Convergence in the $𝑝$ th mean

Convergence in the $𝑝$ th mean is simply convergence in the Lebesgue space $𝐿 𝑝 (𝜉, ℙ)$ of real random variables. prob If $(𝑋 𝑖) \infty 𝑖 = 1$ is a sequence of real random variables and $𝑋$ is a real random variable, for each of which the $𝑝$ -norm $𝔼 [| 𝑋 | 𝑝]$ exists, then $𝑋 ∙ 𝐿 𝑝 \to 𝑋$ iff

l i m 𝑖 \to \infty 𝔼 [| 𝑋 𝑖 - 𝑋 | 𝑝] = 0

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Convergence in the $𝑝$ th mean

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Convergence in the 𝑝th mean

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Convergence in the $𝑝$ th mean