaire HAL et detiée au dépôt et à la diffuio de documet cietifique de iveau recherche publié ou o émaat de établiemet d eeigemet et de recherche fraçai ou étrager de laboratoire public ou privé.

2 Vol A tail boud for um of idepedet radom variable : applicatio to the ymmetric Pareto ditributio Chritophe Cheeau Laboratoire de Mathématique Nicola Oreme Uiverité de Cae Bae-Normadie Campu II Sciece Cae Frace. cheeau@math.uicae.fr Abtract: I thi ote we prove a boud of the tail probability for a um of idepedet radom variable. It ca be applied uder mild aumptio; the variable are ot aumed to be almot urely abolutely bouded or admit fiite momet of all order. Moreover i ome cae it i igificatly better tha the boud obtaied via the tadard Markov iequality. To illutrate thi reult we ivetigate the boud of the tail probability for a um of weighted i.i.d. radom variable havig the ymmetric Pareto ditributio. AMS 2 ubject claificatio: 6E15. Keyword ad phrae: Tail boud ymmetric Pareto ditributio. 1. MOTIVATION Let Y i i N be idepedet radom variable. For ay N we wih to determie the mallet equece of fuctio p t uch that P Y i t p t t [ [. Thi problem i well-kow; umerou reult exit. The mot famou of them i the Markov iequality. Uder mild aumptio o the momet of the X i it give a polyomial boud p t. I may cae thi boud ca be improved. For itace if the X i are almot urely abolutely bouded or admit fiite momet of all order ad thee momet atify ome iequalitie the Bertei iequalitie provide better reult. The obtaied boud p t are expoetial. See Petrov 1995 ad Pollard 1984 for further detail ad complete bibliography. I thi ote we preet a ew iequality which provide a boud p t of the form p t = v t+w t where v t i polyomial ad w t i expoetial. It ca be applied uder mild aumptio o the X i ; a for the Markov iequality oly kowledge of the order of a fiite momet i required. The mai iteret of our iequality i that it ca be applied whe the Bertei coditio are 1 imart-geeric ver. 27/9/18 file: Sum-Pareto.tex date: November 2 27

3 Chritophe Cheeau/A tail boud for um of idepedet radom variable 2 ot atified ad ca give better reult tha the Markov iequality. I order to illutrate thi we ivetigate the boud of the tail probability for a um of weighted i.i.d. radom variable havig the ymmetric Pareto ditributio. Thi i particularly iteretig becaue the exact expreio of the ditributio of uch a um i really difficult to idetify. See for itace Ramay 26. Moreover there are ome applicatio i ecoomic actuarial ciece urvival aalyi ad queuig etwork. The ote i orgaized a follow. Sectio 2 preet the mai reult. I Sectio 3 we illutrate the ue of thi reult by coiderig the ymmetric Pareto ditributio. The techical proof are potpoed to Sectio MAIN RESULT Theorem 2.1 below preet a boud of the tail probability for a um of idepedet radom variable. A metioed i Sectio 1 it require kowledge oly of the order of a fiite momet. Theorem 2.1. Let Y i i N be idepedet radom variable. We uppoe that for ay N ad ay i {1... } we have w.l.o.g. EY i = there exit a real umber p 2 uch that for ay N ad ay i {1...} we have E Y i p <. The for ay t > ad ay N we have P Y i t C p t p max r p tr 2 t p/2 + exp t b where for ay u {2p} r u t = Y E i u 1 { Yi 3b ad C p = 2 2p+1 max p p p p/2+1 e p x p/2 1 1 x p dx. b = E Yi 2 The proof of Theorem 2.1 ue trucatio techic the Roethal iequality ad oe of the Bertei iequalitie. See Roethal 197 ad Petrov Clearly Theorem 2.1 ca be applied for a wide cla of radom variable. However if the variable are almot urely abolutely bouded or have fiite momet of all order the Bertei iequalitie ca give more optimal reult tha 2.1. But whe thee coditio are ot atified Theorem 2.1 become of iteret. Thi fact i illutrated i Sectio 3 below for the ymmetric Pareto ditributio. Other example ca be tudied i a imilar fahio. 3. APPLICATION: SYMMETRIC PARETO DISTRIBUTION Propoitio 3.1 below ivetigate the boud of the tail probability for a um of weighted i.i.d. radom variable havig the ymmetric Pareto ditributio. imart-geeric ver. 27/9/18 file: Sum-Pareto.tex date: November 2 27

