Inequalities compared

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=efor some k =r}=d for given positive e and d, 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=e}=d. We compare these inequalities and illustrate them with numerical results for a fixed choice ofe and d. 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.