Electronic Journal of Probability

Uniformly Accurate Quantile Bounds Via The Truncated Moment Generating Function: The Symmetric Case

Michael Klass and Krzysztof Nowicki

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Let $X_1, X_2, \dots$ be independent and symmetric random variables such that $S_n = X_1 + \cdots + X_n$ converges to a finite valued random variable $S$ a.s. and let $S^* = \sup_{1 \leq n \leq \infty} S_n$ (which is finite a.s.). We construct upper and lower bounds for $s_y$ and $s_y^*$, the upper $1/y$-th quantile of $S_y$ and $S^*$, respectively. Our approximations rely on an explicitly computable quantity $\underline q_y$ for which we prove that $$\frac 1 2 \underline q_{y/2} < s_y^* < 2 \underline q_{2y} \quad \text{ and } \quad \frac 1 2 \underline q_{ (y/4) ( 1 + \sqrt{ 1 - 8/y})} < s_y < 2 \underline q_{2y}. $$ The RHS's hold for $y \geq 2$ and the LHS's for $y \geq 94$ and $y \geq 97$, respectively. Although our results are derived primarily for symmetric random variables, they apply to non-negative variates and extend to an absolute value of a sum of independent but otherwise arbitrary random variables.

Article information

Electron. J. Probab., Volume 12 (2007), paper no. 47, 1276-1298.

Accepted: 16 October 2007
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks 60E15: Inequalities; stochastic orderings
Secondary: 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46B09: Probabilistic methods in Banach space theory [See also 60Bxx]

Sum of independent rv's tail distributions tail distributions tail probabilities quantile approximation Hoffmann-Jo rgensen/Klass-Nowicki Inequality

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Klass, Michael; Nowicki, Krzysztof. Uniformly Accurate Quantile Bounds Via The Truncated Moment Generating Function: The Symmetric Case. Electron. J. Probab. 12 (2007), paper no. 47, 1276--1298. doi:10.1214/EJP.v12-452. https://projecteuclid.org/euclid.ejp/1464818518

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  • H. Cramer, On a new limit in theory of probability, in Colloquium on the Theory of Probability, (1938), Hermann, Paris. Review number not available.
  • F. Esscher, On the probability function in the collective theory of risk, Skand. Aktuarietidskr. 15 (1932), 175–-195. Review number not available.
  • Hahn, Marjorie G.; Klass, Michael J. Uniform local probability approximations: improvements on Berry-Esseen. Ann. Probab. 23 (1995), no. 1, 446–463.
  • Hahn, Marjorie G.; Klass, Michael J. Approximation of partial sums of arbitrary i.i.d. random variables and the precision of the usual exponential upper bound. Ann. Probab. 25 (1997), no. 3, 1451–1470.
  • Fuk, D. H.; Nagaev, S. V. Probabilistic inequalities for sums of independent random variables.(Russian) Teor. Verojatnost. i Primenen. 16 (1971), 660–675. (45 #2772)
  • Hitczenko, Pawel; Montgomery-Smith, Stephen. A note on sums of independent random variables. Advances in stochastic inequalities (Atlanta, GA, 1997), 69–73, Contemp. Math., 234, Amer. Math. Soc., Providence, RI, 1999.
  • Hitczenko, Pawel; Montgomery-Smith, Stephen. Measuring the magnitude of sums of independent random variables. Ann. Probab. 29 (2001), no. 1, 447–466.
  • Jain, Naresh C.; Pruitt, William E. Lower tail probability estimates for subordinators and nondecreasing random walks. Ann. Probab. 15 (1987), no. 1, 75–101.
  • Klass, M. J. Toward a universal law of the iterated logarithm. I. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 36 (1976), no. 2, 165–178. (54 #3822)
  • Klass, Michael J.; Nowicki, Krzysztof. An improvement of Hoffmann-Jørgensen's inequality. Ann. Probab. 28 (2000), no. 2, 851–862.
  • Klass, Michael J.; Nowicki, Krzysztof. An optimal bound on the tail distribution of the number of recurrences of an event in product spaces. Probab. Theory Related Fields 126 (2003), no. 1, 51–60.
  • Kolmogoroff, A. Über das Gesetz des iterierten Logarithmus.(German) Math. Ann. 101 (1929), no. 1, 126–135.
  • Latal a, Rafal. Estimation of moments of sums of independent real random variables. Ann. Probab. 25 (1997), no. 3, 1502–1513.
  • Nagaev, S. V. Some limit theorems for large deviations.(Russian) Teor. Verojatnost. i Primenen 10 1965 231–254. (32 #3106)
  • Prokhorov, Yu. V. An extremal problem in probability theory. Theor. Probability Appl. 4 1959 201–203. (22 #12587)