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I have the following stochastic sequence $t_n$ with the following recurrent equation:

$$t_n = \max(t_{n-1} - \tau, 0) + a_n,$$

where $\tau$ is known constant, $a_n$ are independent and identically distributed random variables with mean $\mu_0$ and variance $\sigma_0^2$.

I would like to know stationary conditions for this process. Any references are also welcome since I don't know how this process is properly called in the literature.

0x2207
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    In dynamical systems, the phenomenon is called steady state. I am familiar with deterministic systems, where the equations remain the same. And so one can analytically determine the steady state. But for stochastic system, the steady state at one iteration may no longer be a steady state in the next, since the equations are changing independently. So I'm not sure what would be an example of stationary conditions for stochastic systems. – Tony Mathew Jun 04 '23 at 11:45
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    Imagine stochastic equation $y_n = \phi y_{n-1} + a_n$, where $\phi$ is constant and $a_n$ are i.i.d. One can show that if $|\phi|<1$ then the process is stationary, i.e. $y_n$ have the same mean and variance at every step $n$, otherwise ($|\phi|>1$) the process is not stationary, i.e. the variance grows unlimited as $n\rightarrow\infty$ – 0x2207 Jun 04 '23 at 11:53

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