= Fourth-moment tightness of polygonal random walks
{title2=$\mathbb E|S_t^N-S_s^N|^4\le\max(3,m_4)|t-s|^2$}
For centered unit-variance <independent and identically distributed> increments with $m_4=\mathbb E\xi^4<\infty$, an interpolated increment is $N^{-1/2}\sum_jc_j\xi_j$. Here $c_j$ is the length of the overlap of $(Ns,Nt)$ with the $j$th unit cell, so $0\le c_j\le1$ and $\sum c_j=N(t-s)$. <Independence> gives its <fourth moment> as $N^{-2}[3(\sum c_j^2)^2+(m_4-3)\sum c_j^4]$. This is at most $\max(3,m_4)N^{-2}(\sum c_j^2)^2$, and $\sum c_j^2\le N(t-s)$ gives the bound, including intervals crossing a mesh point. A uniform fourth-moment continuity criterion yields <tightness of probability measures> on $C_0[0,1]$.
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