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).

1. For the first level $\ell=1$, let $\{X_t^{(\ell)}\}_{t=1}^{T}$ be a (discrete-time) random walk.

2. 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.

3. 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?

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).

1. For the first level $\ell=1$, let $\{X_t^{(\ell)}\}_{t=1}^{T}$ be a (discrete-time) random walk.

2. 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.

3. 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?

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).

1. For the first level $\ell=1$, let $\{X_t^{(\ell)}\}_{t=1}^{T}$ be a (discrete-time) random walk.

2. 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.

3. 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?

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).

1. For the first level $\ell=1$, let $\{X_t^{(\ell)}\}_{t=1}^{T}$ be a (discrete-time) random walk.

2. 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.

3. 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?