Brownian Motion in Space - Tables of Suprema

Tables of Distribution Functions of Suprema of Brownian Motion on a Line or in 2-Space

TABLE OF CONTENTS

Introduction

It is well known that the brownian bridge and its projections often appear as limiting processes in the study of goodness-of-fit (or model-checking) problems. In particular, empirical and related processes typically converge in distribution to a brownian bridge, while the same processes with parameters estimated from the sample converge to projections thereof. [Descriptions of estimated processes as projections are given in Khmaladze (1979) and Khmaladze & Koul (2003).] However brownian motion in one or many dimensions also appears as such a limiting process. In particular, this is the case when empirical processes (with or without estimated parameters) are replaced by innovative martingales. [See Khmaladze (1981, 1993), Nitikin (1995), Khmaladze & Koul (2003), Stute (), Koenker (2003), ...]

In one dimension, the distribution function of the supremum of the absolute value of brownian motion on the interval [0,1] may be calculated analytically. It is also easy to calculate the distribution function of the supremum of a normalised poisson process with very high intensity. [See Khmaladze & Shinjikashvili (2001).]

In higher dimensions, however, there is an added complication introduced by the copula function which describes the dependence of the brownian motion in the different dimensions. For the independent case in two dimensions, there is an analytical solution [see e.g. Resnick]. Currently though (mid-2003) these calculations are yet to be carried out.

Further, when there is a non-trivial copula function, there is no such analytical expression yet found. This then forces the use of numerical analysis to determine the behaviour of the distribution function of the supremum.

Some of the outcomes from such numerical analysis appear below.
Table 1 shows the calculated distribution functions of the supremum of a normalised poisson process on the interval [0,1], namely | ξn(t) - nt | / √n for n=1000 and n=5000. In addition a graphical comparison of the two distribution functions is provided to show how little they differ.
Table 1a shows the recently calculated distribution function of the supremum of the absolute value of brownian motion on the interval [0,1].
Table 2 is the distribution function of the supremum of a normalised two-dimensional independent poisson process on [0,1]2, namely | ξn(s,t) - nst | / √n, obtained from a simulation of 20,000 replications with n=5000. In addition there is provided a graphical comparison of the distribution functions for various values of n (n=40, 50, 100, 200 and 5000) to show convergence as n increases. Further, for interest sake, the locations of the suprema are shown graphically.
Table 2a is the same table from a more recent simulation of 50,000 replications with n=50,000.
Table 3 is the distribution function of the supremum of a normalised two-dimensional poisson process with copula function on [0,1]2, namely | ξnH(s,t) - nH(s,t) | / √n with copula function:
H(s,t) = F(F-1(s), F-1(t); r)
where F-1(x) is the quantile function (inverse) of the standard normal distribution function and F(x, y; r) is the bi-variate normal distribution function with mean 0, variances 1 and correlation coefficient r. The intensity is set to n=5000, and the correlation coefficient r is given the two values r=0.5 and r=-0.5, with 20,000 replications in both cases. The distribution functions corresponding to the two different values of r are shown graphically, as well as the locations of the suprema and a graphical comparison of these two with the uncorrelated case.

One-dimensional case

The distribution function of supremum over t in [0,1] of |W(t)| is the limit as n tends to infinity of, and is approximated by,
P{ | ξn(t) - nt | / √n < x for all 0 < t < 1},
where ξn(t) with t in [0,1] is a Poisson process with intensity n.

These tables have been calculated by Eka Shinjikashvili for n=1000 and n=5000.

For a description of the algorithm used, see E. Khmaladze and E. Shinjikashvili, Calculation of Noncrossing Probabilities for Poisson Processes and its Corollaries Advances in Applied Probability, Vol. 33, No. 3. (Sep., 2001), pp. 702-716.

For convenience, the values of the distribution function corresponding to a selection of p-values have been calculated by linear interpolation.

