Our starting point is the following well-known theorem from probability: Let X1, …, Xn be independent random variables with finite second moments, and let Sn=∑i=1nXi. Then Var(Sn)=∑i=1nVar(Xi). (1) If we suppose that each Xi has mean zero, Xi = 0, then (1) becomes ESn2=∑i=1nEXi2. (2) This equality generalizes easily to vectors in a Hilbert space ℍ with inner product 〈·, ·〉: If the Xi's are independent with values in ℍ such that Xi = 0 and ‖Xi‖2 < ∞, then ‖Sn‖2=〈Sn,Sn〉=∑i,j=1n〈Xi,Xj〉, and since 〈Xi, Xj〉 = 0 for i ≠ j by independence, E‖Sn‖2=∑i,j=1nE〈Xi,Xj〉=∑i=1nE‖Xi‖2. (3) What happens if the Xi's take values in a (real) Banach space (, ‖ · ‖)? In such cases, in particular when the square of the norm ‖ · ‖ is not given by an inner product, we are aiming at inequalities of the following type: Let X1, X2, …, Xn be independent random vectors with values in (, ‖ · ‖) with Xi = 0 and ‖Xi‖2 < ∞. With Sn≔∑i=1nXi we want to show that E‖Sn‖2≤K∑i=1nE‖Xi‖2 (4) for some constant K depending only on (, ‖ · ‖). For statistical applications, the case (B,‖⋅‖)=lrd≔(ℝd,‖⋅‖r) for some r ∈ [1, ∞] is of particular interest. Here the r-norm of a vector x ∈ ℝd is defined as ‖x‖r≔{(∑j=1d|xj|r)1/rif1≤r≤∞,max1≤j≤d|xj|ifr=∞. (5) An obvious question is how the exponent r and the dimension d enter an inequality of type (4). The influence of the dimension d is crucial, since current statistical research often involves small or moderate “sample size” n (the number of independent units), say on the order of 102 or 104, while the number d of items measured for each independent unit is large, say on the order of 106 or 107. The following two examples for the random vectors Xi provide lower bounds for the constant K in (4): Example 1.1 (A lower bound in lrd) Let b1, b2, …, bd denote the standard basis of ℝd, and let e1, e2, …, ed be independent Rademacher variables, i.e., random variables taking the values +1 and −1 each with probability 1/2. Define Xi≔ eibi for 1 ≤ i ≤ n ≔ d. Then Xi = 0, ‖Xi‖r2=1, and ‖Sn‖r2=d2/r=d2/r−1∑i=1n‖Xi‖r2. Thus any candidate for K in (4) has to satisfy K ≥ d2/r−1.

中文翻译：

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重新审视涅米洛夫斯基的不等式

我们的出发点是以下著名的概率定理：设 X1, …, Xn 是具有有限二阶矩的独立随机变量，令 Sn=∑i=1nXi。那么 Var(Sn)=∑i=1nVar(Xi)。(1) 如果我们假设每个 Xi 的均值为 0，Xi = 0，则 (1) 变为 ESn2=∑i=1nEXi2。(2) 这种等式很容易推广到希尔伯特空间 ℍ 中的向量，其内积为 ℍ ℍ：如果 Xi 与ℍ 中的值无关，使得 Xi = 0 且 ‖Xi‖2 < ∞，则 ‖Sn‖2 =〈Sn,Sn〉=∑i,j=1n〈Xi,Xj〉，并且由于 〈Xi, Xj〉 = 0 for i ≠ j 独立，E‖Sn‖2=∑i,j=1nE〈Xi， Xj〉=∑i=1nE‖Xi‖2。(3) 如果 Xi 在（真实的）Banach 空间 (, ‖ · ‖) 中取值会发生什么？在这种情况下，特别是当范数 ‖ · ‖ 的平方不是由内积给出时，我们的目标是以下类型的不等式：令 X1, X2, ..., Xn 是独立的随机向量，其值在 (, ‖ · ‖) 中，Xi = 0 且‖Xi‖2 < ∞。当 Sn≔∑i=1nXi 时，我们想证明 E‖Sn‖2≤K∑i=1nE‖Xi‖2 (4) 对于一些仅依赖于 (, ‖ · ‖) 的常数 K。对于统计应用，对于某些 r ∈ [1, ∞] 的情况 (B,‖⋅‖)=lrd≔(ℝd,‖⋅‖r) 特别有趣。这里向量 x ∈ ℝd 的 r 范数定义为 ‖x‖r≔{(∑j=1d|xj|r)1/rif1≤r≤∞,max1≤j≤d|xj|ifr=∞。(5) 一个明显的问题是指数 r 和维度 d 如何进入类型 (4) 的不等式。维度 d 的影响至关重要，因为当前的统计研究通常涉及小或中等“样本量”n（独立单位的数量），比如 102 或 104 个，而每个独立单位测量的项目数量 d单位很大，比如 106 或 107。下面两个随机向量 Xi 的例子为 (4) 中的常数 K 提供了下界： 例 1.1（lrd 中的一个下界） 设 b1, b2, …, bd 表示 ℝd 的标准基，令 e1, e2 , ..., ed 是独立的 Rademacher 变量，即随机变量取值 +1 和 -1，每个的概率为 1/2。定义 Xi≔ eibi 为 1 ≤ i ≤ n ≔ d。那么Xi = 0，‖Xi‖r2=1，‖Sn‖r2=d2/r=d2/r−1∑i=1n‖Xi‖r2。因此，(4) 中 K 的任何候选必须满足 K ≥ d2/r-1。