TAIL BOUNDS FOR SUMS OF GEOMETRIC AND EXPONENTIAL VARIABLES

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1 TAIL BOUNDS FOR SUMS OF GEOMETRIC AND EXPONENTIAL VARIABLES SVANTE JANSON Abstract. We gve explct bounds for the tal probabltes for sums of ndependent geometrc or exponental varables, possbly wth dfferent parameters. 1. Introducton and notaton Let X n X, where n 1 and X, 1,..., n, are ndependent geometrc random varables wth possbly dfferent dstrbutons: X Ge wth 0 < 1,.e., PX k 1 k 1, k 1, 2, Our goal s to estmate the tal probabltes PX x. Snce X s ntegervalued, t suffces to consder nteger x. However, t s convenent to allow arbtrary real x, and we do so. We defne µ : E X E X 1, 1.2 : mn. 1.3 We shall see that plays an mportant role n our estmates, whch roughly speakng show that the tal probabltes of X decrease at about the same rate as the tal probabltes of Ge,.e., as for the varable X wth smallest and thus fattest tal. Recall the smple and well-known fact that 1.1 mples that, for any non-zero z such that z 1 < 1, E z X z k PX k k1 z 1 1 z z For future use, note that snce x ln1 x s convex on 0, 1 and 0 for x 0, ln1 x x ln1 y, 0 < x y < y Date: 28 June, 2014; typo corrected 24 September, Partly supported by the Knut and Alce Wallenberg Foundaton. 1

2 2 SVANTE JANSON Remark 1.1. The theorems and corollares below hold also, wth the same proofs, for nfnte sums X X, provded E X p 1 <. Acknowledgement. Ths work was ntated durng the 25th Internatonal Conference on Probablstc, Combnatoral and Asymptotc Methods for the Analyss of Algorthms, AofA14, n Pars-Jusseu, June 2014, n response to a queston by Donald Knuth. I thank Donald Knuth and Coln McDarmd for helpful dscussons. 2. Upper bounds for the upper tal We begn wth a smple upper bound obtaned by the classcal method of estmatng the moment generatng functon or probablty generatng functon and usng the standard nequalty an nstance of Markov s nequalty PX x z x E z X, z 1, 2.1 or equvalently PX x e tx E e tx, t Cf. the related Chernoff bounds for the bnomal dstrbuton that are proved by ths method, see e.g. [3, Theorem 2.1], and see e.g. [1] for other applcatons of ths method. See also e.g. [2, Chapter 2] or [4, Chapter 27] for more general large devaton theory. Theorem 2.1. For any p 1,..., p n 0, 1] and any λ 1, PX λµ e p µλ 1 ln λ. 2.3 Proof. If 0 t <, then e t 1 + t > 0, and thus by 1.4, E e tx e t p 1 + t 1 t Hence, f 0 t < mn, then E e tx E e tx 1 t and, by 2.2, PX λµ e tλµ E e tx exp tλµ + ln 1 t. 2.6 By 1.5 and 0 < / 1, we have, for 0 t <, ln 1 t p ln 1 t. 2.7 Consequently, 2.6 yelds PX λµ exp tλµ ln exp tλµ µ ln 1 t 1 t. 2.8