A comparison [PDF, 6KB]) of the two graphs shows that the values are very close. Similarly, comparing the two tables below, it can be seen that the values differ only in the 3rd digit.
n=1000 (graph [PDF, 6KB]) n=5000 (graph [PDF, 6KB])
x Pr x Pr
0.1 0.000 0.1 0.000
0.2 0.000 0.2 0.000
0.3 0.000 0.3 0.000
0.4 0.001 0.4 0.001
0.5 0.010 0.5 0.010
0.6 0.044 0.6 0.042
0.7 0.107 0.7 0.104
0.8 0.190 0.8 0.187
0.9 0.283 0.9 0.280
1.0 0.376 1.0 0.373
1.1 0.464 1.1 0.461
1.2 0.545 1.2 0.542
1.3 0.617 1.3 0.615
1.4 0.680 1.4 0.678
1.5 0.736 1.5 0.734
1.6 0.783 1.6 0.782
1.7 0.824 1.7 0.823
1.8 0.858 1.8 0.857
1.9 0.887 1.9 0.886
1.9598 0.9000 (linear interpolation)
2.0 0.910 2.0 0.910
2.1 0.930 2.1 0.929
2.2 0.945 2.2 0.945
2.2416 0.9500 (linear interpolation)
2.3 0.958 2.3 0.957
2.4 0.968 2.4 0.967
2.4958 0.9750 (linear interpolation)
2.5 0.976 2.5 0.975
2.6 0.982 2.6 0.981
2.7 0.986 2.7 0.986
2.8 0.990 2.8 0.990
2.8054 0.9900 (linear interpolation)
2.9 0.993 2.9 0.993
3.0 0.995 3.0 0.995

One-dimensional case in the limit

The distribution function of supremum over t in [0,1] of |W(t)| has been recently (October, 2006) calculated from the formula
FD(x)=4/π n=0∑∞ (-1)n/(2n+1) exp{-π2(2n+1)2/8x2}
These values are presented below to 6 decimal places. As above, the values of the distribution function corresponding to a selection of p-values have also been calculated. Note that for values of x less than 5, it is only necessary to sum the first 5 terms of the formula to obtain a result accurate to 8 decimal places.
limiting case (graph [PDF, 6KB])
x Pr
0.1 0.000000
0.2 0.000000
0.3 0.000001
0.4 0.000570
0.5 0.009157
0.6 0.041362
0.7 0.102674
0.8 0.185242
0.9 0.277614
1.0 0.370777
1.1 0.459269
1.2 0.540358
1.3 0.612990
1.4 0.677027
1.5 0.732785
1.6 0.780806
1.7 0.821739
1.8 0.856279
1.9 0.885134
1.959964 0.900000
2.0 0.908999
2.1 0.928542
2.2 0.944386
2.241403 0.950000
2.3 0.957104
2.4 0.967210
2.497705 0.975000
2.5 0.975161
2.6 0.981355
2.7 0.986132
2.8 0.989779
2.807034 0.990000
2.9 0.992537
3.0 0.994600
3.1 0.996130
3.2 0.997251
3.3 0.998066
3.4 0.998652
3.5 0.999069
3.6 0.999364
3.7 0.999569
3.8 0.999711
3.9 0.999808
4.0 0.999873
4.1 0.999917
4.2 0.999947
4.3 0.999966
4.4 0.999978
4.5 0.999986
4.6 0.999992
4.7 0.999995
4.8 0.999997
4.9 0.999998
5.0 0.999999

2-dimensional uncorrelated

The distribution function of supremum over (s,t) in [0,1]2 of |W(s,t)| is the limit as n tends to infinity of, and is approximated by,
P{ | ξn(s,t) - nst | / √n < x for all 0 < s,t < 1}
where ξn(s,t) is a Poisson process on [0,1]2 with expected value E[ ξn(s,t) ] = nst.

The table below has been simulated with m=20,000 replications for n=5000.

In addition, we illustrate the rate of convergence to the limit as n tends towards infinity with the graph of edfs [PDF, 82KB] for n=40, 50, 100, 200 and 5000, for m=20,000 replications.

n=5000 (graph of edf [PDF, 18KB]; plot of distribution of locations of suprema [PDF, 580KB])

