Abstract
We establish upper and lower bounds with matching leading terms for tails of weighted sums of two-sided exponential random variables. This extends Janson’s recent results for one-sided exponentials.
TT’s research supported in part by NSF grant DMS-1955175.
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References
B. Bercu, B. Delyon, E. Rio, Concentration Inequalities for Sums and Martingales. SpringerBriefs in Mathematics (Springer, Cham, 2015)
S. Boucheron, G. Lugosi, P. Massart, Concentration Inequalities. A Nonasymptotic Theory of Independence. With a Foreword by Michel Ledoux (Oxford University Press, Oxford, 2013)
Y. Chen, K.W. Ng, Q. Tang, Weighted sums of subexponential random variables and their maxima. Adv. Appl. Probab. 37(2), 510–522 (2005)
A. Eskenazis, On extremal sections of subspaces of Lp. Discret. Comput. Geom. 65(2), 489–509 (2021)
A. Eskenazis, P. Nayar, T. Tkocz, Gaussian mixtures: entropy and geometric inequalities. Ann. Probab. 46(5), 2908–2945 (2018)
S. Foss, T. Konstantopoulos, S. Zachary, Discrete and continuous time modulated random walks with heavy-tailed increments. J. Theor. Probab. 20(3), 581–612 (2007)
S. Foss, D. Korshunov, S. Zachary, An Introduction to Heavy-Tailed and Subexponential Distributions, 2nd edn. Springer Series in Operations Research and Financial Engineering (Springer, New York, 2013)
E.D. Gluskin, S. Kwapień, Tail and moment estimates for sums of independent random variables with logarithmically concave tails. Studia Math. 114(3), 303–309 (1995)
P. Hitczenko, S. Montgomery-Smith, A note on sums of independent random variables, in Advances in Stochastic Inequalities (Atlanta, GA, 1997). Contemp. Math., vol. 234, Amer. Math. Soc., Providence, RI (1999), pp.69–73
G. Jameson, A simple proof of Stirling’s formula for the gamma function. Math. Gaz. 99(544), 68–74 (2015)
S. Janson, Tail bounds for sums of geometric and exponential variables. Stat. Probab. Lett. 135, 1–6 (2018)
J. Li, M. Madiman, A combinatorial approach to small ball inequalities for sums and differences. Comb. Probab. Comput. 28(1), 100–129 (2019)
P. Nayar, T. Tkocz, The unconditional case of the complex S-inequality. Israel J. Math. 197(1), 99–106 (2013)
P. Nayar, T. Tkocz, S-inequality for certain product measures. Math. Nachr. 287(4), 398–404 (2014)
P. Nayar, T. Tkocz, On a convexity property of sections of the cross-polytope. Proc. Am. Math. Soc. 148(3), 1271–1278 (2020)
H.H. Nguyen, V.H. Vu, Small ball probability, inverse theorems, and applications, in Erdös Centennial. Bolyai Society Mathematical Studies, vol. 25 (János Bolyai Mathematical Society, Budapest, 2013), pp. 409–463
K. Oleszkiewicz, Precise moment and tail bounds for Rademacher sums in terms of weak parameters. Israel J. Math. 203(1), 429–443 (2014)
G. Paouris, P. Valettas, A Gaussian small deviation inequality for convex functions. Ann. Probab. 46(3), 1441–1454 (2018)
G. Paouris, P. Valettas, Variance estimates and almost Euclidean structure. Adv. Geom. 19(2), 165–189 (2019)
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Li, J., Tkocz, T. (2023). Tail Bounds for Sums of Independent Two-Sided Exponential Random Variables. In: Adamczak, R., Gozlan, N., Lounici, K., Madiman, M. (eds) High Dimensional Probability IX. Progress in Probability, vol 80. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-26979-0_5
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