StatlectThe Digital Textbook
Index

Asymptotic theory

Learn the basics of asymptotic theory: how sequences of random variables and random vectors are characterized, how their convergence is defined, what conditions guarantee their convergence.

Stochastic sequences

Sequences of random vectors

Generalizes the concepts introduced in the lecture on sequences of random variables

Sequences of random variables

Various definitions, including IID, stationary, ergodic, mixing

Modes of stochastic convergence

Almost sure convergence

Convergence of almost all the sequences of real numbers obtained by fixing a sample point

Pointwise convergence

Convergence of all the sequences of real numbers obtained by fixing a sample point

Mean-square convergence

Convergence to zero of the mean-square distance between the limit and the terms of the sequence

Convergence in probability

Convergence to zero of the probability of being far from the limit

Relations among modes of convergence

Relations among different concepts of stochastic convergence

Convergence in distribution

Convergence of the distribution functions of the terms of the sequence to the distribution function of the limit

Theorems on stochastic convergence

Central Limit Theorems

Conditions guaranteeing that the sample average converges to a normal distribution

Laws of Large Numbers

Conditions guaranteeing that the sample average converges to the true mean

Slutsky's theorem

An important application of the Continuous Mapping theorem

Continuous Mapping theorem

Stochastic convergence is preserved by continuous transformations

Delta method

A method used to derive the asymptotic distribution of a function of an asymptotically normal sequence

Empirical distributions

The plug-in principle

How to use the empirical distribution to approximate features of the original distribution

Empirical distribution

The distribution function of the discrete distribution of a sample of observations

Monte Carlo method

Application of the plug-in principle to computer-generated samples

The book

Most of the learning materials found on this website are now available in a traditional textbook format.