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We use the Berry-Esseen theorem to prove the closeness of the two distributions. In the proof of the theorem people have used the notion of test functions (functions which are smooth and fourth derivative is bounded). I want to know the idea behind the use of test functions. How we can think about that because these two notions i.e., close distributions and test functions are very different. Thank you.

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Well, in one proof of the fundamental theorem of algebra people use the notion of winding number. Polynomials and winding numbers are very different. Why should very different things not appear together in a proof? –  joriki Mar 3 '13 at 18:05

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Consider the representation of of the Euclidean distance $$ \|x-y\| = \max_{\|u\| \le 1} \langle{u,x-y\rangle} $$ Note that $f_u(x) = \langle{u,x\rangle}$ is a linear function. If you define the norm of it as $\|f_u\| = \|u\|$, then the above can be written as $$ \| x - y\| = \max_{f \; \text{linear},\; \|f\|\le 1} [f(x) - f(y)] $$

Now, for two measures (or distributions) $\mu$ and $\nu$ you can use the same idea to define a distance $$ d(\nu,\mu) = \sup_{f\; \text{linear},\; \|f\| \le 1} [f(\nu) - f(\mu)] $$ where $\|f\|$ is a suitable norm. Now a general linear function(al) of measures has the form $$ f_\phi(\mu) = \int \phi(x) d\mu(x) $$ for some test function $\phi$. Thus, you get $$ d(\mu,\nu) = \sup_{\phi \,\in\, \mathcal{C},\; \|\phi\|_{\mathcal{C}}\,\le\, 1} \Big|\int \phi d\mu - \int \phi d\nu\Big| $$ where $\mathcal{C}$ is some class of test functions and $\|\cdot\|_{\mathcal{C}}$ is some sort of norm on it. This gives you a general way to define distances between objects if you can define linear fucntionals on them.

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