= 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 $\langle\xi_j\rangle=0$ and $\langle\xi_j^2\rangle=\Delta$. Expanding the square gives
$$
\langle x_m^2\rangle=\sum_j\langle\xi_j^2\rangle+2\sum_{i<j}\langle\xi_i\xi_j\rangle.
$$
Independence makes each cross expectation $\langle\xi_i\rangle\langle\xi_j\rangle=0$. Consequently
$$
\boxed{\langle x_m\rangle=0,\qquad \langle x_m^2\rangle=m\Delta.}
$$
Uncorrelated 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.
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