Question

3. (30 Points). Let $\epsilon_1, \dots, \epsilon_n$ be $n$ independent mean-zero sub-Gaussian random variables and let $x_1, \dots, x_n$ be independent variable with $M = \max_{i=1,\dots,n} |x_i| < \infty$. Assume that $\epsilon_1, \dots, \epsilon_n$ are independent of $x_1, \dots, x_n$. Show that for any $t > 0$, we have $P\left(\left|\sum_{i=1}^n x_i \epsilon_i\right| \ge t\right) \le 2 \exp\left\{ - \frac{ct^2}{nM^2 K^2} \right\}$, where $K = \max_{i=1,\dots,n} ||\epsilon_i||_{\psi_2}$ with $||\cdot||_{\psi_2}$ being the sub-Gaussian norm.

          3. (30 Points). Let $\epsilon_1, \dots, \epsilon_n$ be $n$ independent mean-zero sub-Gaussian random variables and let $x_1, \dots, x_n$ be independent variable with $M = \max_{i=1,\dots,n} |x_i| < \infty$. Assume that $\epsilon_1, \dots, \epsilon_n$ are independent of $x_1, \dots, x_n$. Show that for any $t > 0$, we have

$P\left(\left|\sum_{i=1}^n x_i \epsilon_i\right| \ge t\right) \le 2 \exp\left\{ - \frac{ct^2}{nM^2 K^2} \right\}$,

where $K = \max_{i=1,\dots,n} ||\epsilon_i||_{\psi_2}$ with $||\cdot||_{\psi_2}$ being the sub-Gaussian norm.

$3. (30 Points). Let ϵ1, …, be n independent mean-zero sub-Gaussian random variables and let x1, …, xn be independent variable with M = maxi=1,…,n |xi| < ∞. Assume that ϵ1, …, are independent of x1, …, xn. Show that for any t > 0, we have P(|∑i=1^n xi | ≥ t) ≤ 2 { - (ct^2)/(nM^2 K^2)}, where K = maxi=1,…,n ||||ψ2 with ||·||ψ2 being the sub-Gaussian norm.$

Added by Renee S.

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