Convergence in Lᵖ of Random Vectors

In the setting of Probability Spaces: Notation, let $k\in\mathbb{N}$, let $p\ge1$ be a real number, and let $\lVert\cdot\rVert$ be the Euclidean norm on $\mathbb{R}^{k}$. Let $(Y_{n})_{n\in\mathbb{N}}$ be a sequence of random vectors in $\mathbb{R}^{k}$ and let $Y$ be a random vector in $\mathbb{R}^{k}$, with $\lVert Y_{n}-Y\rVert\in\mathcal{L}^{p}(P)$ for every $n\in\mathbb{N}$.

(i) Convergence in $L^{p}$. $(Y_{n})$ converges to $Y$ in $L^{p}$ if

$$\mathbb{E}\lVert Y_{n}-Y\rVert^{p}\to0\qquad(n\to\infty),$$

each expectation being a real number since $\lVert Y_{n}-Y\rVert\in\mathcal{L}^{p}(P)$, and convergence of real numbers as in Convergent Sequences of Real Numbers (i).