Is there a closed formula for the distribution of $x_t$ in the following random process, describing a random walk on a directed line?

• $x_0 = n$
• $x_t$ is a uniformly random integer between 1 and $x_{t - 1}$

UPD. Even an expression for $Pr[x_t = 1]$ would be of interest.

Is there a closed formula for the distribution of $x_t$ in the following random process, describing a random walk on a directed line?

• $x_0 = n$
• $x_t$ is a uniformly random integer between 1 and $x_{t - 1}$

UPD. Even an expression for $Pr[x_t = 1]$ would be of interest. A closed form approximation up to lower order terms is fine, e.g. $P[X_2 = 1] = \frac{\ln n }{ n} + \frac{c}{n} + o\left(\frac{1}{n}\right)$

Is there a closed formula for the distribution of $x_t$ in the following random process, describing a random walk on a directed line?

• $x_0 = n$
• $x_t$ is a uniformly random integer between 1 and $x_{t - 1}$

Is there a closed formula for the distribution of $x_t$ in the following random process, describing a random walk on a directed line?

• $x_0 = n$
• $x_t$ is a uniformly random integer between 1 and $x_{t - 1}$

UPD. Even an expression for $Pr[x_t = 1]$ would be of interest.

Is there a closed formula for the distribution of $x_t$ in the following random process?

• $x_0$ is a uniformly random integer between 1 and n
• $x_t$ is a uniformly random integer between 1 and $x_{t - 1}$

Is there a closed formula for the distribution of $x_t$ in the following random process, describing a random walk on a directed line?

• $x_0 = n$
• $x_t$ is a uniformly random integer between 1 and $x_{t - 1}$