A well known argument in cosmology gives that the power spectrum ( or structure function ) P ( k ) of mass density fluctuations produced from a uniform initial state by physics which is causal ( i.e .
moves matter and momentum only up to a finite scale ) has the behaviour P ( k ) \propto k ^ { 4 } at small k .
Noting the assumption of analyticity at k = 0 of P ( k ) 
in the standard derivation of this result , we introduce a class of solvable one dimensional models which allows us to study the relation between the behaviour of P ( k ) at small k 
and the properties of the probability distribution f ( l ) for the spatial extent l of mass and momentum conserving fluctuations .
We find that the k ^ { 4 } behaviour is obtained in the case that the first six moments of f ( l ) 
are finite .
Interestingly the condition that the fluctuations be localised - taken to correspond to the convergence of the first two moments of f ( l ) - imposes only the weaker constraint P ( k ) \propto k ^ { n } 
with n anywhere in the range 0 < n \leq 4 .
We interpret this result to suggest that the causality bound will be loosened in this way if quantum fluctuations are permitted .