How do you prove uniformly integrable?
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.
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.
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.
What is an integrable random variable?
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.
How do you measure tightness?
A family Γ of probability measures in P(S) is said to be tight if, given any ε > 0, there exists a compact set K = Kε of S such that: P(K) > 1 − ε, for all P in Γ.
What is the strong law of large numbers?
The strong law of large numbers states that with probability 1 the sequence of sample means S ¯ n converges to a constant value μX, which is the population mean of the random variables, as n becomes very large. This validates the relative-frequency definition of probability.
How do you show a martingale is uniformly integrable?
Definition 2.7. (Mn,Fn)n≥1 is an uniformly integrable martingale, if (Mn,Fn)n≥1 is a martingale, and the family Mn, n ≥ 1 is uniformly integrable. (2.24) Mn = IE[M∞|Fn].
What is uniformly integrable martingale?
Uniform integrable martingales. Definition 2.7. (Mn,Fn)n≥1 is an uniformly integrable martingale, if (Mn,Fn)n≥1 is a martingale, and the family Mn, n ≥ 1 is uniformly integrable. Let (Mn,Fn)n≥1 be uniformly integrable martingale.
What is uniformly bounded sequence?
In mathematics, a uniformly bounded family of functions is a family of bounded functions that can all be bounded by the same constant. This constant is larger than the absolute value of any value of any of the functions in the family.
What is thigh measurement?
Thigh: Measure the circumference of the fullest part of your thigh. Wrap the tape measure around your thigh from front to back and then around to the front. Wrap the tape measure around the widest part of your upper arm from front to back and around to the start point.
What is a tight sequence?
Theorem 1.8 A sequence of probability measures {πn}∞ n=1 on (Dd,Dd) is tight if and only if. ∀T > 0,ϵ> 0 there exists Kϵ,δϵ > 0 such that: (i) lim supn→∞ πn(x ∈ Dd : ||x||T ≥ Kϵ) < ϵ
How do casinos use the law of large numbers?
The law is basically that if one conducts the same experiment a large number of times the average of the results should be close to the expected value. Furthermore, the more trails conducted the closer the resulting average will be to the expected value. This is why casinos win in the long term.