In probability theory, the dimension doubling theorems are two results about the Hausdorff dimension of an image of a Brownian motion.
In their core both statements say, that the dimension of a set
under a Brownian motion doubles almost surely.
The first result is due to Henry P. McKean jr and hence called McKean's theorem (1955).
The second theorem is a refinement of McKean's result and called Kaufman's theorem (1969) since it was proven by Robert Kaufman.
be a Brownian motion in dimension
be a Brownian motion in dimension
, we have The difference of the theorems is the following: in McKean's result the
-null sets, where the statement is not true, depends on the choice of
Kaufman's result on the other hand is true for all choices of
This means Kaufman's theorem can also be applied to random sets