In the book "The Mathematical Experience" it says:
"An infinite [binary] sequence $x_1, x_2, \ldots$ is called random in the sense of von Mises if every infinite sequence $x_{n_1}, x_{n_2}, \ldots$ extracted from it and determined by a policy or rule R is $\infty$-distributed. Now comes the shocker. It has been established by Joseph Doob that there are no sequences that are random in the sense of von Mises."
A sequence on $\{H,T\}$ is $\infty$-distributed if for each positive integer $k$ and sequence $\vec y \in \{H,T\}^k$ the set $\{n\in {\mathbb N} \colon \langle x_{n},\dots,x_{n+k-1}\rangle=\vec y\}$ has density $2^{-k}$.
But the definition of von Mises seems so natural to me that if a sequence does not satisfy it then the sequence is not random.
