Bất đẳng thức hàm số mũ cho các khoảnh khắc bậc cao của biến ngẫu nhiên Gaussian
X1,…,XnX1,…,XnX_1,\ldots, X_nnnnX∼N(0,σ2)X∼N(0,σ2)X \sim N(0, \sigma^2)P(∣∣1n∑j=1nXj∣∣>t)P(∣∣1n∑j=1n(X2j−EX2j)∣∣>t)≤2exp(cnt2) and≤2exp(cnmin{t2,t}).P(|1n∑j=1nXj|>t)≤2exp(cnt2) andP(|1n∑j=1n(Xj2−EXj2)|>t)≤2exp(cnmin{t2,t}).\begin{align} \mathbb{P}\Bigl( \Bigl|\frac{1}{n}\sum_{j=1}^n X_j \Bigl| >t\Bigr) &\leq 2 \exp( cnt^2)~~\text{and}\\ \mathbb{P}\Bigl( \Bigl|\frac{1}{n}\sum_{j=1}^n (X_j^2 - \mathbb{E}X_j^2) \Bigl| >t\Bigr) &\leq 2 \exp( cn\min \{t^2, t\}). \end{align}P(∣∣1n∑j=1n(X4j−EX4j)∣∣>t)≤2exp(cnt)?P(|1n∑j=1n(Xj4−EXj4)|>t)≤2exp(cnt)?\begin{align} \mathbb{P}\Bigl( \Bigl|\frac{1}{n}\sum_{j=1}^n (X_j^4 - \mathbb{E}X_j^4) \Bigl| >t\Bigr) \leq 2 \exp( cnt)? \end{align}YYYP(|Y|>t)≤2exp(ctα)P(|Y|>t)≤2exp(ctα)\mathbb{P}(|Y| > t) \leq 2 \exp(c t^{\alpha})α>0α>0\alpha > 0∑ni=1Yi∑i=1nYi\sum_{i=1}^n …