x Pr{sup|W|≤x} Pr{sup|W|≤x} x
0.00 0.000 0.00 0.624
0.04 0.000 0.01 0.841
0.08 0.000 0.02 0.886
0.12 0.000 0.03 0.916
0.16 0.000 0.04 0.942
0.20 0.000 0.05 0.952
0.24 0.000 0.06 0.982
0.28 0.000 0.07 0.998
0.32 0.000 0.08 1.01
0.36 0.000 0.09 1.03
0.40 0.000 0.10 1.04
0.44 0.000 0.11 1.05
0.48 0.000 0.12 1.07
0.52 0.000 0.13 1.08
0.56 0.000 0.14 1.09
0.60 0.000 0.15 1.10
0.64 0.000 0.16 1.11
0.68 0.000 0.17 1.12
0.72 0.001 0.18 1.13
0.76 0.002 0.19 1.15
0.80 0.004 0.20 1.16
0.84 0.010 0.21 1.17
0.88 0.018 0.22 1.18
0.92 0.031 0.23 1.19
0.96 0.049 0.24 1.20
1.00 0.071 0.25 1.21
1.04 0.100 0.26 1.22
1.08 0.131 0.27 1.23
1.12 0.167 0.28 1.24
1.16 0.202 0.29 1.25
1.20 0.244 0.30 1.25
1.24 0.285 0.31 1.27
1.28 0.325 0.32 1.28
1.32 0.366 0.33 1.29
1.36 0.405 0.34 1.29
1.40 0.444 0.35 1.30
1.44 0.480 0.36 1.31
1.48 0.516 0.37 1.32
1.52 0.550 0.38 1.33
1.56 0.583 0.39 1.34
1.60 0.614 0.40 1.35
1.64 0.643 0.41 1.36
1.68 0.672 0.42 1.38
1.72 0.697 0.43 1.39
1.76 0.721 0.44 1.40
1.80 0.744 0.45 1.41
1.84 0.765 0.46 1.42
1.88 0.784 0.47 1.43
1.92 0.802 0.48 1.44
1.96 0.819 0.49 1.45
2.00 0.835 0.50 1.46
2.04 0.850 0.51 1.47
2.08 0.864 0.52 1.48
2.12 0.876 0.53 1.50
2.16 0.887 0.54 1.51
2.20 0.898 0.55 1.52
2.24 0.907 0.56 1.53
2.28 0.916 0.57 1.54
2.32 0.925 0.58 1.56
2.36 0.932 0.59 1.57
2.40 0.939 0.60 1.58
2.44 0.945 0.61 1.59
2.48 0.951 0.62 1.61
2.52 0.956 0.63 1.62
2.56 0.961 0.64 1.64
2.60 0.965 0.65 1.65
2.64 0.969 0.66 1.66
2.68 0.972 0.67 1.68
2.72 0.975 0.68 1.69
2.76 0.978 0.69 1.71
2.80 0.980 0.70 1.73
2.84 0.983 0.71 1.74
2.88 0.984 0.72 1.76
2.92 0.986 0.73 1.78
2.96 0.988 0.74 1.79
3.00 0.989 0.75 1.81
3.04 0.991 0.76 1.83
3.08 0.992 0.77 1.85
3.12 0.993 0.78 1.87
3.16 0.994 0.79 1.89
3.20 0.995 0.80 1.91
3.24 0.995 0.81 1.94
3.28 0.996 0.82 1.96
3.32 0.996 0.83 1.99
3.36 0.997 0.84 2.01
3.40 0.997 0.85 2.04
3.44 0.997 0.86 2.07
3.48 0.998 0.87 2.10
3.52 0.998 0.88 2.13
3.56 0.998 0.89 2.17
3.60 0.998 0.90 2.21
3.64 0.999 0.91 2.25
3.68 0.999 0.92 2.30
3.72 0.999 0.93 2.35
3.76 0.999 0.94 2.41
3.80 0.999 0.95 2.47
3.84 0.999 0.96 2.55
3.88 0.999 0.97 2.65
3.92 1.00 0.98 2.79
3.96 1.00 0.99 3.02
4.00 1.00 1.00 87.5

2-dimensional uncorrelated - newer result

The table below has been simulated with m=50,000 replications for n=50,000.

