POL 571: Convergence of Random Variables

9 CONVERGENCE IN PROBABILITY 111 9 Convergence in probability The idea is to extricate a simple deterministic component out of a random situation This is typically possible when a large number of random eﬀects cancel each other out, so some limit is involved The general situation, then, is the following: given a sequence of random variables,

Topic 7 Convergence in Probability

In general, convergence will be to some limiting random variable However, this random variable might be a constant, so it also makes sense to talk about convergence to a real number There are several diﬀerent modes of convergence (i e , ways in which a sequence may converge) We begin with convergence in probability Deﬁnition 7 1 The

7 Convergence in Probability

In general, convergence will be to some limiting random variable However, this random variable might be a constant, so it also makes sense to talk about convergence to a real number There are several diﬀerent modes of convergence We begin with convergence in probability Deﬁnition 7 1 The sequence {X n} converges in probability to X

Various Modes of Convergence

If r =2, it is called mean square convergence and denoted as X n m s → X Relationship among various modes of convergence [almost sure convergence] ⇒ [convergence in probability] ⇒ [convergence in distribution] ⇑ [convergence in Lr norm] Example 1 Convergence in distribution does not imply convergence in probability ⇒ Let Ω = {ω1

POL 571: Convergence of Random Variables

with convergence in probability) Some people also say that a random variable converges almost everywhere to indicate almost sure convergence The notation X n a s → X is often used for al-most sure convergence, while the common notation for convergence in probability is X n →p X or

Basic Probability Theory on Convergence

Basic Probability Theory on Convergence Definition 1 (Convergencein probability) Asequence of random variable(Yn: n = 1;2;:::) is said to converge in probability to another random variable Y, all de ned on Rd if for every ϵ > 0; P(∥Yn Y∥ > ϵ) 0: We denote this phenomenon by YnP Y Definition 2 (Convergence in law/distribution)

Modes of Convergence

n2N is said to converge in probability to X, denoted X n P n1 X, if for every ">0, P(jX n Xj>") n1 0 In other words, for any xed ">0, the probability that the sequence deviates from the supposed limit Xby more than "becomes vanishingly small We now seek to prove that a s convergence implies convergence in probability Theorem 2 If X n

[PDF] POL 571: Convergence of Random Variables

9 Convergence in probability

Topic 7 Convergence in Probability

7 Convergence in Probability

Various Modes of Convergence

POL 571: Convergence of Random Variables

Basic Probability Theory on Convergence

Modes of Convergence