## The Annals of Probability

### 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 $l\leqi\leq n)$ 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 ${f_ {ij}(x, y)}_{1\leq i,j\leq n$ be any array of real valued measurable functions.We present a method of obtaining the order of magnitude of

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 $g_{1ij}(X_i ,\tilde\mathbf{x},\tilde\mathbf{y})$ and $g_{2ij}(Y_j,\tilde\mathbf{x},\tilde\mathbf{y})$, it is proved that

where the latter is an explicitly computable quantity. For any $\tilde\mathbf{x}^0$ and $\tilde\mathbf{y}^0$ which come within a factor of two of minimizing $\Phi(\mathbf{f}^(s), \mathbf{X,Y}\tilde\mathbf{x,y})$ it is shown that

#### Article information

Source
Ann. Probab., Volume 28, Number 4 (2000), 1884-1907.

Dates
First available in Project Euclid: 18 April 2002

https://projecteuclid.org/euclid.aop/1019160512

Digital Object Identifier
doi:10.1214/aop/1019160512

Mathematical Reviews number (MathSciNet)
MR1813847

Zentralblatt MATH identifier
1110.60300

#### Citation

Klass, Michael J.; Nowicki, Krzysztof. A symmetrization-desymmetrization procedure for uniformly good approximation of expectations involving arbitrary sums of generalized U -statistics. Ann. Probab. 28 (2000), no. 4, 1884--1907. doi:10.1214/aop/1019160512. https://projecteuclid.org/euclid.aop/1019160512

#### References

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