### The law of large numbers overview

I have already several posts about the law of large numbers:

- start with the intuition, which is illustrated using Excel;
- simulations in Excel show that convergence is not as fast as some textbooks claim;
- to distinguish the law of large numbers from the central limit theorem read this;
- the ultimate purpose is the application to simple regression with a stochastic regressor.

Here we busy ourselves with the proof.

### Measuring deviation of a random variable from a constant

Let be a random variable and some constant. We want a measure of differing from the constant by a given number or more. The set where differs from by or more is the outside of the segment , that is, .

Now suppose

### Convergence to a spike formalized

Once again, check out the idea. Consider a sequence of random variables

**Definition**. Let *converges to in probability* or, alternatively,

*consistently estimates*

### The law of large numbers in its simplest form

Let

(1)

and its variance tends to zero

(2)

Now

Since this is true for any

### Final remarks

The above proof applies in the next more general situation.

**Theorem**. Let

This statement is often used on the Econometrics exams of the University of London.

In the unbiasedness definition the sample size is *fixed*. In the consistency definition it *tends to infinity*. The above theorem says that unbiasedness for all

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