be expected to be inadequate.

The assertion that the central limit theorem guarantees that $P(\Delta x, t)$ is a Gaussian in $\Delta x$ is sometimes describes as the “Gaussian Approximation”. Experiments such as those summarized above prove that this assertion is incorrect. The central limit theorem (for random variables) and Doob’s theorem (for random processes) are well-known. Where do their invocations go wrong? The central limit theorem and Doob’s theorem are statements about the sum of a large number of identically distributed, independent random processes. As applied to the diffusion problem, the displacement $\Delta x$ during experimentally accessible times can be expressed as the sum of a large number of far smaller steps $\delta x$, each taken during a far smaller time interval $\delta t$. If a random variable $\Delta x$ is the sum of a large number of identically distributed *independent* variables, i. e., if $\delta x(t)$ is a Markoff process, it must in general be the case that $P(\Delta x, t)$ is a Gaussian in $\Delta x$. This rationale fails because it refers to a sum of *independent* random processes. The process that generates $\delta x$ for probes in a viscoelastic fluid is not a Markoff process, because 