 Solution

Write the one-dimensional random walk endpoint as $x_m={\sum }_{j=1}^m{\xi }_j$, starting at the origin. Assume unbiased, independent increments, each with $⟨{\xi }_j⟩=0$ and $⟨{\xi }_j^2⟩=\Delta$. Expanding the square givesIndependence makes each cross expectation $⟨{\xi }_i⟩⟨{\xi }_j⟩=0$. ConsequentlyUncorrelated increments with the same second moments would suffice. Unbiasedness alone does not remove correlations: persistent or anticorrelated steps need not satisfy this result. The symbol $\Delta$ is already the mean-square step size, with units of length squared.