User:DavidS2013/sandbox

Proof: The Gaussian random walk can be thought of as the sum of series of independent and identically distributed random variables, X_i 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(μ_X + μ_Y, σ²_X + σ²_Y) (see here).

In our case, μ_X = μ_Y = 0 and σ²_X = σ²_Y = 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${\sqrt {n}}$.