x Pr{sup|W|≤x} Pr{sup|W|≤x} x
0.00 0.000 0.00 0.596
0.04 0.000 0.01 0.839
0.08 0.000 0.02 0.882
0.12 0.000 0.03 0.911
0.16 0.000 0.04 0.934
0.20 0.000 0.05 0.952
0.24 0.000 0.06 0.973
0.28 0.000 0.07 0.988
0.32 0.000 0.08 1.00
0.36 0.000 0.09 1.02
0.40 0.000 0.10 1.03
0.44 0.000 0.11 1.04
0.48 0.000 0.12 1.06
0.52 0.000 0.13 1.07
0.56 0.000 0.14 1.08
0.60 0.000 0.15 1.09
0.64 0.000 0.16 1.10
0.68 0.000 0.17 1.11
0.72 0.001 0.18 1.12
0.76 0.002 0.19 1.14
0.80 0.005 0.20 1.15
0.84 0.010 0.21 1.16
0.88 0.019 0.22 1.17
0.92 0.033 0.23 1.18
0.96 0.053 0.24 1.19
1.00 0.078 0.25 1.20
1.04 0.109 0.26 1.21
1.08 0.141 0.27 1.22
1.12 0.176 0.28 1.23
1.16 0.212 0.29 1.24
1.20 0.252 0.30 1.25
1.24 0.293 0.31 1.26
1.28 0.333 0.32 1.27
1.32 0.375 0.33 1.28
1.36 0.415 0.34 1.29
1.40 0.454 0.35 1.30
1.44 0.488 0.36 1.31
1.48 0.522 0.37 1.32
1.52 0.556 0.38 1.33
1.56 0.588 0.39 1.33
1.60 0.621 0.40 1.35
1.64 0.649 0.41 1.35
1.68 0.676 0.42 1.36
1.72 0.703 0.43 1.37
1.76 0.726 0.44 1.39
1.80 0.747 0.45 1.40
1.84 0.769 0.46 1.41
1.88 0.788 0.47 1.42
1.92 0.807 0.48 1.43
1.96 0.825 0.49 1.44
2.00 0.841 0.50 1.45
2.04 0.855 0.51 1.47
2.08 0.870 0.52 1.48
2.12 0.882 0.53 1.49
2.16 0.892 0.54 1.50
2.20 0.903 0.55 1.51
2.24 0.912 0.56 1.53
2.28 0.918 0.57 1.54
2.32 0.926 0.58 1.55
2.36 0.934 0.59 1.56
2.40 0.941 0.60 1.57
2.44 0.947 0.61 1.59
2.48 0.952 0.62 1.60
2.52 0.957 0.63 1.61
2.56 0.961 0.64 1.63
2.60 0.964 0.65 1.64
2.64 0.968 0.66 1.65
2.68 0.972 0.67 1.67
2.72 0.974 0.68 1.69
2.76 0.977 0.69 1.70
2.80 0.980 0.70 1.72
2.84 0.982 0.71 1.73
2.88 0.984 0.72 1.75
2.92 0.985 0.73 1.77
2.96 0.987 0.74 1.79
3.00 0.988 0.75 1.80
3.04 0.990 0.76 1.82
3.08 0.992 0.77 1.84
3.12 0.993 0.78 1.86
3.16 0.994 0.79 1.88
3.20 0.995 0.80 1.90
3.24 0.996 0.81 1.93
3.28 0.996 0.82 1.95
3.32 0.997 0.83 1.97
3.36 0.997 0.84 2.00
3.40 0.998 0.85 2.03
3.44 0.998 0.86 2.05
3.48 0.998 0.87 2.08
3.52 0.998 0.88 2.12
3.56 0.998 0.89 2.15
3.60 0.999 0.90 2.19
3.64 0.999 0.91 2.23
3.68 0.999 0.92 2.28
3.72 0.999 0.93 2.33
3.76 0.999 0.94 2.39
3.80 0.999 0.95 2.46
3.84 0.999 0.96 2.55
3.88 0.999 0.97 2.66
3.92 0.999 0.98 2.81
3.96 1.00 0.99 3.03
4.00 1.00 1.00 4.70

2-dimensional correlated

The distribution function of supremum over (s,t) in [0,1]2 of |WH(s,t)| is the limit as n tends to infinity of, and is approximated by,
P{ | ξnH(s,t) - nH(s,t) | / √n < x for all 0< s,t < 1}
where ξnH(s,t) is a Poisson process on [0,1]2 with expected value:
E[ ξnH(s,t) ] = nH(s,t).
Here H(s,t) is the following copula function:
H(s,t) = F(F-1(s), F-1(t); r)
where F-1(x) is the quantile function (inverse) of the standard normal distribution function and F(x, y; r) is the bi-variate normal distribution function with mean 0, variances 1 and correlation coefficient r.
  • n=5000, r=0.5 (graph of edf [PDF, 20KB]; plot of distribution of locations of suprema [PDF, 580KB])
  • n=5000, r=-0.5 (graph of edf [PDF, 20KB]; plot of distribution of locations of suprema [PDF, 580KB])
  • n=5000, r=-0.5, 0 and 0.5 (a comparison of three different correlation coefficients) [PDF, 50KB]; which illustrates that the graphs are remarkably similar for quite dissimilar copula functions.

