U statistics and a new paper by Terence Tao

Terence Tao has a new paper out that relates to a couple things I've written about recently.

Elementary symmetric polynomials came up when developing the general equations for tangent sum andhyperbolic tangent sum. The latter post goes into more detail.

Before that, means of symmetric functions, not necessarily elementary polynomials or even polynomials, came up in the context of U-statistics.

Now combine these and you have means of elementary symmetric polynomials. This is what Tao just wrote about. Here is his blog post announcing his paper.

There are several inequalities satisfied by means of elementary symmetric named after Maclauren and Newton. If the inequalities don't go all the way back to these two men, presumably they are a direct continuation of work they started.

The Maclauren and Newton inequalities require arguments to be non-negative; Tao allows arguments to possibly be negative.

U-statistics are not necessarily the means of elementary symmetric polynomials. But for those U-statistics that are, Tao's new paper may imply new results. That is, Tao's new paper has implications for statistics.

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