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.
Let be a probability space. For a -dimensional Brownian motion and a set we define the image of under , i.e.
Let be a Brownian motion in dimension . Let , then
-almost surely.
Let be a Brownian motion in dimension . Then -almost surely, for any set , 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 simultaneously. This means Kaufman's theorem can also be applied to random sets .