= Scaling classification of a continuous-time symmetric simple random walk
{title2=$n^{-1/2}X_{nt}\Rightarrow B_t\quad\text{in finite-dimensional distributions}$}
For the unit-rate <continuous-time symmetric simple random walk>, the processes $n^{-\alpha}X_{nt}$ have three finite-dimensional regimes. For $0<\alpha<1/2$ no probability limit is possible, because the one-time characteristic functions tend to zero off the origin and are discontinuous there. At $\alpha=1/2$ the limit is standard <Brownian motion>. For $\alpha>1/2$ it is the identically zero <Lévy process>. This follows from $n(1-\cos(n^{-\alpha}\theta))\sim(\theta^2/2)n^{1-2\alpha}$ and independent increments. Thus a nonzero limit uniquely determines the exponent, whereas merely asking for a Lévy limit does not.
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