Integrated random-walk limit ($n^{-3/2}{\sum }_{k=1}^n{S}_k\stackrel{d}{\to }{\int }_0^1{B}_{t}dt$)

For a random walk with independent identically distributed steps of mean zero and variance one, the Donsker invariance principle and the continuous mapping theorem for integration give this limit. The integral of the polygonal interpolation differs from the right-endpoint sum by $S_n/(2n^{3/2})$, whose squared expectation is $1/(4n^2)$. The Slutsky theorem removes that error. The limiting variable is the time-one value of integrated Brownian motion, with law $N(0,1/3)$.