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Probability · Lecture 64 of 76 · 5:45

Convergence in Probability and in the Mean, Part 2

Study guide

What this lecture covers

This short problem continues the previous part's comparison of random variable sequences by introducing a new, stronger notion of convergence: convergence in mean square. It proves that mean-square convergence implies convergence in probability, then uses the earlier sequence Y_n to show the converse is false.

It follows directly from Part 1's analysis of X_n and Y_n and their convergence in probability. After watching, you should be able to state the definition of mean-square convergence, prove it implies convergence in probability using the Markov inequality, and explain why the reverse implication fails.

Key ideas

Mean-square convergence: X_n converges to c in mean square if E[(X_n - c)^2] -> 0 as n -> infinity; when c equals a constant mean, this is equivalent to the variance of X_n going to 0.
Mean-square implies convergence in probability: squaring the inequality |X_n - c| > epsilon and applying the Markov inequality to (X_n - c)^2 shows P(|X_n - c| > epsilon) -> 0 whenever E[(X_n - c)^2] -> 0.
The converse is false: convergence in probability does not imply mean-square convergence.
Counterexample: Y_n (value n with probability 1/n, else 0) converges to 0 in probability, but E[Y_n^2] = n, which diverges, so Y_n does not converge to 0 in mean square.
Hierarchy of convergence notions: mean-square convergence is strictly stronger than convergence in probability.

Before you watch

Watch Part 1 of this problem first, which defines X_n and Y_n and establishes their convergence in probability.
Review the Markov inequality and the definition of variance.

Check your understanding

How does squaring the inequality |X_n - c| > epsilon let you apply the Markov inequality to prove convergence in probability?
Why does Y_n's convergence in probability to 0 not carry over to mean-square convergence?
If a sequence has a constant mean, how does mean-square convergence relate to its variance?

Vocabulary

mean-square convergence (phrase): A type of convergence where the average squared difference from a target value goes to zero.
Mean-square convergence is stronger than convergence in probability.
implies (verb): Logically leads to or guarantees another result.
Mean-square convergence implies convergence in probability.
converse (noun): The reverse direction of a logical statement.
The converse of this implication turns out to be false.
counterexample (noun): A specific case that shows a general claim is false.
Y_n is a counterexample to the converse statement.
hierarchy (noun): An ordering of concepts from weaker to stronger.
There is a hierarchy of convergence notions in probability.
strictly stronger (phrase): Guaranteeing more than another condition, without being guaranteed by it.
Mean-square convergence is strictly stronger than convergence in probability.
convergence in probability (phrase): A sequence of random variables getting closer to a fixed value with higher and higher chance.
Convergence in probability is the weaker of the two notions compared here.
Markov inequality (phrase): A rule that bounds the chance a non-negative variable is large, using its expected value.
The proof applies the Markov inequality to a squared difference.
inequality (noun): A mathematical statement that one quantity is greater or less than another.
Squaring the original inequality lets you use the Markov bound.
sequence (noun): An ordered list of values, one after another.
X_n and Y_n are sequences of random variables.
diverge (verb): To grow without settling toward any fixed value.
E[Y_n^2] diverges even though Y_n converges to 0 in probability.
mean (noun): The average value of a random variable.
When c equals a constant mean, mean-square convergence matches variance going to 0.
establish (verb): To prove or firmly show that something is true.
The proof establishes that mean-square convergence implies convergence in probability.
notion (noun): A particular idea or concept, often one of several related ones.
This problem compares two different notions of convergence.
comparison (noun): An examination of how two things are similar or different.
The problem continues the earlier comparison of convergence notions.
stronger (adjective): Guaranteeing more, or implying more other results.
Mean-square convergence is the stronger notion of the two.
weaker (adjective): Guaranteeing less, or implying fewer other results.
Convergence in probability is the weaker notion here.
apply (verb): To use a method or rule in a specific situation.
We apply the Markov inequality to the squared difference.
definition (noun): A precise statement of exactly what a term means.
You should be able to state the definition of mean-square convergence.
square (verb): To multiply a number or expression by itself.
Squaring the inequality lets the Markov inequality apply.
prove (verb): To show with certainty, using logical steps, that something is true.
The problem proves that mean-square convergence implies convergence in probability.
state (verb): To say or write something clearly and precisely.
You should be able to state the definition of mean-square convergence.
reverse (adjective): Going in the opposite direction of the original statement.
The reverse implication turns out to be false.
fail (verb): To not succeed or not hold true.
The reverse implication fails, as the counterexample shows.
carry over (phrasal verb): To continue applying from one situation to another.
Convergence in probability does not carry over to mean-square convergence.

Chapters

0:00 Introduction to mean square
1:13 Mean square implies prob.
3:55 Counterexample to reverse

From the YouTube description

MIT 6.041SC Probabilistic Systems Analysis and Applied Probability, Fall 2013
View the complete course: http://ocw.mit.edu/6-041SCF13
Instructor: Kuang Xu

License: Creative Commons BY-NC-SA
More information at http://ocw.mit.edu/terms
More courses at http://ocw.mit.edu

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