# Potential uses for viewing discrete wavelets constructed by filter banks as hierarchical random walks.

I have some weak memory that some sources I have encountered a long time ago make some connection between random walks and wavelets, but I am quite sure it is not in the same sense.

What I was thinking is to in the matrices representing filtering operations replace every $a\in \mathbb{R}$: $$\cases {\phantom{-}a\to \begin{bmatrix}a&0\\0&a\end{bmatrix}\\-a\to \begin{bmatrix}0&a\\a&0\end{bmatrix}}$$ Assuming filter taps are real numbers to start with, we will then have made our matrix positive in the sense that each element $\geq 0$, furthermore we can choose to normalize so that the matrix becomes stochastic. All operations will then preserve the positivity because they will be permutations and subsamplings concatenated with new filterings which are in turn constructed in the same way.

And more generally : not necessarily filter banks, but any sequence of convolutions (assuming filters and signals consisting of real numbers) could be translated in the same way.

What would the implications of such a construction be? What could it be useful for? What would "happen" to the signal in this new probabilistic sense and how to make use of it?

Own work / example: The Haar system is usually considered one of the easiest to learn when introduced to the subject. It has a lowpass and a highpass filter of two taps each ( sum and difference ):

$$\left[\begin{array}{rr}1&1\\1&-1\end{array}\right] \to \frac{1}{2}\cdot \left[\begin{array}{rr|rr}1&0&1&0\\0&1&0&1\\\hline 1&0&0&1\\0&1&1&0\end{array}\right]$$

where the factor $\frac 1 2$ is normalization factor to get a stochastic matrix and the horizontal/vertical lines are just to make the block structure clearer. The stationary distribution (eigenvalue 1) has 1/4 in each bin.