Abstract
The term law has various meanings within probability.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Etemadi, N. (1983): “On the Laws of Large Numbers for Nonnegative Random Variables,” J. Multivariate Analysis, 13, pp. 187–193.
- 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.
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.
See Mitzenmacher, M. and E. Upfal (2005) for illustrative applications in the context of machine learning.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this chapter
Cite this chapter
Bhattacharya, R., Waymire, E.C. (2016). Classical Zero–One Laws, Laws of Large Numbers and Large Deviations. In: A Basic Course in Probability Theory. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-47974-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-47974-3_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-47972-9
Online ISBN: 978-3-319-47974-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)