# Asymptotic upper densities in infinite binary stochastic processes

Consider an infinite binary process $$X=X_1,X_2,\ldots$$ (with corresponding probability $$P$$). For some bits $$1$$ is less probable than $$0$$. I am interested in the following asymptotic upper density : $$\limsup_{n\to\infty}\frac{|\{i\leq n:X_i=1\wedge (P(X_i=1|X_1^{i-1})<1/2)\} |}{|\{i\leq n:(P(X_i=1|X_1^{i-1})<1/2)\}|}$$ In other words, we look at those bits of the realization where $$1$$ is less probable and ask: what is the upper density of the ratio of such bits being one to all such bits?

For a well-behaving process (for example, i.i.d. or ergodic) such upper density will be equal or lesser than $$1/2$$ (almost surely). My intuition is that it should be true in general case as well, but I am struggling to prove it.

Perhaps there are some exotic counterexamples?

• Certainly it would be easy to attain 1/2 by having a sequence of independent but not identically distributed 0–1 valued random variables, with probability of 1 approaching 1/2 from below. – Anthony Quas Mar 11 at 4:16
• OK, that seems to be right, but I am more interested in going above $1/2$. I will edit the question. Thanks – Tomek Steifer Mar 12 at 10:31