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The symbol $\perp\!\!\!\!\perp$ in statistics is a way to denote statistical independence of a collection of random variables. I have seen two forms of it. The first is highly suitable in writing about independence between two variables:

$$X\perp\!\!\!\!\perp Y$$

The second works for two variables, but is especially suited for multiple variables:

$$\perp\!\!\!\!\perp \left[ X_1, \cdots, X_n \right]$$

What these symbols mean, for a collection of (let's say real-valued) random variables $\{ X_j \}_{j=1}^n$, is

$$\perp\!\!\!\!\perp \left[ X_1, \cdots, X_n \right] \iff F_{X_1, \cdots, X_n} (x_1, \cdots, x_n) = \prod_{j=1}^n F_{x_j}(x_j)$$

where $F_{X_1, \cdots, X_n}$ is the joint cumulative distribution function and the $F_{x_j}$ is the marginal cumulative distribution function for the $j$th random variable.

What is the earliest-known publication that used the $\perp\!\!\!\!\perp$ symbol (either as a binary or n-ary relation) to denote statistical independence between/among random variables?

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  • $\begingroup$ I occasionally see $X_1 \perp\!\!\!\!\perp \cdots \perp\!\!\!\!\perp X_n$, or similar, so the former notation is capable of illustrating the n-ary case. $\endgroup$
    – Galen
    May 17, 2022 at 21:29
  • $\begingroup$ Note that I am asking about $\perp\!\!\!\!\perp$, which has two vertical bars, rather than $\perp$ which has one vertical bar. $\endgroup$
    – Galen
    May 17, 2022 at 21:35

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An early reference that uses this notation and (at a minimum) introduced many people to this notation is the following publication.

Dawid, A. P. “Conditional Independence in Statistical Theory.” Journal of the Royal Statistical Society. Series B (Methodological) 41, no. 1 (1979): 1–31. http://www.jstor.org/stable/2984718.

Have you found earlier uses?

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