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On domination of tail probabilities of (super)martingales: Explicit bounds

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Abstract

Let X be a random variable with survival function G(x)=ℙ{X≥x}. Let α > 0. Consider a transform, say T α, of survival functions G↦T α G defined by

$$(T_\alpha G)(x)\mathop = \limits^{def} \mathop {\inf }\limits_{h:h < x} (x - h)^{ - \alpha } E(X - h)^\alpha \mathbb{I}\{ X \geqslant h\} $$

. In this paper, we examine the properties of the transform as well as provide explicit expressions for T α G in some special cases. Our motivation to study the properties of the transform comes from the theory of inequalities for tail probabilities of sums of independent random variables, martingales, super-martingales, etc. The transform is a commonly used tool in this field, and up to date it has led to most advanced or new-type inequalities. For statistical applications, such as construction of upper confidence bounds, particularly, in problems related to the auditing mathematics, computable and as precise as possible inequalities are required. This is the main motivation to write this paper. We examine general properties of the transform and consider special cases of normal, exponential, Bernoulli, binomial, uniform, and Poisson tails. These types and other survival functions appear in known applications. Until now only the values of α = 1, 2, 3 were actually used, and we concentrate our attention to these three special cases.

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Akademijos 4, LT-08663, Vilnius

    V. Bentkus

  2. Vilnius Pedagogical University, Studentų 39, LT-08106, Vilnius, Lithuania

    V. Bentkus

  3. University of Nijmegen, Postbus 9010, 6500 GL, Nijmegen, The Netherlands

    N. Kalosha & M. van Zuijlen

Authors
  1. V. Bentkus
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  2. N. Kalosha
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  3. M. van Zuijlen
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Additional information

This research was supported by the Technology Foundation STW, applied science division of NWO and the technology program of the Ministry of Economic Affairs.

Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 1, pp. 3–54, January–March, 2006.

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Bentkus, V., Kalosha, N. & van Zuijlen, M. On domination of tail probabilities of (super)martingales: Explicit bounds. Lith Math J 46, 1–43 (2006). https://doi.org/10.1007/s10986-006-0011-5

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  • DOI: https://doi.org/10.1007/s10986-006-0011-5

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