4 Chritophe Cheeau/A tail boud for um of idepedet radom variable 3 Propoitio 3.1. Let > 2 ad X i i N be i.i.d. radom variable with the probability deity fuctio fx = 2 1 x 1 1 { x 1}. Let a i i N be a equece of ozero real umber uch that a i <. The for ay N ay t 3b ρ where ρ = a i 1/ ad ay p 2 we have P a i X i t K p t 2p+ b p a i + exp t b where b = 2 a2 i K p = 3 p max p 2 2p+1 max p p p p/2+1 e p x p/2 1 1 x p dx. 2 p/2 C p ad C p = Notice that ice the ditributio of the variable i ymmetric the cotat C p aociated to the Roethal iequality ca be improved. For it optimal form we refer to Ibragimov ad Sharakhmetov I the literature there exit everal reult o the approximatio of the tail probability of a um of i.i.d. radom variable havig the ymmetric Pareto ditributio. But to our kowledge cotrary to Propoitio 3.1 thee reult are aymptotic i.e. t. See for itace Goovaert Kaa Laeve Tag ad Veric 25. Illutratio. Here we coider a imple example to compare the preciio betwee 3.1 ad the boud obtaied via the Markov iequality. Let > 2 ad X i i N be i.i.d. radom variable with the probability deity fuctio fx = 2 1 x 1 1 { x 1}. For ay iteger uch that 1/2 1/ log 1/2 > 2 3/2 3 1/2 ad ay p max 2 2 if we take t = t = 2 3/2 log 1/2 the we ca balace the two term of the boud i 3.1; there exit two cotat Q 1 > ad Q 2 > uch that P X i t Q 1 1 /2 log p+/2 + 1 /2 Q 2 1 / Uder the ame framework for ay p < the Markov iequality combied with the Roethal iequality ee Lemma 4.1 below implie the exitece of two cotat R 1 > ad R 2 > uch that P X i t t p E p X i R 1 t p p/2 R 2 log p/ Therefore for large eough the rate of covergece i 3.2 i really fater tha thoe i 3.3. I thi cae 3.1 give a better reult tha the Markov iequality. imart-geeric ver. 27/9/18 file: Sum-Pareto.tex date: November 2 27

5 Chritophe Cheeau/A tail boud for um of idepedet radom variable 4 4. PROOFS. Proof of Theorem 2.1. Let N. For ay t > we have P Y i t = P Y i EY i t U + V where ad U = P V = P Y i 1 { Yi 3b E Y i 1 { Yi 3b t 2 Y i 1 { Yi < 3b E Y i 1 { Yi < 3b t. 2 Let u boud U ad V i tur. The upper boud for U. The Markov iequality yield U 2 p t p E p Y i 1 { Yi 3b E Y i 1 { Yi 3b. 4.1 Now let u itroduce the Roethal iequality. See Roethal 197. Lemma 4.1 Roethal iequality. Let p 2 ad X i i N be idepedet radom variable uch that for ay N ad ay i {1... } we have EX i = ad E X i p <. The we have p E X i c p max E X i p E X 2 p/2 i where c p = 2 max p p p p/2+1 e p x p/2 1 1 x p dx. For ay i {1... } et Z i = Y i 1 { Yi 3b E Y i 1 { Yi 3b. Sice EZ i = ad E Z i p 2 p E Y i p 1 { Yi 2 p E Y 3b i p < Lemma 4.1 applied to the idepedet variable Z i i N give p E Z i c p max E Z i p E Z 2 p/2 i 4.2 where c p = 2max p p p p/2+1 e p x p/2 1 1 x p dx. imart-geeric ver. 27/9/18 file: Sum-Pareto.tex date: November 2 27

