In this influential paper from Bin Yu:

B. Yu. Assouad, Fano, and Le Cam. In D. Pollard, E. Torgersen, and G. Yang, editors, Festschrift for Lucien Le Cam, pages 423–435. Springer, 1997.

The author at page 431 starts when detailing a concrete example of the methods described in the first part of the paper defines a set of functions

$$g_j(x) = cm^{-2}g(mx - x_j), j=1,\dots,m$$

where $g$ is twice differentiable on $[0, 1]$ for which

$$\int_{0}^{1}g(x)dx=0, \int_{0}^{1}g^2(x)dx=a > 0, \text{ and } \int_{0}^{1}(g'(x))^2dx=b > 0$$

and $[0,1]$ is divided into $m$ disjoint intervals of size $1/m$ and $x_1, \dots, x_m$ denote their centers and $c$ is some constant.

What I don't understand: The $g_j$ defined in this way will have $mx - x_j$ to get out of the domain of definition of $g$. Am I missing something ?


Two remarks: The author meant to write

$$g\Big(m(x - x_j)\Big)$$

instead of

$$g(mx - x_j)$$

and $g$ has constant value $0$ outside of $[0, 1]$.

With these two points, the rest of the example is consistent.

For information, a similar work is done in these lecture notes: http://www.stat.cmu.edu/~larry/=sml/minimax.pdf


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