Abstract
In this paper, we obtain a Bernstein-type concentration inequality and McDiarmid’s inequality under upper probabilities for exponential independent random variables. Compared with the classical result, our inequalities are investigated under a family of probability measures, rather than one probability measure. As applications, the convergence rates of the law of large numbers and the Marcinkiewicz–Zygmund-type law of large numbers about the random variables in upper expectation spaces are obtained.
Highlights
Concentration inequalities are useful technique tools for studying the limit theory in the classical probability and statistics, which describe the bounds of a random variable deviating from some value
The law of large numbers, central limit theorem, and law of the iterated logarithm could all be regarded as derivative results of concentration inequalities
The Bernstein-type inequality plays an important role among concentration inequalities especially, which provides a bound for the sum of independent random variables deviating from its mean value, while McDiarmid’s inequality is established to bound the deviations for Doob martingale in the probability space
Summary
Concentration inequalities are useful technique tools for studying the limit theory in the classical probability and statistics, which describe the bounds of a random variable deviating from some value. The law of large numbers, central limit theorem, and law of the iterated logarithm could all be regarded as derivative results of concentration inequalities. The Bernstein-type inequality plays an important role among concentration inequalities especially, which provides a bound for the sum of independent random variables deviating from its mean value, while McDiarmid’s inequality is established to bound the deviations for Doob martingale in the probability space. Is a sequence of independent, zero-mean random variables defined in the probability space (Ω, F , P). Denote by Sn the partial sum of this sequence, namely
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