A symmetrization-desymmetrization procedure for uniformly good approximation of expectations involving arbitrary sums of generalized U-statistics

Abstract

Let $\Phi$ be a symmetric function, nondecreasing on $[0,\infty)$ and satisfying a $\Delta_2$ growth condition, $(X_1, Y_1), (X_2, Y_2),\ldots,( X_n,Y_n)$ be independent random vectors such that (for each $1\le i\le n)$ either $Y_i =X_i$ or $Y_i$ is independent of all the other variates, and the marginal distributions of $\{X_i\}$ and $\{Y_j\}$ are otherwise arbitrary. Let $\{f_ {ij}(x, y)\}_{1\le i,j\le n}$ be any array of real valued measurable functions.We present a method of obtaining the order of magnitude of $$E\Phi\left( \sum_{1 \le i, j \le n}f_{ij}(X_i,Y_j)\right).$$ The proof employs a double symmetrization, introducing independent copies $\{\tilde{X}_i,\tilde{Y}_j\}$ of $\{X_i,Y_j\}$, and moving from summands of the form $f _{ij}(X_i, Y_j)$ to what we call $f_{ij}^{(s)}(X_i,Y_j,\tilde{X}_i,\tilde{Y}_j)$. Substitution of fixed constants $\tilde{x}_i$ and $\tilde{y}_ j$ for $\tilde{X}_ i$ and $\tilde{Y}_ j$ results in $f_{ij}^{(s)}(X_i,Y_j,\tilde{x}_i,\tilde{y}_j)$, which equals $f_{ij}(X_i,Y_j)$ adjusted by a sum of quantities of first order separately in ${X_i}$ and ${Y_j}$. Introducing further explicit first-order adjustments, call them $g_{1ij}(X_i,\mathbf{\tilde{x}},\mathbf{\tilde{y}})$ and $g_{2ij}(Y_j,\mathbf{\tilde{x}},\mathbf{\tilde{y}})$, it is proved that $$\begin{equation} E\Phi\left( \sum_{1 \le i, j \le n}\left(f^{(s)}_{ij}(X_i,Y_j,\tilde{x}_i,\tilde{y}_j)−g_{1ij}(X_i,\mathbf{\tilde{x}},\mathbf{\tilde{y}})−g_{2ij}(Y_j,\mathbf{\tilde{x}},\mathbf{\tilde{y}})\right)\right)\\ \le_\alpha E\Phi\left(\sqrt{\sum_{1 \le i, j \le n}\left(f^{(s)}_{ij}(X_i,Y_j,\tilde{x}_i,\tilde{y}_j)\right)^2}\right)\approx_\alpha \Phi\left(\mathbf{f}^{(s)},\mathbf{X},\mathbf{Y},\mathbf{\tilde{x}},\mathbf{\tilde{y}}\right) \end{equation}$$ where the latter is an explicitly computable quantity. For any $\mathbf{\tilde{x}}^0$ and $\mathbf{\tilde{y}}^0$ which come within a factor of two of minimizing $\Phi(\mathbf{f}^{(s)}, \mathbf{X},\mathbf{Y},\mathbf{\tilde{x}},\mathbf{\tilde{y}})$ it is shown that $$ \begin{equation} \left.\begin{aligned} E\Phi &\left( \sum_{1 \le i, j \le n}f_{ij}(X_i,Y_j)\right)\\ \approx_\alpha & \max\left\{\Phi(\mathbf{f}^{(s)},\mathbf{X},\mathbf{Y},\mathbf{\tilde{x}}^0,\mathbf{\tilde{y}}^0),E\Phi \left(\sum_{1 \le i, j \le n}\Big(f_{i,j}(X_i,\tilde{y}^0_j)+f_{ij}(\tilde{x}^0_i,Y_j)\\ −f_{ij}(\tilde{x}^0_i,\tilde{y}^0_j)+g_{1ij}(X_i, \tilde{x}^0_i,\tilde{y}^0_j)+g_{2ij}(Y_j,\tilde{x}^0_i,\tilde{y}^0_j)\Big)\right)\right\}, \end{aligned}\right. \end{equation} $$ which is computable (approximable)in terms of the underlying random variables. These results extend to the expectation of $\Phi$ of a sum of functions of $k$-components.