Let's consider N i.i.d continuous random variables from some arbitrary distribution. Why do we have to approximate the distribution of the sample mean using the CLT? Why can't we explicitly compute its distribution using convolutions and then study its properties?
2026-04-01 18:10:17.1775067017
Use of convolutions to compute the distribution of the sample mean?
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Because convolutions of a large number of functions are often not so easy to manipulate. For example, if $(X_n)$ is i.i.d. uniform on $(0,1)$, then the density of $S_n=X_1+X_2+\cdots+X_n$ has $n$ different (polynomial) expressions on the intervals $(k-1,k)$ for $1\leqslant k\leqslant n$ and it is impossible to use them to prove any sensible asymptotics about $S_n$.