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 process starts from zero (does it matter?). 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.

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.

Given the following stochastic process

\begin{equation} x_t = \frac{u_t}{\sum_{s=1}^{t-1} u_s} \end{equation}

where $u_t \overset{i.i.d.}{\sim} \mathcal{N}(0,\sigma^2)$, $\sigma^2<\infty$, 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.

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 process starts from zero (does it matter?). 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.