Addendum to “An extension of an inequality of Hoeffding to unbounded random variables”: The non-i.i.d. case |
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Authors: | V Bentkus |
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Institution: | (1) Vilnius Pedagogical University, Studentu 39, LT-08106 Vilnius, Lithuania |
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Abstract: | Let S
n = X
1 + ⋯ + X
n be a sum of independent random variables such that 0 ⩽ X
k ⩽ 1 for all k. Write {ie237-01} and q = 1 − p. Let 0 < t < q. In our recent paper 3], we extended the inequality of Hoeffding (6], Theorem 1) {fx237-01} to the case where X
k are unbounded positive random variables. It was assumed that the means {ie237-02} of individual summands are known. In this
addendum, we prove that the inequality still holds if only an upper bound for the mean {ie237-03} is known and that the i.i.d.
case where {ie237-04} dominates the general non-i.i.d. case. Furthermore, we provide upper bounds expressed in terms of certain
compound Poisson distributions. Such bounds can be more convenient in applications. Our inequalities reduce to the related
Hoeffding inequalities if 0 ⩽ X
k ⩽ 1. Our conditions are X
k ⩾ 0 and {ie237-05}. In particular, X
k can have fat tails. We provide as well improvements comparable with the inequalities in Bentkus 2]. The independence of
X
k can be replaced by super-martingale type assumptions. Our methods can be extended to prove counterparts of other inequalities
in Hoeffding 6] and Bentkus
The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No T-25/08. |
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Keywords: | Hoeffding’ s inequalities probabilities of large deviations bounds for tail probabilities bounded and unbounded random variables supermartingales |
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