Probability Inequalities

  • Roberto Tempo
  • Giuseppe Calafiore
  • Fabrizio Dabbene
Part of the Communications and Control Engineering book series (CCE)


This chapter addresses the issue of finite sample size in probability estimation, that is, the so-called sample complexity. The main objective in this context is to analyze rigorously the reliability of the probabilistic estimates introduced in Chap.  7, for a finite sample size. This issue is crucial in the development of randomized algorithms for uncertain systems and control, and makes a clear distinction from the asymptotic methods which are instead based on the laws of large numbers. Specifically, the chapter includes Markov, Chebychev and Hoeffding inequalities. The additive and multiplicative Chernoff bounds are subsequently derived and the sample complexity for estimation of extrema is also studied.


Sample Complexity Classical Inequality Nonnegative Random Variable Markov Inequality Probability Inequality 
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Copyright information

© Springer-Verlag London 2013

Authors and Affiliations

  • Roberto Tempo
    • 1
  • Giuseppe Calafiore
    • 2
  • Fabrizio Dabbene
    • 1
  1. 1.CNR - IEIITPolitecnico di TorinoTurinItaly
  2. 2.Dip. Automatica e InformaticaPolitecnico di TorinoTurinItaly

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