An explicit version of the Chebyshev-Markov-Stieltjes inequalities and its applications

Journal of Inequalities and Applications

Table 2 Simulation of generalized LS bounds for arbitrary sequences

Normal distribution
n	Order of approximation										\(\boldsymbol {\max\|\frac{x_{i}-\mu}{\sigma}\|}\)
n	1	2	3	4	5	6	7	8	9	10	\(\boldsymbol {\max\|\frac{x_{i}-\mu}{\sigma}\|}\)
10²	9.950	3.607	2.780	2.590	2.520	2.479	2.454	2.438	2.432	2.431	2.431
10³	31.61	6.712	4.346	3.687	3.472	3.411	3.394	3.391	3.390	3.390	3.390
10⁴	100	11.91	6.327	4.838	4.263	4.023	3.926	3.889	3.874	3.868	3.865
10⁵	316.2	21.18	9.256	6.462	5.510	5.202	5.119	5.100	5.095	5.094	5.094

Laplace distribution
n	Order of approximation										\(\boldsymbol {\max\|\frac{x_{i}-\mu}{\sigma}\|}\)
n	1	2	3	4	5	6	7	8	9	10	\(\boldsymbol {\max\|\frac{x_{i}-\mu}{\sigma}\|}\)
10²	9.950	4.302	3.739	3.677	3.670	3.670	3.670	3.670	3.670	3.670	3.670
10³	31.61	8.897	6.661	6.063	5.809	5.693	5.633	5.597	5.574	5.557	5.542
10⁴	100	14.74	8.720	7.196	6.687	6.528	6.483	6.469	6.465	6.464	6.463
10⁵	316.2	26.74	13.54	10.49	9.355	8.849	8.602	8.479	8.411	8.373	8.333

Pearson type VII distribution ( α = 3)
n	Order of approximation										\(\boldsymbol {\max\|\frac{x_{i}-\mu}{\sigma}\|}\)
n	1	2	3	4	5	6	7	8	9	10	\(\boldsymbol {\max\|\frac{x_{i}-\mu}{\sigma}\|}\)
10²	9.950	7.665	7.646	7.646	7.646	7.646	7.646	7.646	7.646	7.646	7.646
10³	31.61	22.16	21.80	21.79	21.79	21.79	21.79	21.79	21.79	21.79	21.79
10⁴	100	88.94	88.80	88.80	88.80	88.80	88.80	88.80	88.80	88.80	88.80
10⁵	316.2	227.6	225.2	225.2	225.2	225.2	225.2	225.2	225.2	225.2	225.2

Bowers distribution
n	Order of approximation										\(\boldsymbol {\max\|\frac{x_{i}-\mu}{\sigma}\|}\)
n	1	2	3	4	5	6	7	8	9	10	\(\boldsymbol {\max\|\frac{x_{i}-\mu}{\sigma}\|}\)
10²	9.950	6.291	6.063	6.050	6.050	6.050	6.050	6.050	6.050	6.050	6.050
10³	31.61	13.71	11.21	10.36	10.04	9.930	9.911	9.907	9.907	9.907	9.907
10⁴	100	38.67	34.27	33.36	33.26	33.25	33.25	33.25	33.25	33.25	33.25
10⁵	316.2	118.1	103.5	100.5	100.0	99.93	99.92	99.92	99.92	99.92	99.92

Cauchy distribution
n	Order of approximation										\(\boldsymbol {\max\|\frac{x_{i}-\mu}{\sigma}\|}\)
n	1	2	3	4	5	6	7	8	9	10	\(\boldsymbol {\max\|\frac{x_{i}-\mu}{\sigma}\|}\)
10²	9.950	9.269	9.260	9.260	9.260	9.260	9.260	9.260	9.260	9.260	9.260
10³	31.61	30.08	30.07	30.07	30.07	30.07	30.07	30.07	30.07	30.07	30.07
10⁴	100	95.31	95.28	95.28	95.28	95.28	95.28	95.28	95.28	95.28	95.28
10⁵	316.2	291.6	291.1	291.1	291.1	291.1	291.1	291.1	291.1	291.1	291.1