Does anyone know where and when Cantelli's inequality was originally published? Strangely enough, I have not been able to find this information online.
Probability Inequalities of the Tchebycheff Type (1961) contains two inequalities C1 and C2 that are referred to as "Cantelli inequalities", with references:
[C1] F.P. Cantelli Intorno ad un teorema fondamentale della teoria del rischio, Bolletino dell Associazione degli Attuari Italiani, pages 123 (1910).
[C2] F.P. Cantelli, Sui confini della probabilità, Atti del Congresso Internazionale dei Matematici 6, 4759 (1928)
The identity given in Wikipedia is C2, and can be found on page 53 of the 1928 paper:
The C1 inequality from the 1910 paper is different, it involves second and fourth moments.
I do note that the 1910 paper is cited as the source of the C2 inequality at several places, for example in Wikipedia, but I have not been able to locate that paper to see if this earlier attribution is correct.
The 1928 paper is explicitly a review article, and Cantelli complains that few have read his older papers and have rederived some of his results without giving him due credit. In this respect Cantelli mentions the 1910 paper referred to above, as well as a 1911 paper that I have also not been able to locate:
[C3] F.P. Cantelli, Intorno ad un problema di calcolo delle probabilità, Giornale di matematiche di Battaglini, 49, 341352 (1911).

$\begingroup$ Thank you Carlo. I'll try to get that paper by Cantelli. Is the inequality there in the form it is known now? I saw the paper by Regazzini, but was not able to find Cantelli's inequality there. $\endgroup$ – Iosif Pinelis Jan 5 '18 at 19:02

$\begingroup$ I have now gotten the La tendenza paper, but I don't see anything there resembling Cantelli's inequality. I don't know Italian, and yet, the paper seems to discuss properties of convergence of sequences of r.v.'s to a constant (in probability or, equivalently, in distribution). Am I missing something? $\endgroup$ – Iosif Pinelis Jan 5 '18 at 19:47

$\begingroup$ I gave the wrong Cantelli paper, it's hopefully correct now. $\endgroup$ – Carlo Beenakker Jan 5 '18 at 21:42