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Suppose that $X = (X_1,X_2,\cdots ,X_n)$ is a vector, where $X_i, i=1,2,\cdots ,n$ are independent and sub-gaussian random variables satisfying $\mathbb{E}[X_i^2] = 1$. Prove or disprove the following claim: $\left| \mathbb{E}\|X\|_2-\sqrt{n} \right| = o(1)$ as $n \to \infty$.

This problem is from the book High Dimensional Probability by Vershynin (P52). It seems that to prove this result one needs to refine the analysis in the book, but I do not know how such refinement can be made. Can anyone help please?

j200932
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  • As stated, the claim is obviously false: take $X_i=X_1$ for all $i$ as a counterexample. Perhaps you are assuming the $X_i$ are independent? – whuber Jul 10 '20 at 15:02
  • @whuber Oh yes I forgot to write this condition. Thanks for your remind. – j200932 Jul 11 '20 at 02:03

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