## Saturday, September 1, 2012

### Probability 1- The uniform integrability

In this series I am reviewing some of my lecture notes of the first graduate course in probability.

Uniform Integrability - The definition

The sequence, $X_n$, is called uniformly integrable if:

$\lim_{A \to \infty} \sup_n \mathbb{E} [|{X_n}| \mathbb{1}_{|{X_n}|>A}] = 0$

What does this formula tells us?

Lets start with the $|{X_n}| \mathbb{1}_{|{X_n}|>A}$ term. This term actually means that we “reset” $|{X_n}|$ to zero where ever it is below $A$ and leave it untouched where ever it is above $A$. Look at the figure below:

Next, look at $\mathbb{E} [|{X_n}| \mathbb{1}_{|{X_n}|>A}]$. This is the aria under the graph of $|{X_n}| \mathbb{1}_{|{X_n}|>A}$ (the aria relative to the probability measure in question $dP(\omega)$ here I suppose that the probability space is, say, $\omega \in [0,M]$ with uniform probability measure so that $dP(\omega)= \frac{1}{M}$). The supremum $\sup_n$ picks the $n$ for which the corresponding graph gives the biggest area for fixed $A$.

Finally, by taking the limit $\lim_{A \to \infty} \sup_n \mathbb{E} [|{X_n}| \mathbb{1}_{|{X_n}|>A}]$, we examine what happens with that area when $A$ grows bigger and bigger.

Here are two sequences of random variables, first is not uniformly integrable and the second is:

This is not uniformly integrable sequence, since no matter how big $A$ grows, there is always some $n$ for which $\mathbb{E} [|{X_n}| \mathbb{1}_{|{X_n}|>A}]$ equals 1, for big enough $n$, $|{X_n}| \mathbb{1}_{|{X_n}|>A}$ is just $|{X_n}|$. So $\sup_n \mathbb{E} [|{X_n}| \mathbb{1}_{|{X_n}|>A}] = \sup_n \mathbb{E} [|{X_n}|] = \sup_n 1 = 1$.

The sequence below is a uniformly integrable one:

By the way, do notice that both sequence converge to zero in probability ( $\lim_{n \to \infty} P(X_n – 0) = 1)$ ).

Some Intuition:

So what we got here? The intuition is complicated, I haven't found any elegant way to describe it. The restrain imposed by uniform integrability is related to the behavior of the sequence in the limit of $n$, and to the way the sequence goes to the infinity:

• It is easy to see that if the average aria below the graph of $X_n$ is running away to infinity as $n$ grow, then it is not uniformly integrable.

On the other hand -

• Any finite sequence is always uniformly integrable (since for any fixed $n$ the $P(X_n >A)$ inevitably decreases as $A$ grows big)
• Any bounded sequence [bounded  in a sense opposite to unbounded sequence. Unbounded that is as $n$ grows bigger, the random variables, $X_n$, can receive bigger and bigger values with some probability] is uniformly integrable. But the inverse is not true -

Here, I tried to draw unbounded sequence that is uniformly integrable since the average shrinks as $n$ grows bigger.

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