Inequalities compared

A.H. Hoekstra and J. Th. Runnenburg

Statistica Neerlandica, 1980, vol. 34, issue 2, 67-82

Abstract: We know that the partial means mrof a sequence of i.i.d. standardized random variables tend to 0 with probability 1. If we want P{mk≥εfor some k ≥r}≤δ for given positive ε and δ, how large should we take r? Several (strong) inequalities for the distribution of partial sums providing an answer to this question can be found in the literature (Hájek‐RényiRobbins, Khan). Furthermore there exist wellknown (weak) inequalities (Bienaymé‐Chebyshev, Bernstein, Okamoto) that give us values of rfor which P{mr≥ε}≤δ. We compare these inequalities and illustrate them with numerical results for a fixed choice ofε and δ. After a general survey and introduction in section 1, the normal and the binomial distribution are considered in more detail in the sections 2 and 3, while in section 4 it is shown that the strong inequality essentially due to Robbinscan give an inferior result for particular distributions.

Date: 1980
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