= Continuous-time symmetric simple random walk
{title2=$\mathbb E e^{i\theta X_t}=e^{-t(1-\cos\theta)}$}
= Poissonized simple symmetric random walk
{c}
{synonym}
At unit jump rate, this is $X_t=\sum_{k=1}^{N_t}\xi_k$, where $N$ is a rate-one <Poisson process> and the independent marks are uniform signs. Equivalently it is the difference of two independent rate-$1/2$ Poisson processes. It is a <Lévy process> with characteristic function $\exp\{t(\cos\theta-1)\}$, mean zero, <variance> $t$, and <Lévy characteristic exponent> $1-\cos\theta$ in the negative-exponent convention. A general total jump rate rescales time.
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