Are all random variables integrable?

Are all random variables integrable?

A random variable X is called “integrable” if E|X| < ∞ or, equivalently, if X ∈ L1; it is called “square integrable” if E|X|2 < ∞ or, equivalently, if X ∈ L2. Integrable random variables have well-defined finite means; square-integrable random variables have, in addition, finite variance.

Is random variable bounded?

Bounded in probability. Definition: A sequence of random variables is bounded in probability if for any , there exist and such that P { ‖ x n ‖ > M } < ϵ for all .

What does it mean for a random variable to be integrable?

A random variable is said to be integrable if its expected value exists and it is well-defined.

How do you test uniform integrability?

If { X i : i ∈ I } is uniformly integrable and is a nonempty subset of , then { X j : j ∈ J } is uniformly integrable. If the random variables in the collection are dominated in absolute value by a random variable with finite mean, then the collection is uniformly integrable.

Why is uniform integrability important?

In mathematics, uniform integrability is an important concept in real analysis, functional analysis and measure theory, and plays a vital role in the theory of martingales. The definition used in measure theory is closely related to, but not identical to, the definition typically used in probability.

Is a bounded random variable Subgaussian?

-sub-Gaussian random variable. We get our desired inequality by exponentiating both sides.

How do you prove uniform integrability?

Which of the following are continuous random variables?

A continuous random variable is one which takes an infinite number of possible values. Continuous random variables are usually measurements. Examples include height, weight, the amount of sugar in an orange, the time required to run a mile. A continuous random variable is not defined at specific values.

What is the difference between discrete random variable and continuous random variable?

A random variable is a variable whose value is a numerical outcome of a random phenomenon. A discrete random variable X has a countable number of possible values. Example: Let X represent the sum of two dice. A continuous random variable X takes all values in a given interval of numbers.

Are martingales uniformly integrable?

Since all backward martingales are uniformly integrable (why?) and the sequence {An}n∈−N0 is uniformly dominated by A−∞ ∈ L1 – and therefore uniformly integrable – we conclude that {Xn}n∈−N0 is also uniformly integrable.

How do you prove uniforms integrable?

If the random variables in the collection are dominated in absolute value by a random variable with finite mean, then the collection is uniformly integrable. Suppose that is a nonnegative random variable with E ( Y ) < ∞ and that | X i | ≤ Y for each i ∈ I . Then.