Convergence concepts in probability MOC

Convergence in the $𝑝$ th 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 $𝑋 𝑛 : 𝜉 \to ℝ$ define a sequence of real random variables and $𝑋 : 𝜉 \to ℝ$ be a real random variable. Then #m/thm/prob

Convergence almost surely to $𝑋$ implies Convergence in probability to $𝑋$

Proof of 1

Let $𝐴 𝑖 = {| 𝑋 𝑖 - 𝑋 | > 𝜖}$ be an event, and $𝐵 𝑗 = ⋃ \infty 𝑖 = 𝑗 𝐴 𝑖$ . Then clearly $𝐴 𝑖 \subseteq 𝐵 𝑖$ for all $𝑖 \in ℕ$ . Now $| 𝑋 𝑖 - 𝑋 | > 𝜖$ implies $s u p 𝑗 \geq 𝑖 | 𝑋 𝑗 - 𝑋 | > 𝜖$ , so

l i m 𝑖 \to \infty ℙ (| 𝑋 𝑖 - 𝑋 | < 𝜖) \geq l i m 𝑖 \to \infty ℙ (s u p 𝑗 \geq 𝑖 | 𝑋 𝑗 - 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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