exercise:Afff89fe59

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Recall that for any [math]q \ge 1[/math], the [math]\ell_q[/math] norm of a vector [math]x \in \R^n[/math] is defined by

[[math]] |x|_q=\Big(\sum_{i=1}^n |x_i|^q\Big)^{\frac1q}\,. [[/math]]

Let [math]X=(X_1, \ldots, X_n)[/math] be a vector with independent entries such that [math]X_i[/math] is sub-Gaussian with variance proxy [math]\sigma^2[/math] and [math]\E(X_i)=0[/math].

1. Show that for any [math]q\ge2[/math], and any [math]x \in \R^d[/math],
   [[math]] |x|_2\le |x|_qn^{\frac12-\frac1q}\,, [[/math]]
   and prove that the above inequality cannot be improved
2. Show that for for any [math]q \gt 1[/math],
   [[math]] \E|X|_q\le 4\sigma n^{\frac{1}{q}}\sqrt{q} [[/math]]
3. Recover from this bound that
   [[math]] \E\max_{1\le i\le n} |X_i|\le 4e\sigma\sqrt{\log n}\,. [[/math]]