Inequalities in probability MOC

Convergence concepts in probability MOC

• Convergence in the pth mean (Lebesgue space)
• Convergence almost surely (Convergence almost everywhere)
• Sure convergence (Pointwise convergence)
• Convergence in probability (Ky Fan metric)
• Convergence in distribution

Relation between convergence concepts

Let define a sequence of real random variables and be a real random variable. Then prob

1. Convergence almost surely to implies Convergence in probability to

Proof of 1

Let be an event, and . Then clearly for all . Now implies , so

for every , proving ^R1.

Convergence theorems

• Kolmogorov’s law
• Khinchin’s law
• Empirical cumulative distribution function
• Central limits theorem

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