    In the location plots, red points are those locations for which sup|WH| = sup WH and green points are those ones for which sup|WH| = -inf WH.

    These tables have been simulated with m=20,000 replications for n=5000.
    r = 0.5
    		r = -0.5
    x Pr{sup|WH|≤x} Pr{sup|WH|≤x} x x Pr{sup|WH|≤x} Pr{sup|WH|≤x} x
    0.00 0.000 0.00 0.592 0.00 0.000 0.00 0.677
    0.04 0.000 0.01 0.781 0.04 0.000 0.01 0.875
    0.08 0.000 0.02 0.825 0.08 0.000 0.02 0.918
    0.12 0.000 0.03 0.857 0.12 0.000 0.03 0.949
    0.16 0.000 0.04 0.882 0.16 0.000 0.04 0.977
    0.20 0.000 0.05 0.902 0.20 0.000 0.05 0.996
    0.24 0.000 0.06 0.921 0.24 0.000 0.06 1.02
    0.28 0.000 0.07 0.937 0.28 0.000 0.07 1.03
    0.32 0.000 0.08 0.951 0.32 0.000 0.08 1.05
    0.36 0.000 0.09 0.964 0.36 0.000 0.09 1.06
    0.40 0.000 0.10 0.977 0.40 0.000 0.10 1.08
    0.44 0.000 0.11 0.991 0.44 0.000 0.11 1.09
    0.48 0.000 0.12 1.00 0.48 0.000 0.12 1.10
    0.52 0.000 0.13 1.02 0.52 0.000 0.13 1.12
    0.56 0.000 0.14 1.03 0.56 0.000 0.14 1.13
    0.60 0.000 0.15 1.04 0.60 0.000 0.15 1.14
    0.64 0.000 0.16 1.05 0.64 0.000 0.16 1.15
    0.68 0.001 0.17 1.06 0.68 0.000 0.17 1.16
    0.72 0.002 0.18 1.07 0.72 0.000 0.18 1.17
    0.76 0.006 0.19 1.08 0.76 0.001 0.19 1.18
    0.80 0.014 0.20 1.09 0.80 0.002 0.20 1.19
    0.84 0.025 0.21 1.10 0.84 0.005 0.21 1.20
    0.88 0.040 0.22 1.11 0.88 0.011 0.22 1.21
    0.92 0.060 0.23 1.12 0.92 0.021 0.23 1.22
    0.96 0.086 0.24 1.13 0.96 0.034 0.24 1.24
    1.00 0.118 0.25 1.14 1.00 0.053 0.25 1.25
    1.04 0.152 0.26 1.15 1.04 0.074 0.26 1.26
    1.08 0.191 0.27 1.16 1.08 0.101 0.27 1.27
    1.12 0.230 0.28 1.17 1.12 0.133 0.28 1.28
    1.16 0.270 0.29 1.18 1.16 0.168 0.29 1.29
    1.20 0.308 0.30 1.19 1.20 0.206 0.30 1.30
    1.24 0.346 0.31 1.20 1.24 0.244 0.31 1.31
    1.28 0.385 0.32 1.21 1.28 0.284 0.32 1.31
    1.32 0.424 0.33 1.22 1.32 0.325 0.33 1.32
    1.36 0.461 0.34 1.23 1.36 0.366 0.34 1.33
    1.40 0.497 0.35 1.24 1.40 0.406 0.35 1.34
    1.44 0.533 0.36 1.25 1.44 0.446 0.36 1.35
    1.48 0.566 0.37 1.26 1.48 0.483 0.37 1.36
    1.52 0.597 0.38 1.27 1.52 0.519 0.38 1.37
    1.56 0.625 0.39 1.29 1.56 0.553 0.39 1.38
    1.60 0.654 0.40 1.29 1.60 0.588 0.40 1.39
    1.64 0.679 0.41 1.31 1.64 0.621 0.41 1.40
    1.68 0.704 0.42 1.32 1.68 0.649 0.42 1.42
    1.72 0.725 0.43 1.33 1.72 0.677 0.43 1.42
    1.76 0.746 0.44 1.34 1.76 0.703 0.44 1.43
    1.80 0.767 0.45 1.35 1.80 0.728 0.45 1.44
    1.84 0.787 0.46 1.36 1.84 0.751 0.46 1.46
