Convergence in Probability and in the Mean, Part 1

What this lecture covers

This is a worked problem comparing two sequences of random variables, X_n and Y_n, that both place most of their mass near 0 but differ in what happens on rare events. It asks whether each sequence converges in probability, using the Chebyshev inequality where it works and the distribution directly where the inequality is too weak, and closes by testing whether convergence in probability implies convergence of expected values.

It follows directly from the course's treatment of the Chebyshev inequality and convergence in probability. After watching, you should be able to compute expected value and variance for such sequences, apply Chebyshev's inequality to prove convergence in probability, recognize when that inequality is uninformative, and explain why convergence in probability does not guarantee convergence in expectation.

Key ideas

• Two constructed sequences: X_n takes value 1 with probability 1/n and value 0 otherwise; Y_n takes value n with probability 1/n and value 0 otherwise, so both are usually 0 but Y_n's rare event is much larger.
• Moments diverge: E[X_n] = 1/n -> 0 and Var(X_n) -> 0, while E[Y_n] = 1 for every n and Var(Y_n) = n - 1 -> infinity.
• Chebyshev works for X_n: because its variance shrinks fast enough, Chebyshev's inequality directly shows X_n converges to 0 in probability.
• Chebyshev fails for Y_n: its variance diverges, so the Chebyshev bound becomes useless and cannot establish convergence.
• Direct calculation still works: computing P(|Y_n| >= epsilon) directly from the distribution shows it equals 1/n, which goes to 0, proving Y_n also converges to 0 in probability.
• Convergence in probability does not imply convergence in expectation: Y_n converges to 0 in probability while E[Y_n] stays at 1 forever, because a shrinking-probability event can carry an unboundedly large value.

Before you watch

• Review the Chebyshev inequality and the definition of convergence in probability from the lecture on the weak law of large numbers.
• Be comfortable computing expected value and variance for a discrete random variable with two possible values.

Check your understanding

1. Why does the Chebyshev inequality prove convergence in probability for X_n but not for Y_n?
2. How is P(|Y_n| >= epsilon) computed directly from Y_n's distribution, and why does this succeed where Chebyshev's inequality fails?
3. Why can a sequence of random variables converge in probability to 0 while its expected value stays fixed at a nonzero number?
4. What property of Y_n's rare event (its size versus its probability) causes the expected value to stay constant as n grows?