Convergence in Probability, Example

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?