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Probability · Lecture 65 of 76 · 7:36

Convergence in Probability, Example

Convergence in Probability Example on YouTube

Study guide

What this lecture covers

This problem tests convergence in probability on three sequences built from a uniform random variable X on [-1, 1]: the raw sequence X_i, the scaled sequence Y_i = X_i / i, and the powered sequence Z_i = X_i^i. It works through each case to build intuition for when convergence in probability holds and how to prove it.

It builds directly on the definition of convergence in probability from earlier lectures. After watching, you should be able to recognize when a sequence of identically distributed random variables fails to converge, and prove convergence for sequences that shrink toward 0 by bounding tail probabilities.

Key ideas

  • X_i does not converge: since each X_i has the same uniform distribution on [-1, 1] regardless of i, the distribution never concentrates around any single number, so X_i does not converge in probability.
  • Y_i = X_i / i converges to 0: because |Y_i| <= 1/i, the probability of Y_i deviating from 0 by more than any fixed epsilon is eventually 0 once i > 1/epsilon.
  • Z_i = X_i^i converges to 0: bounding P(|Z_i| >= epsilon) reduces to showing epsilon^(1/i) -> 1 as i -> infinity, using the fact that a^x -> 1 as x -> 0 for positive a.
  • General technique: proving convergence in probability for a shrinking sequence means bounding the tail probability directly from the sequence's definition and distribution, rather than relying on Chebyshev's inequality.

Before you watch

  • Review the definition of convergence in probability and how to bound a tail probability from a known distribution.
  • Be comfortable with the uniform distribution and basic limit properties of exponentials.

Check your understanding

  1. Why does X_i fail to converge in probability even though each X_i is bounded between -1 and 1?
  2. How does the bound |Y_i| <= 1/i lead directly to a proof that Y_i converges to 0 in probability?
  3. Why does the proof for Z_i = X_i^i split into cases based on whether epsilon is greater than or less than 1?

Vocabulary

scaled sequence (phrase)
A sequence of variables each divided or multiplied by a changing factor.
Y_i is a scaled sequence built by dividing X_i by i.
powered sequence (phrase)
A sequence of variables each raised to a changing power.
Z_i is a powered sequence, X_i raised to the power i.
concentrate (verb)
To cluster closely around a single value.
The distribution of X_i never concentrates around one number.
bound (verb)
To set a limit on how large or small something can be.
We bound the tail probability directly from the distribution.
eventually (adverb)
At some point after enough time or steps have passed.
The probability eventually becomes 0 once i is large enough.
convergence in probability (phrase)
The idea that a sequence of random variables gets closer and closer to a fixed value with high probability.
The problem tests convergence in probability for three different sequences.
sequence (noun)
An ordered list of numbers or variables, one after another.
X_i, Y_i, and Z_i are each a sequence indexed by i.
tail probability (phrase)
The chance that a variable takes a value far from its typical range.
We bound the tail probability P(|Z_i| >= epsilon).
deviate (verb)
To move away from an expected or central value.
Y_i is unlikely to deviate from 0 by more than epsilon.
epsilon (noun)
A small positive number used to describe how close a value must be to count as converged.
We check whether the probability of exceeding epsilon goes to 0.
uniform distribution (phrase)
A pattern where every value in a range is equally likely.
X is uniform on the interval [-1, 1].
threshold (noun)
A fixed value used as a boundary for comparison.
We compare epsilon to the threshold value of 1.
shrink (verb)
To become smaller over time.
Y_i is a sequence designed to shrink toward 0.
limit (noun)
The value a sequence approaches as it goes on forever.
As i goes to infinity, epsilon^(1/i) approaches a limit of 1.
proof (noun)
A logical argument that shows a statement must be true.
The lecture gives a short proof for each of the three sequences.
technique (noun)
A specific method used to solve a type of problem.
The general technique bounds the tail probability directly.
rely on (phrasal verb)
To depend on something in order to work.
The proof doesn't rely on Chebyshev's inequality.
Chebyshev's inequality (phrase)
A rule that bounds the probability of being far from the mean using the variance.
This proof avoids using Chebyshev's inequality.
exponent (noun)
The power a number is raised to.
Z_i raises X_i to the exponent i.
positive (adjective)
Greater than zero.
The rule a^x -> 1 holds for any positive a.
approach (verb)
To get closer and closer to a value without necessarily reaching it.
epsilon^(1/i) approaches 1 as i grows.
fixed (adjective)
Set and not changing.
We check convergence for any fixed epsilon.
guarantee (verb)
To make certain that something will happen.
The bound guarantees the probability goes to 0.
straightforward (adjective)
Simple and direct, without complications.
The proof for Y_i is fairly straightforward.
arbitrarily (adverb)
By any amount you choose, without a fixed limit.
The probability becomes arbitrarily small as i increases.

Chapters

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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