∑나는에이나는엑스나는

$\sum _{\mathrm{나는}}{\mathrm{에이}}_{\mathrm{나는}}{\mathrm{엑스}}_{\mathrm{나는}}$

$\sum_i a_iX_i$± 1

$\pm 1$

$\pm 1$ 엑스나는

${\mathrm{엑스}}_{\mathrm{나는}}$

$X_i$σ= ∥ a ∥2

$\sigma =”\mathrm{에이}{”}_2$

$\sigma = \|a\|_2$t ≥ 2

$티\ge 2$

$t \ge 2$

$$\begin{eqnarray*} \mathbf{E}\left[\left(\sum_i a_iX_i\right)^t\right] &=& \sum_{i_1,\ldots,i_t} \left(\prod_{j=1}^t a_{i_j}\right) \mathbf{E}\left[\prod_{j=1}^t X_{i_j}\right]\\ &\le& \sum_{i_1,\ldots,i_t} \left(\prod_{j=1}^t |a_{i_j}|\right) \mathbf{E}\left[\prod_{j=1}^t X_{i_j}\right]\\ &=& \sum_{\substack{i_1<\ldots< i_m\\ r_1,\ldots,r_m\\ \sum_j r_j = t\\ \forall j\ r_j > 0}} \binom{t}{r_1,\ldots,r_m}\left(\prod_{j=1}^m |a_{i_j}|^{r_j}\right)\left(\prod_{j=1}^m \mathbf{E}[X_{i_j}^{r_j}]\right) \end{eqnarray*}$$

아르 자형j

${\mathrm{아르\; 자형}}_j$

$r_j$0

$0$

$0$1

$1$

$1$엑스나는

${\mathrm{엑스}}_{\mathrm{나는}}$

$X_i$지나는

${지}_{\mathrm{나는}}$

$G_i$아르 자형j

${\mathrm{아르\; 자형}}_j$

$r_j$0

$0$

$0$아르 자형j

${\mathrm{아르\; 자형}}_j$

$r_j$ 1

$1$

$1$

$$\mathbf{E}\left[\left(\sum_i a_iX_i\right)^t\right] \le \mathbf{E}\left[\left(\sum_i |a_i|G_i\right)^t\right]$$

2

$2$

$2$∑나는| 에이나는| 지나는

$\sum _{\mathrm{나는}}|{\mathrm{에이}}_{\mathrm{나는}}|{지}_{\mathrm{나는}}$

$\sum_i |a_i| G_i$∥ a ∥2

$”\mathrm{에이}{”}_2$

$\|a\|_2$티

$티$

$t$∥ a ∥티2⋅ t ! / ( 2t / 2⋅ ( t / 2 ) ! )

$”\mathrm{에이}{”}_2^{티}\cdot 티!/(2^{티/2}\cdot (티/2)!)$

$\|a\|_2^t \cdot t! / (2^{t/2} \cdot (t/2)!)$∥ a ∥티2티t / 2

$”\mathrm{에이}{”}_2^{티}{티}^{티/2}$

$\|a\|_2^tt^{t/2}$

$$\mathbf{Pr}\left[\left|\sum_i a_iX_i\right| > \lambda\right] < 2^{O(t)}\cdot \lambda^{-t}\cdot \|a\|_2^t t^{t/2}$$ t = λ2/ ( C⋅ ∥ a ∥22)

$티={\lambda }^2/(\mathrm{기음}\cdot ”\mathrm{에이}{”}_2^2)$

$t = \lambda^2 / (C\cdot \|a\|_2^2)$기음

$\mathrm{기음}$

$C$e x p (−Ω( λ2/ ∥a ∥22) )

$\mathrm{이자형}\mathrm{엑스}\mathrm{피}(-\mathrm{\Omega }({\lambda }^2/”\mathrm{에이}{”}_2^2))$

$\mathrm{exp}(-\Omega(\lambda^2 / \|a\|_2^2))$엑스나는

${\mathrm{엑스}}_{\mathrm{나는}}$

$X_i$

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답변