Convergence in r-th mean can be thought of as convergence of integrals (where of course, in probability theory, integrals are equivalent to expectations). Convergence in r-th mean is equivalent to Convergence in Norm on [[lp-spaces|\(L^p\) Spaces]]
Definition (Convergence in r-th Mean) A sequence of random variables \(\{X_n\}_{n\in\mathbb{N}}\) on probability space \((\Omega, \mathcal{F}, \mathbb{P})\) is said to converge in r-th mean to \(X\) for \(r\geq 1\), denoted \(X_n\stackrel{L^r}{\rightarrow}X\) \((n\rightarrow\infty)\) if \[ \lim_{ n \to \infty } \mathbb{E}|X_{n}-X|^r = 0, \] and \(\mathbb{E}|X_{n}|^r<\infty\) for all \(n\).
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