Let’s suppose that we have a random vector $X_0 \sim U(-1,1)^3$ which we transform using a sequence of random matrices $M_i \sim U(-1,1)^{3 \times 3}$. We may then define the following:

\begin{equation} \pi_{n} = \prod_{i=1}^n M_i \end{equation}

\begin{equation} X_n = \pi_n X_0 \end{equation}

Now, based on numerical experiments I conjecture that $\{X_i\}_{i=1}^n$ behaves like a Brownian type motion and:

\begin{equation} \forall C \in [0, \infty) , \lim_{N \to \infty} P(\lVert X_N -X_0 \rVert \geq C \quad i.o.) = 1 \end{equation}

In order to proceed with a mathematical analysis of this problem it’s useful to note that the product of two uniformly distributed random variables $x_1, x_2 \sim U(0,1)$ aren’t uniform. In fact, it can be easily shown that:

\begin{equation} P(0 \leq x_1 \cdot x_2 \leq z) = z - z\ln z \end{equation}

I shall definitely return to this problem next week and one motivation for it is that I think $\{X_i\}_{i=1}^n$ may be used to approximate any space of continuous functions with compact support in $\mathbb{R}^3$ as $N \to \infty$. Assuming this is true, I’m certain it generalises readily to $\mathbb{R}^3$.

Below are references which I have yet to consult that may be useful for further investigations.

1. J.R. Ipsen. Products of Independent Gaussian Random Matrices. 2015.
2. Vladislav Kargin. Products of Random Matrices: Dimension and Growth in Norm. 2010.
3. Carl P. Dettmann & Orestis Georgiou. Product of n independent Uniform Random Variables. 2009.