# Convergence Theorems: MCT, DCT and Fatou

Why limits and expectations do not always commute: escaping mass, monotone and dominated convergence, Fatou's lemma and convergence modes. Interactive lab.

## About this tool

Can you take a limit inside an expectation? Not always. Double your stake on a fair coin until you win and almost every outcome ends at a profit of 1, yet every stage has mean 0: the losses are packed into a set that shrinks to nothing, and the limit cannot see it. These pages show where the mass goes, which theorems stop it escaping, and why sequences can converge in different senses: almost surely, in probability or in mean.

## Frequently asked questions

### What do the monotone and dominated convergence theorems say?

Monotone convergence: if nonnegative functions increase to a limit, their integrals increase to the integral of the limit. Dominated convergence: if the functions converge and all stay below one fixed integrable function, the integrals converge to the integral of the limit. Both give conditions under which the limit and the expectation can be swapped. Fatou's lemma needs only that the functions are nonnegative, and it gives just an inequality: the expectation of the limit is at most the limit of the expectations.

### Where does the mass go in the spike example?

The spike n·1[0, 1/n), which equals n on the interval from 0 to 1/n and 0 elsewhere, has mean 1 for every n, but at any fixed outcome it is eventually 0. All its mass sits in a shrinking set. In the limit the function is 0 almost surely, so its mean is 0, which is below 1. Fatou's inequality holds with room to spare. There is no integrable function above all the spikes, so dominated convergence does not apply.

### What should I try on the pages?

On the first page switch between the five sequences and compare the mean of the limit with the limit of the means: the doubling strategy and the spike lose mass, while a dominated sequence or an increasing one do not. On the modes page compare the typewriter, the shrinking spike and a sample mean of coin flips, and see which of the three senses of convergence each one satisfies.

### Does convergence almost surely imply convergence in mean?

No, as the spike shows: it converges to 0 almost surely, but its mean stays at 1. The typewriter sequence goes the other way: it shrinks in probability and in mean while failing to converge at any single outcome. Going from almost sure convergence to convergence in mean needs a roof (domination) or uniform integrability. Convergence in mean always implies convergence in probability, but not the reverse without extra conditions.

## Learn the theory

- [Convergence theorems](https://quantsandbox.dev/guide/probability/convergence-theorems): When the limit of integrals is the integral of the limit: escaping mass, monotone convergence, Fatou's lemma, dominated convergence and series of integrals.
- [Modes of convergence and the law of large numbers](https://quantsandbox.dev/guide/probability/modes-of-convergence): Almost sure, in probability, in mean and in distribution: how they differ, the Borel–Cantelli lemma, uniform integrability and the laws of large numbers.
- [The law of large numbers and the central limit theorem](https://quantsandbox.dev/guide/stochastic/law-of-large-numbers-clt): Why averages settle down, how fast, and why the normal distribution appears: the two theorems behind Monte Carlo and the Brownian limit.
- [Conditional expectation and martingales](https://quantsandbox.dev/guide/stochastic/martingales): Conditional expectation given the past, fair games, why no betting strategy beats a martingale, and discounted prices under Q.

Open the interactive tool: https://quantsandbox.dev/convergence