3 TAIL BOUNDS FOR GEOMETRIC AND EXPONENTIAL VARIABLES 3 Choosng t 1 λ 1 whch s optmal n 2.8, we obtan 2.3. As a corollary we obtan a bound that s generally much cruder, but has the advantage of not dependng on the s at all. Corollary 2.2. For any p 1,..., p n 0, 1] and any λ 1, PX λµ λe 1 λ eλe λ. 2.9 Proof. Use µ 1/ for each, and thus µ 1 n 2.3. Alternatvely, use t 1 λ 1 /µ n 2.8. The bound n Theorem 2.1 s rather sharn many cases. Also the cruder 2.9 s almost sharp for n 1 a sngle X and small p 1 ; n ths case µ 1/p 1 and PX λµ 1 p 1 λµ 1 exp λ + Oλp Nevertheless, we can mprove 2.3 somewhat, n partcular when mn s not small, by usng more careful estmates. Theorem 2.3. For any p 1,..., p n 0, 1] and any λ 1, PX λµ λ 1 1 λ 1 ln λµ The proof s gven below. We note that Theorem 2.3 mples a mnor mprovement of Corollary 2.2: Corollary 2.4. For any p 1,..., p n 0, 1] and any λ 1, Proof. Use 2.11 and 1 µ e p µ e 1. PX λµ e 1 λ We begn the proof of Theorem 2.3 wth two lemmas yeldng a mnor mprovement of 2.1 usng the fact that the varables are geometrc. The lemmas actually use only that one of the varables s geometrc. Lemma 2.5. For any ntegers j and k wth j k, PX j 1 j k PX k For any real numbers x and y wth x y, PX x 1 x y+1 PX y Proof.. We may wthout loss of generalty assume that p 1. Then, for any ntegers, j, k wth j k, PX j X X 1 PX 1 j 1 j 1 +, 2.15 and smlarly for PX k X X 1. Snce j 1 + j k + k 1 +, t follows that PX j X X 1 1 j k PX k X X for every, and thus 2.13 follows by takng the expectaton.

4 4 SVANTE JANSON. For real x and y we obtan from 2.13 PX x PX x 1 x y PX y 1 x y+1 PX y Lemma 2.6. For any x 0 and z 1 wth z1 < 1, PX x 1 z1 z x E z X Proof. Snce z 1, 2.13 mples that for every k 1, X 1 E z X Ez X 1{X k} E z k + z 1 z 1{X j k} E z k 1{X k} + z 1 z k PX k + z 1 jk z j 1{X j + 1} jk z j PX j + 1 jk z k PX k 1 + z 1 z j k 1 j+1 k jk z k PX k 1 + z 11 p 1 z1 z k PX k 1 z The result 2.18 follows when x k s a postve nteger. The general case follows by takng k max x, 1 snce then PX x PX k. Proof of Theorem 2.3. We may assume that < 1. Otherwse every 1 and X 1 a.s., so X n µ a.s. and the result s trval. We then choose.e., z : z 1 λ1 λ λ λ1, λ 1 λ ; 2.21 note that z 1 1 so z 1 and z 1 > 1 1 for every. Thus, by 1.4, E z X E z X z z 1 /. 2.22

5 TAIL BOUNDS FOR GEOMETRIC AND EXPONENTIAL VARIABLES 5 By 2.22, 2.7 wth t 1 z 1 < and 2.21, ln E z X ln 1 1 z 1 ln 1 1 z 1 Furthermore, by 2.20, p ln 1 λ 1 µ ln 1 µ ln λ. λ λ 1 1 z1 Hence, Lemma 2.6, 2.20 and 2.23 yeld where ln PX λµ ln λ λµ ln z + ln E z X λ /λ 1 λ ln λ λµ ln λ λ1 + µ ln λ 1 ln λ + λµ ln1 + µfλ, 2.25 fλ : λ ln λ + ln λ λ 1 λ lnλ + λ ln λ ln We have f1 ln1 and, for λ 1, usng 1.5, f λ lnλ + ln λ ln 1 p 1 λ λ ln Consequently, by ntegratng 2.27, for all λ 1, fλ ln1 ln λ ln1, 2.28 and the result 2.11 follows by Remark 2.7. Note that for large λ, the exponents above are roughly lnear n λ, whle for λ 1+o1 we have λ 1 ln λ 1 2 λ 12 so the exponents are quadratc n λ 1. The latter s to be expected from the central lmt theorem. However, f λ 1 + ε wth ε very small and the central lmt theorem s applcable, then PX 1 + εµ s roughly exp ε 2 µ 2 /2σ 2, where σ 2 Var X n Var X n 1. Hence, n ths case the p 2 exponents n 2.3 and 2.11 are asymptotcally too small by a factor of rougly, for small, µ µ 2 /σ 2 p n p 2, 2.29 whch may be much smaller than 1. p 1 p 2 /n 1/3. n p 1 For example f p 2 p n and

