Background:
Martin-Löf's way of defining randomness of finite strings (over a finite alphabet such as $\{0,1\}$) and infinite sequences uses a generalized notion of a statistical test.
Often, when this idea is presented, there is a background assumption that strings should be treated as if they were generated by independent, uniformly distributed trials. That is, strings of length $n$ are treated as having probability $Q^n$, where $Q$ is the size of the alphabet. This is so, for example, in Calude's Information and Randomness.
However, Martin-Löf's 1966 paper was more general, allowing the randomness to be relative to any probability distribution over strings. Li and Vitanyi's An Introduction to Kolmogorov Complexity and Its Applications captures this idea in its definition of a Martin-Löf test (section 2.4.1, p. 135, 3rd ed.), where one of the requirements for a Martin-Löf test is that the sum of probabilities of strings $x$ (of a particular length) that fail the randomness test $\delta$ at level $m$--i.e. the probability of strings for which $\delta(x)\geq m$--should be less than or equal to a small number $2^{-m}$ (analogous to an $\alpha$ level in statistics):
$$\sum_x \mathsf{P}[x|l(x)=n \mbox{ and } \delta(x)\geq m] \leq 2^{-m}$$
The Question:
I'm interested in thinking about this idea for cases in which the probability distribution over strings is not an independent, uniform distribution. For example, in a random walk, perhaps assymetric, the probability of reaching element $k$ at step $t$ varies conditional on the previous element at $t-1$. This kind of example seems entirely consistent with the general idea of a Martin-Löf test relative to a probability distribution over strings, but I have not found any examples illustrating this kind of idea.
Can you point me to articles or papers that give examples of Martin-Löf/Kolmogorov/Chaitin randomness relative to a distribution on strings that doesn't assume independence? Or if you care to provide an illustration yourself, that would be great, but extended discussions in publications, blog post, etc. would be particularly helpful. Thanks. (My level of sophistication in this area is not very high, but I'm willing to get what I can from literature that's strictly over my head.)
(I've looked for such examples in Li and Vitanyi, and in Martin-Löf's paper, but have not noticed any. I have looked in other books, but either tests are specified only relative to uniform distributions, or the machinery in the examples used is too complex for me to figure out quickly what I'm looking at--and I suspect that the examples only involve a uniform distribution anyway.)