# Problem statement

Let $\mathbb{X} \subset \mathbb{R}^{k}$ and let $p:\mathbb{X}\times\mathbb{X}\rightarrow\mathbb{R}$ be a density kernel on $\mathbb{X}$. We assume the following model for the stochastic process \begin{equation} X_{t+1}\sim p(X_t, y)dy \quad (t\geq1), \end{equation} which defines a discrete time Markov process ${(X_t)}_{t\geq1}$ on $\mathbb{X}$. Let $X_0$ be a given initial state, and let $(y_i)_{i=1}^{T}$ be an observation, or realization, of that Markov process.

Furthermore, assume that we have observed another $n$ independent paths ${(x_{i}^{j})}_{i=1}^{T}$, where $j=1,\dots, n$, from the discrete time Markov process.

Would anyone be able to tell me what can be said about the equation \begin{equation} \frac{1}{n}\sum_{j=1}^{n}\prod_{i=1}^{T}p(x_{i-1}^{j}, y_i), \end{equation} and in particular what happens in the limit as $n\rightarrow\infty$?

This has been proposed to me as a way of estimating the likelihood, or density, of the observed path $(y_i)_{i=1}^{T}$ for the assumed model of the stochastic process, with the motivation that it propagates the uncertainties of the model, but I have been unsuccessful in understanding the equation.

I hope that this is clear enough to at least give a hint which may lead me down the proper path.

# Insights (update)

I have found a paper by Stachurski et. al (2008), which give some insight for the problem.

The marginal distribution of $y$ at time $t$, $\mathbb{P}(X_t = y)$, can be written (assuming a given initial condition) as \begin{equation} \psi_{t}(y) = \int p(x,y)\psi_{t-1}(x)dx, \end{equation} and we define an estimate of this as \begin{equation} \psi_{t}^{n}(y) \equiv \frac{1}{n}\sum_{i=1}^{n}p(x_{t-1}^{i}, y), \end{equation} where ${(x_{t}^{i})}_{i=1}^{n}$ are $n$ i.i.d realizations, or observations, of the Markov process at time $t$.

By the law of large numbers we get \begin{equation} \frac{1}{n}\sum_{i=1}^{n}p(x_{t-1}^{i}, y) \rightarrow \mathbb{E}p(x_{t-1}^{i}, y) = \int p(x,y)\psi_{t-1}(x)dx = \psi_t(y), \end{equation} as $n\rightarrow\infty$ (Stachurski, 2008).

However, this only helps us get insight into estimates of the marginal probability distributions of the discrete time Markov process, and not the joint probability distributions.

We could make the assumption that ${(X_t)}_{t\geq1}$ are i.i.d (which is incorrect by the definition of the model, but lets see what happens), which would allow me to write \begin{equation} \label{eq:likelihood_1} \mathbb{P}(X_1=x_1, \dots, X_T=x_T) \approx \prod_{i=1}^{T} \psi_{i}^{n}(x_i) = \prod_{i=1}^{T} \frac{1}{n}\sum_{j=1}^{n}p(x_{t-1}^{j}, x_t), \end{equation} which looks very similar to the equation I have an interest in, the major two differences being that the product is in a different place and the way we draw ${(x_{t}^{i})}_{i=1}^{n}$.

While this helps give some insight into the problem, I am still unable to understand the equation of interest. Namely, \begin{equation} \frac{1}{n}\sum_{j=1}^{n}\prod_{i=1}^{T}p(x_{i-1}^{j}, y_i). \end{equation}