elementary number theory - How often does $pi$ contain a sequence of digits $x$ at decimal place $x$?

Today, I came across an interesting bit of trivia. The decimal expansion of $\pi$ contains the sequence of digits $79873884$ starting at decimal place $79873884$.

This is not unique, since another (perhaps trivial) example would be $1$ at decimal place $1$. But I wonder if there are more of these occurences.

Can anything be said about the frequency of these sequences? Are there infinitely many?