6 Chritophe Cheeau/A tail boud for um of idepedet radom variable 5 It follow from 4.1 ad 4.2 that U 2 p t p c p max E Z i p E Zi 2 2 2p t p c p max E Y i p 1 { Yi 3b p/2 E Yi 2 1 { Yi 3b = C p t p max r p tr 2 t p/2 4.3 where C p = 2 2p c p. The upper boud for V. Let u preet oe of the Bertei iequalitie. See for itace Petrov Lemma 4.2 Bertei iequality. Let X i i N be idepedet radom variable uch that for ay N ad ay i {1... } we have EX i = ad X i M <. The for ay λ > ad ay N we have λ 2 P X i λ exp 2d 2 + λm 3 where d 2 = EX2 i. For ay i {1... } et Z i = Y i 1 { Yi < 3b E Y i 1 { Yi < 3b. Sice EZ i = ad Z i Y i 1 { Yi < 3b + E Y i 1 { Yi < 3b 6b t Lemma 4.2 applied with the idepedet variable Z i i N ad the parameter λ = t 2 ad M = 6b t give V exp 8 Y V i 1 { Yi < 3b t 2 + t 6b. 6 t Sice Y V i 1 { Yi < 3b E Yi 2 = b it come V exp t b Puttig 4.3 ad 4.4 together we obtai the iequality P Y i t U + V C p t p max r p t r 2 t p/2 + exp t2. 16b Theorem 2.1 i proved. imart-geeric ver. 27/9/18 file: Sum-Pareto.tex date: November 2 27

7 Chritophe Cheeau/A tail boud for um of idepedet radom variable 6. Proof of Propoitio 3.1. Let N. Set for ay i {1... } Y i = a i X i. Clearly Y i i N are idepedet radom variable uch that EY i = a i EX i = ad EY i 2 = 2 a2 i <. I order to apply Theorem 2.1 let u boud the term r u t = Y E i u 1 { Yi 3b = a i u E X i u 1 { } for ay u {2p} ad ay p max X i 3b 2 2. a i t Recall that ρ = a i 1/. Sice t 3b ρ 3b σ where σ = up... a i we have E X i u 1 { } = X i 3b 3b x u 1 dx = a i t a i t u 3b a i t u. Hece r u t = u 3b a i. u t Therefore max r p tr 2 t p/2 p 3b p/2 3b R p max 3b t tρ tρ where R p = max p 2 p/2. 3b 3b p/2 Sice t 3b ρ ad p > 2 we have max tρ tρ 3b tρ. Hece max r p tr 2 t p/2 p 3b R p a i. 4.5 t Puttig 4.5 i Theorem 2.1 we obtai P a i X i t K p t 2p+ b p a i + exp t2 16b where b = 2 a2 i K p = 3 p max p 2 = p/2 C p ad C p = 2 2p+1 max p p p p/2+1 e p x p/2 1 1 x p dx. Propoitio 3.1 i proved. Referece Goovaert M. Kaa R. Laeve R. Tag Q. ad Veric R. 25. The Tail Probability of Dicouted Sum of Pareto-like Loe i Iurace. Scadiavia Actuarial Joural Iue 6 November 25 pp imart-geeric ver. 27/9/18 file: Sum-Pareto.tex date: November 2 27

8 Chritophe Cheeau/A tail boud for um of idepedet radom variable 7 Ibragimov R. ad Sharakhmetov Sh O a exact cotat for the Roethal iequality Theory Probab. Appl. 42 pp Petrov V. V Limit Theorem of Probability Theory Claredo Pre Oxford. Pollard D Covergece of Stochatic Procee Spriger New York. Ramay Coli M. 26 The ditributio of um of certai i.i.d. Pareto variate. Commu. Stat. Theory Method. 35 No.1-3 pp Roethal H. P O the ubpace of L p p 2 paed by equece of idepedet radom variable Irael Joural of Mathematic 8: pp imart-geeric ver. 27/9/18 file: Sum-Pareto.tex date: November 2 27