English: A plot of the average of n simulated independent Bernoulli trials (each taking a value of +/- 1) to show the law of large numbers and the law of the iterated logorithm.

The graphical difficulty here is that we want to show details for very large values of n. If the y-axis were left untransformed, all we could see is that it converges to zero. If we used a log transformation it would be upset with the negative values. So the y is transformed by something that is very close to a log transformation, namely, y^.12. (The .12 looks nice. Other small powers look about the same.)

Humans are very good at seeing "straight lines". So if we transform the x-axis to be almost log also we can make the upper and lower envelopes look almost straight. The correct power is -1/2 of the power used for the y axis.

So the plot is x^-.06 vs y^.12. These are basically "log(x)" vs "log(y)". But by using these values, the natural square root behaviour now is a straight line. If you have suggestions for improving the image, let me know and I can generate a new one.

A detailed view of the right hand most "pixel" is: