Asymptotic theory

It is often the case that the aggregate behavior of combined effect is easier to describe than the individual activities.

Brownian Motion is an example of a process that is complex at the molecular scale, but gives way to some reasonably easy to use aggregate physical properties at the large scale.

One aim of statistics is to describe the behaviour of MLE \hat{\theta} as the sample size goes to $$\infty$$

behavior as $$n \rightarrow \infty$$ is described as asymptotic

modes of convergence

In statistics, the point of doing asymptotic theory is not just aesthetic, but also very practical. When n is sufficiently large, the asymptotic properties of an estimator will hold, approximately, when it’s obtained from a sample of size n. In other words, asymptotic theory provides the theoretical justification for approximations that can be applied to practical problems involving finite values of n, for example the calculation of confidence limits and hypothesis tests, as described in Unit 6.

For this reason, asymptotic theory occupies a central place in mathematical statistics. Arguably, it provides much of the unifying theory that gives statistics its coherence as a discipline, underpins its everyday practice, and provides its scientific rationale. The aim of this unit is to describe some of the key features of asymptotic statistical theory, particularly as it relates to inferences based on the likelihood.

Law of Large numbers
the settling down phemomenon is the law of large numbers

weak law of large numbers

$$Xbar = 1/n S_n goes to \mu$$

Chebyshev inequality
$$P(|X|\geq a) \leq \frac{E\{X^2\}}{a^2}$$