User:DavidS2013/sandbox

Proof: The Gaussian random walk can be thought of as the sum of series of independent and identically distributed random variables, Xi from the inverse cumulative normal distribution with mean equal zero and σ of the original inverse cumulative normal distribution plus the mean, μ, of the original inverse cumulative normal distribution:
 * Z = μ + $$\sum_{i=0}^n {X_i}$$,

but we have the distribution for the sum of two independent normally distributed random variables, Z = X + Y, is given by
 * N(μX + μY, σ2X + σ2Y) (see here).

In our case, μX = μY = 0 and σ2X = σ2Y = σ2 yield
 * N(0, 2σ2).

By induction, for n steps we have
 * Z ~ μ + N(0, nσ2) ~ N(μ, nσ2).

But for the Gaussian random walk, this is just the standard deviation of the random walk distribution after n steps. Since the root mean square(rms) translation distance is one standard deviation of the resultant distribution of the random walk, there is 68.27% probability that the rms translation distance after n steps will fall between ± σ$$\sqrt{n}$$. Likewise, there is 50% probability that the translation distance after n steps will fall between ± 0.6745σ$$\sqrt{n}$$.