{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,6]],"date-time":"2026-06-06T17:11:19Z","timestamp":1780765879586,"version":"3.54.1"},"reference-count":33,"publisher":"Mathematical Society of Japan (JST)","issue":"0","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Japanese journal of mathematics :transactions and abstracts"],"published-print":{"date-parts":[[1941]]},"DOI":"10.4099\/jjm1924.18.0_261","type":"journal-article","created":{"date-parts":[[2017,10,19]],"date-time":"2017-10-19T22:28:17Z","timestamp":1508452097000},"page":"261-301","source":"Crossref","is-referenced-by-count":82,"title":["On stochastic processes (I)"],"prefix":"10.4099","volume":"18","author":[{"given":"Kiyosi","family":"IT\u00d4","sequence":"first","affiliation":[{"name":"Mathematical Institute, Faculty of Science, Imperial University of Tokyo"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"2576","reference":[{"key":"1","unstructured":"(1) See P. Lévy: Théorie de l'addition des variables aléatoires (1937), p. 180 Th. 54. 1."},{"key":"2","unstructured":"(1) and see also A. Khintchine: Déduction nouvelle d'une formule de M. Paul Lévy. Bull. Univ. Etat Moscou, Sér. Int. Sect. A. Math. et Mechan. 1 (1937)."},{"key":"3","unstructured":"(2) See J. L. Doob: Stochastic processes depending on a continuous parameter, Th. 3.8 (Transactions of the American Mathematical Society, Vol. 42)."},{"key":"4","unstructured":"(3) See P. Lévy: Théorie de l'addition des variables aléatoires (1937), p. 186 Th. 55. 1."},{"key":"5","unstructured":"(4) See J. L. Doob: See P. Lévy: Théorie de l'addition des variables aléatoires (1937), p. 186 Th. 55. 1."},{"key":"6","unstructured":"(5) Let A (ω) be a proposition depending on an element ω in a probability field (Ω, P). Then we shall denote the set of the ω-elements for which A (ω) is true by A itself, if we have no fear of confusions; this notation may be consistent with the conventional use."},{"key":"7","unstructured":"(6) || || means the euclidian metric."},{"key":"8","unstructured":"(7) By [ab) we shall denote the interval closed on the left and open on the right. [ab], (ab] or (ab) may also be understood similarly."},{"key":"9","unstructured":"(8) E (+) a means the set E(x+a; x∈E)."},{"key":"10","unstructured":"(9) See J. L. Doob: Stochastic processes with an integral-valued parameter, Th. 1.1 (Transactions of the American Mathematical Society, Vol. 44)."},{"key":"11","unstructured":"(9) and E. Hopf: Ergodentheorie, p. 4, Mass und Integral in Produkträumen (Ergebnisse der Mathematik, Vol. 5, No. 2). The. word “separable” used here is explained at the end of this paper."},{"key":"12","unstructured":"(10) In this paper we shall make use of the term “monotone increasing” in the sense of “monotone non-decreasing”."},{"key":"13","unstructured":"(11) See P. Lévy: See P. Lévy: Théorie de l'addition des variables aléatoires (1937), p. 28, foot-note (1)."},{"key":"14","unstructured":"(12) See P. Lévy: See P. Lévy: Théorie de l'addition des variables aléatoires (1937), p. 90."},{"key":"15","unstructured":"(13) See P. Lévy: See P. Lévy: Théorie de l'addition des variables aléatoires (1937), p. 63."},{"key":"16","unstructured":"(14) See P. Lévy: See P. Lévy: Théorie de l'addition des variables aléatoires (1937), p. 19, Th. 29, 1."},{"key":"17","unstructured":"(15) See A. Kolmogoroff: Grundbegriffe der Wahrscheinlichkeitsrechnung, p. 25 (Ergebnisse der Mathematik Vol. 2, No. 3)."},{"key":"18","unstructured":"(16) Let Mα<\/sub>(ω) be a proposition where α runs over a parameter domain A, and ω over a probability field (Ω, P). If P(∏ Mα)=1, then we say that Mα<\/sub> holds for any α In A with probability 1, or that, with probability 1, Mα<\/sub> holds for any α in A. If P(Mα<\/sub>)=1 for any α in A, then we say that Mα<\/sub> holds with probability 1 for any α in A, or that, for any α in A, Mα<\/sub> holds with probability 1."},{"key":"19","unstructured":"(17) (xSi+1<\/sub><\/sub>*-xSi<\/sub><\/sub>*∈E) means the ω*-set E (ω*; xSi+1<\/sub><\/sub>*-xSi<\/sub><\/sub>*∈E). Cf. Let A (ω) be a proposition depending on an element ω in a probability field (Ω, P). Then we shall denote the set of the ω-elements for which A (ω) is true by A itself, if we have no fear of confusions; this notation may be consistent with the conventional use."},{"key":"20","unstructured":"(18) See A. Kolmogoroff: Grundbegriffe der Wahrscheinlichkeitsrechnung, p. 25 (Ergebnisse der Mathematik Vol. 2, No. 3)."},{"key":"21","unstructured":"(19) See A. Kolmogoroff: Grundbegriffe der Wahrscheinlichkeitsrechnung, p. 25."},{"key":"22","unstructured":"(19) Cf. J. L. Doob: Stochastic processes with an integral-valued parameter, Th. 1. 1."},{"key":"23","unstructured":"(20) See J. L. Doob: Stochastic processes depending on a continuous parameter, Th. 3. 8."},{"key":"24","unstructured":"(21) λ(×)E means E (λa; a∈E)."},{"key":"25","unstructured":"(22), (23) See P. Lévy: Théorie de l'addition des variables aléatoires (1937), p. 192."},{"key":"26","unstructured":"(24) A. Wintner: Asymptotic distributions and infinite convolutions (1937-1938), Lemma 5. 1."},{"key":"27","unstructured":"(25) ∫a<\/sub>b<\/sup> means ∫a+0<\/sub>b+0<\/sup>."},{"key":"28","unstructured":"(26) ∫a<\/sub>∞<\/sup>uβdu=((limb→∞<\/sub> ∫a<\/sub>b<\/sup>uβt<\/sub>du<\/sup>(ω); ω∈Ω); 0≤t≤1)"},{"key":"29","unstructured":"(27) See Definition 1. 2."},{"key":"30","unstructured":"(28) See the beginning of Chapter III."},{"key":"31","unstructured":"(29) ℜ means the real part."},{"key":"32","unstructured":"(30) See J. L. Doob: Stochastic processes depending on a continuous parameter, Th. 3. 9."},{"key":"33","unstructured":"(31), (32) See Definition 1. 11."}],"container-title":["Japanese journal of mathematics :transactions and abstracts"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.jstage.jst.go.jp\/article\/jjm1924\/18\/0\/18_0_261\/_pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2017,10,21]],"date-time":"2017-10-21T07:31:07Z","timestamp":1508571067000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.jstage.jst.go.jp\/article\/jjm1924\/18\/0\/18_0_261\/_article"}},"subtitle":["Infinitely divisible laws of probability"],"short-title":[],"issued":{"date-parts":[[1941]]},"references-count":33,"journal-issue":{"issue":"0","published-print":{"date-parts":[[1941]]}},"URL":"https:\/\/doi.org\/10.4099\/jjm1924.18.0_261","relation":{},"ISSN":["0075-3432","1861-3624"],"issn-type":[{"value":"0075-3432","type":"print"},{"value":"1861-3624","type":"electronic"}],"subject":[],"published":{"date-parts":[[1941]]}}}