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 𝑋𝑛 :𝜉 →ℝ define a sequence of real random variables and 𝑋 :𝜉 →ℝ be a real random variable. Then prob

1. Convergence almost surely to 𝑋 implies Convergence in probability to 𝑋

Proof of 1

Let 𝐴𝑖 ={|𝑋𝑖 −𝑋| >𝜖} be an event, and 𝐵𝑗 =⋃∞𝑖=𝑗𝐴𝑖. Then clearly 𝐴𝑖 ⊆𝐵𝑖 for all 𝑖 ∈ℕ. Now |𝑋𝑖 −𝑋| >𝜖 implies sup𝑗≥𝑖|𝑋𝑗 −𝑋| >𝜖, so

lim𝑖→∞ℙ(|𝑋𝑖−𝑋|<𝜖)≥lim𝑖→∞ℙ(sup𝑗≥𝑖|𝑋𝑗−𝑋|>𝜖)

for every 𝜖 >0, proving ^R1.

Convergence theorems

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

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