    1.88 0.803 0.47 1.37 1.88 0.772 0.47 1.47
    1.92 0.822 0.48 1.38 1.92 0.792 0.48 1.48
    1.96 0.837 0.49 1.39 1.96 0.811 0.49 1.49
    2.00 0.852 0.50 1.40 2.00 0.827 0.50 1.50
    2.04 0.865 0.51 1.41 2.04 0.844 0.51 1.51
    2.08 0.876 0.52 1.43 2.08 0.859 0.52 1.52
    2.12 0.887 0.53 1.44 2.12 0.872 0.53 1.53
    2.16 0.901 0.54 1.45 2.16 0.884 0.54 1.54
    2.20 0.911 0.55 1.46 2.20 0.895 0.55 1.56
    2.24 0.918 0.56 1.47 2.24 0.905 0.56 1.57
    2.28 0.927 0.57 1.48 2.28 0.915 0.57 1.58
    2.32 0.934 0.58 1.50 2.32 0.924 0.58 1.59
    2.36 0.940 0.59 1.51 2.36 0.932 0.59 1.60
    2.40 0.946 0.60 1.52 2.40 0.938 0.60 1.61
    2.44 0.951 0.61 1.54 2.44 0.945 0.61 1.63
    2.48 0.956 0.62 1.55 2.48 0.950 0.62 1.64
    2.52 0.960 0.63 1.57 2.52 0.954 0.63 1.65
    2.56 0.965 0.64 1.58 2.56 0.959 0.64 1.67
    2.60 0.968 0.65 1.59 2.60 0.963 0.65 1.68
    2.64 0.972 0.66 1.61 2.64 0.968 0.66 1.70
    2.68 0.975 0.67 1.62 2.68 0.972 0.67 1.71
    2.72 0.977 0.68 1.64 2.72 0.975 0.68 1.72
    2.76 0.980 0.69 1.66 2.76 0.977 0.69 1.74
    2.80 0.982 0.70 1.67 2.80 0.979 0.70 1.75
    2.84 0.984 0.71 1.69 2.84 0.981 0.71 1.77
    2.88 0.986 0.72 1.71 2.88 0.983 0.72 1.79
    2.92 0.988 0.73 1.73 2.92 0.985 0.73 1.80
    2.96 0.989 0.74 1.75 2.96 0.987 0.74 1.82
    3.00 0.990 0.75 1.77 3.00 0.988 0.75 1.84
    3.04 0.991 0.76 1.79 3.04 0.990 0.76 1.86
    3.08 0.992 0.77 1.81 3.08 0.991 0.77 1.88
    3.12 0.993 0.78 1.82 3.12 0.993 0.78 1.89
    3.16 0.994 0.79 1.85 3.16 0.993 0.79 1.91
    3.20 0.995 0.80 1.87 3.20 0.994 0.80 1.94
    3.24 0.996 0.81 1.89 3.24 0.995 0.81 1.96
    3.28 0.996 0.82 1.92 3.28 0.996 0.82 1.98
    3.32 0.997 0.83 1.94 3.32 0.996 0.83 2.01
    3.36 0.997 0.84 1.97 3.36 0.996 0.84 2.03
    3.40 0.998 0.85 1.99 3.40 0.997 0.85 2.06
    3.44 0.998 0.86 2.02 3.44 0.998 0.86 2.08
    3.48 0.998 0.87 2.06 3.48 0.998 0.87 2.11
    3.52 0.998 0.88 2.09 3.52 0.998 0.88 2.15
    3.56 0.999 0.89 2.13 3.56 0.998 0.89 2.18
    3.60 0.999 0.90 2.16 3.60 0.998 0.90 2.22
    3.64 0.999 0.91 2.20 3.64 0.999 0.91 2.26
    3.68 0.999 0.92 2.25 3.68 0.999 0.92 2.30
    3.72 0.999 0.93 2.30 3.72 0.999 0.93 2.35
    3.76 0.999 0.94 2.36 3.76 0.999 0.94 2.41
    3.80 0.999 0.95 2.43 3.80 0.999 0.95 2.48
    3.84 0.999 0.96 2.52 3.84 0.999 0.96 2.57
    3.88 1.00 0.97 2.62 3.88 1.00 0.97 2.66
    3.92 1.00 0.98 2.76 3.92 1.00 0.98 2.81
    3.96 1.00 0.99 3.00 3.96 1.00 0.99 3.05
    4.00 1.00 1.00 4.33 4.00 1.00 1.00 4.40