6 6 SVANTE JANSON 3. Upper bounds for the lower tal We can smlarly bound the probablty PX λµ for λ 1. We gve only a smple bound correspondng to Theorem 2.1. Note that λ 1 ln λ > 0 for both λ 0, 1 and λ 1,. Theorem 3.1. For any p 1,..., p n 0, 1] and any λ 1, PX λµ e p µλ 1 ln λ. 3.1 Proof. We follow closely the proof of Theorem 2.1. If t 0, then by 1.4, E e tx 1 + t Hence E e tx e t 1 + E e tx t t and, n analogy to 2.2, PX λµ e tλµ E e tx exp tλµ ln 1 + t. 3.4 In analogy wth 2.7, stll by the convexty of ln x, ln 1 + t p ln 1 + t, 3.5 and 3.4 yelds PX λµ exp tλµ ln exp tλµ µ ln Choosng t λ 1 1, we obtan t 4. A lower bound 1 + t. 3.6 We show also a general lower bound for the upper tal probabltes, whch shows that for constant λ > 1, the exponents n Theorems 2.1 and 2.3 are at most a constant factor away from best possble. Theorem 4.1. For any p 1,..., p n 0, 1] and any λ 1, PX λµ 1 1+1/p 1 λ 1µ µ Lemma 4.2. If A 1 and 0 x 1/A, then A x + ln1 x ln 1 Ax 2 /2. 4.2

7 TAIL BOUNDS FOR GEOMETRIC AND EXPONENTIAL VARIABLES 7 Proof. Let fx : A x + ln1 x ln 1 Ax 2 /2. Then f0 0 and f x A 1 1 Ax + 1 x 1 Ax 2 /2 Ax 1 x + Ax 1 Ax /2 for 0 x < 1/A 1, snce then 0 < 1 x 1 Ax 2 /2. Hence fx 0 for 0 x 1/A. Proof of Theorem 4.1. Let ε : 1/ µ. By Theorem 3.1 wth λ 1 ε and Lemma 4.2 wth A µ 1, PX 1 εµ exp µ ε ln1 ε 1 µε 2 Hence, PX 1 εµ 1/2 µ, and by Lemma 2.5, µ. 4.4 PX λµ 1 λ 1+εµ+1 PX 1 εµ 1 λ 1+εµ µ, whch completes the proof snce εµ 1/. 5. Exponental dstrbutons In ths secton we assume that X n X where X, 1,..., n, are ndependent random varables wth exponental dstrbutons: X Expa, wth densty functon a xe ax, x > 0, and expectaton E X 1/a. Thus a can be nterpreted as a rate. The exponental dstrbuton s the contnuous analogue of the geometrc dstrbutons, and the results above have smpler analogues for exponental dstrbutons. We now defne µ : E X E X 1 a, 5.1 a : mn a. 5.2 Theorem 5.1. Let X n X wth X Expa ndependent. For any λ 1, PX λµ λ 1 e a µλ 1 ln λ. 5.3 For any λ 1, we have also the smpler but weaker For any λ 1, v For any λ 1, PX λµ e 1 λ. 5.4 PX λµ e a µλ 1 ln λ. 5.5 PX λµ 1 2ea µ e a µλ

8 8 SVANTE JANSON Proof. Let X N Gea /N be ndependent for N > max a. Then X N /N d d X, where denotes convergence n dstrbuton, and thus X N /N d X, where X N : n XN. Furthermore, µ N : E X N Mν and : mn a /N a /N. The results follow by takng the lmt as N n 2.11, 2.12, 3.1 and 4.1. Alternatvely, we may mtate the proofs above, usng E e tx a /a t for t < a. References [1] Stéphane Boucheron, Gábor Lugos and Pascal Massart, Concentraton Inequaltes, Oxford Unv. Press, Oxford, [2] Amr Dembo and Ofer Zetoun, Large Devatons Technques and Applcatons. 2nd ed., Sprnger, New York, [3] Svante Janson, Tomasz Luczak & Andrzej Rucńsk, Random Graphs. Wley, New York, [4] Olav Kallenberg, Foundatons of Modern Probablty. 2nd ed., Sprnger, New York, Department of Mathematcs, Uppsala Unversty, PO Box 480, SE Uppsala, Sweden E-mal address: svante.janson@math.uu.se URL: svante/