Is there a standard well-known notation for approximation of a distribution?

If a random variable $X$ has exactly standard normal distribution then we write $X \sim \mathcal {N}(0,1)$.

But what symbol should we use instead of "$\sim$" when the real distribution is unknown and we know only its approximation?
For example, sample mean $\overline{X} = \frac{1}{n}\sum_{i=1}^n X_i$ has approximately normal distribution $\mathcal{N}(\mu, \frac{\sigma^2}{n})$ when $n$ is large enough. I saw that some sources use symbol "$\approx$" in the following way: $\,\,\overline{X} \approx \mathcal{N}(\mu, \frac{\sigma^2}{n})$. But I'm not sure that symbol "$\approx$" is a standard well-known notation for approximation of a distribution.

  • $\begingroup$ I don't know what it means to say one distribution is approximately equal to another. I can imagine several definitions in this context. I'd say that you would need to define what you want precisely and introduce whatever notation you want to use in the context of that definition. $\endgroup$
    – lulu
    Mar 25 '18 at 21:12
  • $\begingroup$ You can get an asymptotic definition of $\approx$ if you consider $\sqrt{n}(\overline{X}-\mu)$. $\endgroup$
    – J.G.
    Mar 25 '18 at 21:31
  • $\begingroup$ @J.G. And indeed saying $\sqrt{n}\left(\overline{X} - \mu\right)\xrightarrow{d}\mathcal{N}\left(0,\sigma^2\right)$ amounts to the Central Limit Theorem $\endgroup$
    – Henry
    Mar 25 '18 at 21:32
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    $\begingroup$ How about $X \sim \left[\approx \mathcal{N}\left(0,1\right)\right]$ ?. $\endgroup$ Mar 25 '18 at 23:41
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    $\begingroup$ Not everything has to have a symbol. The English word 'approximately' with an other word or two of qualification or context would work nicely. But I have seen $\bar X \stackrel{aprx}{\sim} \mathsf{Norm}(\mu, \sigma/\sqrt{n})$ and $\bar X \stackrel{\cdot}{\sim} \mathsf{Norm}(\mu, \sigma/\sqrt{n}).$ Not saying I like them, and they are sufficiently rare that they should be explained at least once in the chapter, paper, report. $\endgroup$
    – BruceET
    Mar 26 '18 at 2:01

Instead of stacking together two tildes, you could put a dot above and a dot below the tilde to indicate an approximate distribution(similar to a notation for approximate equal with a dot above and below an equal sign), as shown below:



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