Prove that a process is memoryless (simple example) Given the following stochastic process
\begin{equation}
    x_t = \frac{u_t}{\sum_{s=1}^{t-1} u_s}
\end{equation}
where $u_t \overset{i.i.d.}{\sim} \mathcal{N}(0,\sigma^2)$, $\sigma^2<\infty$, and the first observation is not available (NA) since $x_t$ is basically $\Delta R/R$ where R is a random walk. Prove or disprove that $x_t$ is memoryless. The definition of memoryless should be with respect to the Markov property, but if you know different ways to tackle the problem I would like to see different solutions
Intuitively, I would say that it is indeed memoryless, as at the numerator we have a white noise, and the denominator is still white noise given that the sum of i.i.d. white noises is still white noise. I woul like to see a rigorous proof. Thank you.

Hint: With a bit of recursive algebra it can be shown that:
$$\frac{u_t}{u_1} = x_t \prod_{i=2}^{t-1} (1+x_i),$$
which gives the recursive form:
$$x_t = \frac{u_t}{u_1 \prod_{i=2}^{t-1} (1+x_i)}.$$
Have a go at deriving the above equations and see if this tells you anything about the "memory" of the stochastic process.
