Let us consider the following hierarchical (recursive) random walk model, which is also known as the hierarchical hidden Markov model in computer science (https://en.wikipedia.org/wiki/Hierarchical_hidden_Markov_model).
For the first level $\ell=1$, let $\{X_t^{(\ell)}\}_{t=1}^{T}$ be a (discrete-time) random walk.
For the next level $\ell=2$, we consider a set of $T$ random walks. In particular, the $t$-th $(t \in \{1,\ldots, T\})$ random walk in this set has parameters (especially transition probabilities) depending on the realization of $X_t^{(1)}$ in the first level.
In the same way we recursively generate the next level $\ell = 3$, which contains $T^2$ random walks, and so on to the $L$-th level.
It seems that results for this model is rather sparse in existing probability or statistics literature, or perhaps it has a different name. I would much appreciate any pointers to existing results.
A further question: Consider the concatenation of the $T^L$ random walks in the $L$-th level. Does this sequence have stronger long range correlation in comparison with canonical random walks?
