What is a tail sigma-algebra?

Then T is a σ-algebra, called the tail σ-algebra of (Xn : n ∈ N). It contains the events which depend only on the limiting behaviour of the sequence. Theorem 2.5. 1 (Kolmogorov’s zero-one law). Suppose that (Xn : n ∈ N) is a se- quence of independent random variables.

What is tail sigma-field?

It is easy to see that the tail sigma-field is a subset of the exchangeable sigma-field. For, if A is a tail event, then it is independent of the first n random variables in the underlying sequence, so in particular, is invariant under permutations of initial segments of the sequence.

How do you show that an event is a tail event?

For example, the event that the sequence converges, and the event that its sum converges are both tail events. In an infinite sequence of coin-tosses, a sequence of 100 consecutive heads occurring infinitely many times is a tail event.

What does it mean for sigma algebras to be independent?

Independent σ-Algebras* Definition E. 4.1. Let (Ω,F,μ) be a measure space. Two events A,B∈F are independent if μ(A∩B)=μ(A)μ(B).

What is random variable and its types?

A random variable either has an associated probability distribution (Discrete Random Variable), or a probability density function (Continuous Random Variable). Therefore, we have two types of random variables – Discrete and Continuous.

What is 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.

What is an example of a tail event?

The financial crisis of 2007/08 was a tail event. Coronavirus was a tail event. Many of the every day products you use, services you engage with and companies you work for are tail events. So, yes, tail events are rare but when they happen their impact is huge.

What is tail probability?

The probability that a random variable deviates by a given amount from its expectation is referred to as a tail probability.

What is tail event slack?

The tail event slack of an activity in a network is the slack at the tail (or starting point) of an activity. In other words, tail event slack of an activity in a network is the difference between the latest event time and the earliest event time at its tail (or starting point or node).

How do you prove two sigma algebras are independent?

Definition E. 4.1. Let (Ω,F,μ) be a measure space. Two events A,B∈F are independent if μ(A∩B)=μ(A)μ(B). A finite number of events A1,…,An∈F are independent if μ(A1∩⋯An)=μ(A1)⋯μ(An).

How do you show that a random variable is independent of a sigma algebra?

Let X and Y be random variables on (Ω, F, P). We say these two random variables are independent if the σ-algebras they generate, σ(X) and σ(Y ), are independent. We say that the random variable X is independent of the σ-algebra G if σ(X) and G are independent.