I am interested in studying the limit behavior of a system described by a sequence of variables $(\alpha_t)_{t=0}^\infty$ which take values in $[0,1]$ whose law of motion is as follows: $$\begin{cases}\alpha_0\in [0,1]\\ \alpha_{t+1}=\kappa\alpha_t+(1-\kappa)w_t \end{cases}$$ where $\kappa\in (0,1)$ is a persistence parameter and $w_t$ is the realization of a Bernoulli $W_t$, whose probability of sucess depends on $\alpha_t$: $$W_t\sim \textrm{Bern}(F(\alpha_t))$$ where $F:[0,1]\to [0,1]$ is a surjective and weakly decreasing function The specific shape of $F$ depends on the primitive of the model I am studying, but it is always "step like".
In particular it is always the case that, $$\exists 0<a<b<1\quad st.\quad F(\alpha)=1\quad \forall \alpha\leq a\quad\land\quad F(\alpha)=0\quad \forall \alpha\geq b$$ then, either
- $F$ is strictly decreasing in $(a.b)$ or
- $F$ has a constant piece $(a^{\prime},b^{\prime})$ for some $a^{\prime}>a$ and $b^{\prime}<b$ where $F((a^{\prime},b^{\prime}))=\frac{1}{2}$ and it is strictly decreasing otherwise.
Notice that $\alpha_t$ should rewrite $\alpha_t=\kappa^t\alpha_0+(1-\kappa)\sum_{i=0}^t\kappa^{t-i}w_{i-1}$, whence the title of the question.
My question is: What can I say about the limit behavior of $\alpha$?
It seems clear to me that, doing a qualitatively study of the system, almost surely, $\alpha_t$ will eventually be in $(a-\delta_a,b+\delta_b)$ where $\delta$'s are due to the discrete sizes of the jumps of $\alpha_t$. But beyond this, my null experience with stochastic processes and dynamical systems lets me say nothing.
Another variable of interest for me would be the average of the Bernoullis:
$$\theta_{t}=\frac{\sum_{i=1}^t W_i}{t}$$
Usually, one uses some law of large numbers to study its convergence. Here, the Bernoullis are all correlated and not identically distributed, so the standard results do not apply. Is there any result I could exploit?
Any help or reference on systems of this kind would be really appreciated.
To analyze the limit behavior of $\alpha$, we can first consider its expected value, $E[\alpha_{t+1}]$, given $\alpha_t$:
$$E[\alpha_{t+1} | \alpha_t] = \kappa \alpha_t + (1 - \kappa) E[W_t | \alpha_t]$$
Since $W_t$ is a Bernoulli random variable with success probability $F(\alpha_t)$, we have:
$$E[W_t | \alpha_t] = F(\alpha_t)$$
Now, we can substitute this into the expected value of $\alpha_{t+1}$:
$$E[\alpha_{t+1} | \alpha_t] = \kappa \alpha_t + (1 - \kappa) F(\alpha_t)$$
The given properties of $F(\alpha)$ help us understand how the system behaves. For $\alpha_t \leq a$, $F(\alpha_t) = 1$, and for $\alpha_t \geq b$, $F(\alpha_t) = 0$. In the interval $(a, b)$, $F(\alpha_t)$ is strictly decreasing or has a constant piece $(a', b')$ where $F(\alpha) = \frac{1}{2}$.
For the limit behavior of $\alpha$, we can analyze different cases:
If $\alpha_t$ is such that $\alpha_t \leq a$, then $E[\alpha_{t+1} | \alpha_t] = \kappa \alpha_t + (1 - \kappa)$. As $\kappa \in (0, 1)$, the expected value of $\alpha_{t+1}$ will be greater than $\alpha_t$.
If $\alpha_t$ is such that $\alpha_t \geq b$, then $E[\alpha_{t+1} | \alpha_t] = \kappa \alpha_t$. As $\kappa \in (0, 1)$, the expected value of $\alpha_{t+1}$ will be less than $\alpha_t$.
If $\alpha_t$ is in the interval $(a, b)$, then the expected value of $\alpha_{t+1}$ will depend on the specific shape of $F(\alpha_t)$. For a strictly decreasing $F(\alpha_t)$ in the interval $(a, b)$, the value of $E[\alpha_{t+1} | \alpha_t]$ will lie between $\kappa \alpha_t$ and $$\kappa \alpha_t + (1 - \kappa)$$
Given these cases, the expected value of $\alpha_{t+1}$ given $\alpha_t$ does not necessarily imply convergence of the sequence $\alpha_t$. Furthermore, determining the exact limit behavior of $\alpha$ will depend on the specific shape of $F(\alpha)$ and the initial conditions.
To analyze the limit behavior, you might want to consider Markov chain analysis, particularly for discrete-time stochastic processes. The behavior of the sequence $(\alpha_t)_{t=0}^\infty$ depends on the specific shape of the function $F(\alpha)$ and the properties of the underlying stochastic process.
One possible way to analyze the limit behavior of $\alpha_t$ would be to study its ergodic properties. To do so, you can explore the dynamics of the system by constructing a Markov chain that describes the transitions between different values of $\alpha_t$. This Markov chain would be determined by the function $F(\alpha)$ and the persistence parameter $\kappa$.
You can then investigate the stationary distribution of the Markov chain and study the conditions under which the chain is ergodic. If the Markov chain is ergodic, the system will have a unique stationary distribution, and the sequence $(\alpha_t)_{t=0}^\infty$ will converge to this distribution in the long run, at least in a probabilistic sense.
However, determining the exact limit behavior of $\alpha_t$ will still depend on the specific shape of $F(\alpha)$ and the initial conditions. Further investigation and more rigorous analysis would be needed to make any definitive claims about the convergence of the sequence $(\alpha_t)_{t=0}^\infty$.