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The mathematical description of this is that for each $n$ we can choose some central value $m_n$ (not necessarily unique!) … Why don't we have central limit theorems for other mathematical combinations of numbers such as their product or their maximum? …

By the central limit theorem, means of samples from a population with finite variance approach a normal distribution regardless of the distribution of the population. …

The distribution of either correlation coefficient will depend on the underlying distribution, although both are asymptotically normal because of the central limit theorem. …

It turns out that a lot of nice results which holds for independent random variables (law of large numbers, central limit theorem to name a few) hold for stationary random variables (we should strictly … It turns out that any stationary data can be approximated with stationary ARMA model, thanks to Wold decomposition theorem. …

Dividing both sides by $dt$ and taking the limit as $dt\to 0$ gives $$f_{X+Y}(t) = F_Y(t) - F_Y(t-1).$$ In other words, adding a Uniform $[0,1]$ variable $X$ to any variable $Y$ changes the pdf $f_Y$ … The Central Limit Theorem has little to say here. After all, a sum of iid Binomial variables converges to a Normal distribution, but that sum is always discrete: it never even has a PDF at all! …

If $y_i$ are not normal, but independent, we can get approximate distribution of $\hat\bet$ thanks to the central limit theorem. … The central limit theorem then gives us the following result: $$\sqrt{n}(\hat\bet - \bet) \to \mathcal{N}\left(0, A^{-1} B A^{-1} \right).$$ So from this we see that independence and constant variance …

To make matters worse, it is unsafe to use the Central Limit Theorem as a safety net: for small n we can't rely on the convenient asymptotic normality of the test statistic and t distribution. … CLT for the numerator, and Slutsky’s theorem suggest that eventually the t-statistic should begin to look normal, if the conditions for both hold.) …

limit theorem. … So, it will only work in settings where there is a central limit theorem, but it needs stronger assumptions. …

are histograms of the median positions: Clearly, for a sufficiently large number of atoms, the distribution of their median position begins to look bell-shaped and grows narrower: that looks like a Central … Limit Theorem result, doesn't it? …

And if your data truly is IID, then the central limit theorem on variation of random processes would also suggest that those ranges are a reasonable approximation of the full gradient. …

The smoothing also affects the assumption that the residuals are normally distributed, as smoothing, by the central limit theorem, will make the data more Gaussian. …

Edit: I often hear it said that you can rely on the Central Limit Theorem to take care of non-normal errors - this is not always true (I'm not just talking about counterexamples where the theorem fails … Limit Theorem to give you approximately unbiased inference for realistic finite sample sizes. …

The central limit theorem is less useful than one might think in this context. First, as someone pointed out already, one does not know if the current sample size is "large enough". …

To construct a confidence interval (which is, essentially, what the bootstrap is all about), we can use the central limit theorem, the consistency of empirical quantiles and the delta method as tools to …

$Z_i = 1$ if $X_i \leq x$ and $0$ otherwise), you can directly apply the Central limit theorem to a mean of $Z$'s, and by using the Delta method, turn that into an asymptotic normal distribution for $F_X …