3 added definition of $\infty$-distributed

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.

In the book "The mathematical experienceMathematical Experience" it says: "An infinite 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."
In the book "The mathematical experience" it says: "An infinite 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."