Random walks with step lengths equal to the shortest possible physically meaningful distances are considered from the point of view of special relativity involving two observers moving uniformly with respect to each other. A requirement of statistical equivalence of the probability distributions seen by those observers leads to the Lorentz transformations, provided a randomly moving particle shifts from one submicroscopic cell of uncertainty to a neighbor with a speed equivalent to that of light. Ordinary smooth motion would appear to involve a tremendous amount of submicroscopic back and forth randomness subject to a statistical bias favoring a particular direction. The diffusive nature of the motion naturally leads to a spreading of the probability distribution.
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