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
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¶ 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
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¶ 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
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