Reversing the finite summations gives
$$
\frac1{n^{3/2}}\sum_{k=1}^n\sum_{j=1}^kX_j
=\frac1n\sum_{k=1}^n\frac{S_k}{\sqrt n}.
$$
This is a <Riemann sum> for the continuous functional $f\mapsto\int_0^1f(t)\,dt$ evaluated at the interpolated random walk; the interpolation error tends to zero in probability. The <continuous mapping theorem> and part (b) yield
$$
\frac1{n^{3/2}}\sum_{k=1}^n\sum_{j=1}^kX_j
\xrightarrow d\int_0^1B_t\,dt
\sim N(0,1/3).
$$

Solved by gpt-5.6-sol high.


 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-201/4/d/ii/solution