## Overview

In this lecture {doc}`this lecture <likelihood_ratio_process>` we described a peculiar property of a likelihood ratio process, namely, that it's mean equals one for all $t \geq 0$ despite it's converging to zero almost surely.

In {doc}`this lecture <likelihood_ratio_process>` we described a peculiar property of a likelihood ratio process, namely, that it's mean equals one for all $t \geq 0$ despite it's converging to zero almost surely.

While it is easy to verify that peculiar properly analytically (i.e., in population), it is challenging to use a computer simulation to verify it via an application of a law of large numbers that entails studying sample averages of repeated simulations.

## Mathematical Expectation of Likelihood Ratio

In lecture XXXX, we studied a likelihood ratio $\ell \left(\omega_t\right)$

In {doc}`this lecture <likelihood_ratio_process>`, we studied a likelihood ratio $\ell \left(\omega_t\right)$

$$

\ell \left( \omega_t \right) = \frac{f\left(\omega_t\right)}{g\left(\omega_t\right)}

Our goal is to approximate the mathematical expectation $E \left[ L\left(\omega^t\right) \right]$ well.

In Quantecon Lecture XXXX, we showed that $E \left[ L\left(\omega^t\right) \right]$ equals $1$ for all $t$.

In {doc}`this lecture <likelihood_ratio_process>`, we showed that $E \left[ L\left(\omega^t\right) \right]$ equals $1$ for all $t$.

We want to check out how well this holds if we replace $E$ by with sample averages from simulations.

This turns out to be easier said than done because for