Reference: who introduced the tilde "~" notation to mean "has probability distribution..."?

Question

[Note: although this question has an accepted answer, the investigation is not finished yet. I encourage you to post your findings.]

Who first introduced the notation "$X \sim Q$", meaning that $Q$ is the probability distribution for $X$, and its related meanings?

Classical texts such as Jeffreys's Theory of Probability (3rd ed. 1961, 2nd ed. 1948) and Fisher's Statistical Methods for Research Workers (13th repr. ed. 1963) and Statistical Methods and Scientific Inference (1956) do not seem to use it (in Jeffreys "$\sim$" denotes the logical not).

I checked also the references given in this informative answer, but did not find anything relevant there.

(This notation is by no means universal in recent times or today, see eg Mosteller & Tukey's Data Analysis and Regression (1977) or Jaynes's Probability Theory (2003). In fact it does not make much sense when probability is seen as an extension of formal logic (eg Keynes, Johnson, Jeffreys, Pólya, Hailperin, Jaynes), defined over propositions rather than random variables.)

Edit (2021-06-24): The earliest reference I found so far is a paper by Khatri written in 1965 and published 1967, where the notation is introduced on p. 1854. The earliest textbook reference found so far is by Srivastava & Khatri, 1979, p. 41.

John Aldrich also kindly replied to my inquiry, saying that he never investigated the origin of this notation.

Answer 1

Early uses

There are some earlier uses since 1961 by Ingram Olkin with others.

• Olkin, Ingram, and Robert F. Tate. "Multivariate correlation models with mixed discrete and continuous variables." The Annals of Mathematical Statistics (1961): 448-465.

  $X \sim F(x)$ means that $x$ is distributed according to the d.f. $F(x)$, and $x(n) \to F(x)$ means that the asymptotic d.f. of $x(n)$ is $F(x)$.

• Olkin, Ingram, and Herman Rubin. "A characterization of the Wishart distribution." The Annals of Mathematical Statistics (1962): 1272-1280.

  We write $X \sim \mathcal{W}(\Lambda,p,n), (n > p-1, \Lambda: p \times p, \Lambda > 0 )$, to mean that $X$ is a $p \times p$ symmetric matrix with the density function $$p(X) = c \vert \Lambda \vert ^{n/2} \vert X \vert^{(n-p-1)/2} e^{-(\frac{1}{2}) \text{tr} \Lambda X} $$

digging further you come across the notation in the dissertation "Multivariate Tests of Hypotheses with Incomplete Data" by Raghunandan Prasad Bhargava (student of Olkin) in 1963

We use the notation $Y \sim p(z)$ to mean that the probability density of $Z$ is $p(z)$, and $Y \underset{H}{\sim} p(z)$ to denote that the probability density of $Z$ under the hypothesis $H$ is $p(z)$.

Srivastava is also an alumnus of Stanford university (and used the notation in a paper of 1965).

Before 1961

• Before this time Olkin uses the more verbose descriptions of the form 'Let ... be independent and each distributed as ...'. So it seems like he either started it around 1961 or picked it up from others.

• For Rubin there is neither earlier use of $\sim$ in this way. In one paper he seems to use it to indicate asymptotic equality. (analogous to Bachmann–Landau notations)

  Rubin, Herman. "The estimation of discontinuities in multivariate densities, and related problems in stochastic processes." Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics. University of California Press, 1961.

• Other uses of the symbol $\sim$ are by Kolmogorov in

  Колмогоров, Андрей Николаевич. "Об аналитических методах в теории вероятностей." Успехи математических наук 5 (1938): 5-41.

  I do not know Russian but based on google translate I guess that equation 67 $$P^p_k \sim \frac{A^kp^k}{k!}e^{-Ap}$$ is supposed to mean is approximately distributed as

• A similar use is in

  Колмогоров, Андрей Николаевич. "Обобщение формулы Пуассона на случай выборки из конечной совокупности." Успехи математических наук 6.3 (43 (1951): 133-134.

  In which we have an equation $$P_n(m \vert N,M) \sim P_n^\star(m\vert N , M)$$ and $P^\star$ is a simpler expression that is supposed to approximate to $P$ (in this case the hypergeometric distribution is being approximated)

• Another use by Kolmogorov is in

  Колмогоров, Андрей Николаевич. "Об операциях над множествами." Математический сборник 35.3-4 (1928): 415-422.

  In which the following equation occurs $$\overline{\overline{X}} \sim X$$ Again, I do not know Russian and have to decipher it. To me it seems that it means that the operation denoted by $\overline{\overline{X}}$ will give the same result as the operation denoted by $X$. (this is not anymore about probability).

• Paul Lévy uses the tilde symbol also as 'approximation' or 'asymptotic formula'. For instance in

  Lévy, Paul. "Sur le développement en fraction continue d'un nombre choisi au hasard." Compositio mathematica 3 (1936): 286-303.

  La probabilité de l’existence de $p$ valeurs supérieures à $n$ dans la suite $y_1, y_2, ..., y_n$ est d’autre part $$C_{n}^p \left( \frac{1}{n}\right)^p\left( 1-\frac{1}{n}\right)^{n-p} \sim \frac{1}{ep!} \quad (n \to \infty)$$

  But, here both sides of the binary relation are mathematical expressions. It is not a 'statistical variable' on the left side.

• Lévy uses $\sim$ in stochastic expressions in

  Lévy, Paul. "Wiener's random function, and other Laplacian random functions." Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability. University of California Press, 1951.

  Now it is an asymptotic relation for the case of infinitesimally small steps. For instance Lévy's fundamental stochastic infinitesimal equation (equation 2.1.1) is written as $$\delta X(t) \sim dt \int_{t_0}^t F(t,u) dX(u) + \zeta \sigma(t) \sqrt{dt}$$ and the symbol is explained as

  The symbol $\sim$ means that, as $dt \to 0$, the two first moments of the difference of the two sides are $o(dt)$ or $o\left[d\omega(t) \right]$ [if $\sigma^2(t)dt$ is replaced by $d\omega(t)$].

(in Jeffreys "$∼$" denotes the logical not)

The tilde notation for negation dates at least back to use by Giuseppe Peano in 1987. See Jeff Miller's webpage: https://jeff560.tripod.com/set.html