= Integrated random-walk limit
{title2=$n^{-3/2}\sum_{k=1}^nS_k\xrightarrow d\int_0^1B_t\,dt$}
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 $N(0,1/3)$.
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