Is there a version of the CLT for distributions that don't have a mean, but have a median, and finite moments of order > 1? (if such distributions exist)
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3$\begingroup$ The sampling distribution of the sample median $\hat{M}$ is approximately normal with mean $m$ and variance $\frac{1}{4n(f(m))^2}$ where $n$ is the sample size, $m$ is the population median, and $f$ is the pdf for the observations i.e. $$\hat{M} \sim N(m,1/4n(f(m))^2)$$ This approximation is good for $n\geq 30$. Is this what you're looking for? $\endgroup$– Matthew H.Commented Sep 15, 2020 at 2:52
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6$\begingroup$ There aren't any distributions with finite moments of order >1 but no mean. If any given moment exists, so do all the moments of lower order. $\endgroup$– Thomas LumleyCommented Sep 15, 2020 at 3:40
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1 Answer
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Example: A random variable with a standard Cauchy distribution (t distribution with DF=1) has no mean, but median $\eta = 0$ and $f(\eta)=1/\pi.$ So according to @MatthewHolder's Comment (see Ref.), the median $H$ of a sample of size $n=100$ from this distribution should have $H \stackrel{aprx}{\sim} \mathsf{Norm}(\mu=0,\, \sigma=\pi/20).$
set.seed(2020)
h = replicate(10^5, median(rt(100,1)))
summary(h); sd(h); pi/20
Min. 1st Qu. Median Mean 3rd Qu. Max.
-0.8457575 -0.1043798 0.0006376 0.0013486 0.1067248 0.6956875
[1] 0.1589696 # aprx SD
[1] 0.1570796 # asymptotic SD
hist(h, prob=T, col="skyblue2", ylim=c(0,2.5),
main="Dist'n of Medians of 100 Std Cauchy RVs")
curve(dnorm(x, mean(h), sd(h)), add=T, col="red")
Ref: This relationship is Thm. 7.9.1 in Bain & Englehardt (1992): Intro. to Probability and Math. Stat. (2e). Also, in other math stat books along with discussions of order statistics. Also see this Q&A.