The termlaw has various meanings within probability.

1. 1.

   Etemadi, N. (1983): “On the Laws of Large Numbers for Nonnegative Random Variables,”J. Multivariate Analysis,13, pp. 187–193.

2. 2.

   The refinement presented here is due to Bahadur, R., and Ranga Rao, R. (1960): On deviations of the sample mean,Ann. Math. Stat., v31(4), 1015-1027. This work was preceded by earlier results of Blackwell, D. and Hodges, J.L. (1959): The probability in the extreme tail of a convolution,Ann. Math. Stat., v30, 1113–1120.

3. 3.

   W. Hoeffding (1963): Probability inequalities for sums of bounded random variables,J. of the Am. Stat. Assoc.58(301),1330, obtained this inequality in 1963. There are a number of related “concentration inequalities” of this type in the modern probability literature, expressing the concentration of the distribution of\(\overline{X}\) near the mean.

4. 4.

   See Mitzenmacher, M. and E. Upfal (2005) for illustrative applications in the context of machine learning.

Authors and Affiliations

1. Department of Mathematics, University of Arizona, Tucson, AZ, USA

   Rabi Bhattacharya

2. Department of Mathematics, Oregon State Univeristy, Corvallis, OR, USA

   Edward C. Waymire

Corresponding author

Correspondence toRabi Bhattacharya.

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