A "cousin" of a theorem of Cs\'aki and Fischer

TL;DR

This paper introduces a new correlation-based measure of dependence related to a 1963 theorem, demonstrating its limitations and applying it to enhance a recent example comparing dependence measures.

Contribution

It presents a restricted 'cousin' of Csáki and Fischer's theorem using correlation of indicator functions, including a constructed example and an application to Peyre's 2013 dependence measure comparison.

Findings

The 'cousin' measure has limitations demonstrated by a constructed example.

The measure can be used to embellish existing dependence comparison examples.

A new dependence measure related to classical theorems is introduced.

Abstract

A 1963 theorem of P. Cs\'aki and J. Fischer deals with the "maximal correlation coefficient" in the context of independent pairs of $\sigma$-fields on a probability space. Here a somewhat restricted "cousin" of their result is presented for the same context, but involving in part an analogous measure of dependence based only on correlations of indicator functions. It was first proved by the author in an unpublished 1978 Ph.D. thesis. An example is constructed to show a limitation of this "cousin". Also, this "cousin" is used to trivially embellish a very sharp 2013 example of R. Peyre in connection with the comparison of these two measures of dependence.