Celebratio Mathematica — Doob — Complete Bibliography

[1] J. L. Doob: “The boundary values of an analytic function,” Bull. Am. Math. Soc. 37 : 5 (1931), pp. 339. JFM 57.0396.10 article

[2] J. L. Doob: “The boundary values of analytic functions,” Bull. Am. Math. Soc. 38 : 3 (1932), pp. 188–189. Research announcement for a two-part article in Trans. Am. Math. Soc. 34:1 (1932) and 35:2 (1933). JFM 58.0354.04 article

[3] J. L. Doob: “The boundary values of analytic functions,” Trans. Am. Math. Soc. 34 : 1 (1932), pp. 153–170. A research announcement for this article was published in Bull. Am. Math. Soc. 38:3 (1932). MR 1501633 JFM 58.1077.01 Zbl 0003.40401 article

[4] J. L. Doob: “On a theorem of Gross and Iversen,” Ann. Math. (2) 33 : 4 (October 1932), pp. 753–757. MR 1503089 JFM 58.0331.02 Zbl 0005.25005 article

[5] J. L. Doob: Boundary values of analytic functions. Ph.D. thesis, Harvard University, 1932. Advised by J. L. Walsh. A two part article based on this dissertation was published in Trans. Am. Math. Soc. 34:1 (1932) and 35:2 (1933). MR 2936471 phdthesis

[6] J. L. Doob: “The ranges of analytic functions,” Bull. Am. Math. Soc. 39 : 3 (1933), pp. 195–196. Research announcement for an article published in Ann. Math. 36:1 (1935). JFM 59.0346.07 article

[7] J. L. Doob and B. O. Koopman: “The resolvent of a self-adjoint transformation,” Bull. Am. Math. Soc. 39 : 1 (1933), pp. 872. JFM 59.0428.10 article

[8] J. F. Ritt and J. L. Doob: “Systems of algebraic difference equations,” Bull. Am. Math. Soc. 39 : 7 (1933), pp. 508. JFM 59.0461.04 article

[9] J. L. Doob: “The boundary values of analytic functions, II,” Trans. Am. Math. Soc. 35 : 2 (1933), pp. 418–451. MR 1501694 JFM 59.1030.01 Zbl 0006.35501 article

[10] J. F. Ritt and J. L. Doob: “Systems of algebraic difference equations,” Am. J. Math. 55 : 1–4 (1933), pp. 4. A research announcement for this was published in Bull. Am. Math. Soc. 39:7 (1933). MR 1506981 JFM 59.0456.01 Zbl 0008.01804 article

The object of this paper is to derive an analogue, for systems of algebraic difference equations, of the fundamental theorem in the theory of systems of algebraic differential equations developed by [Ritt 1932]. We introduce the notion of irreducible system of algebraic difference equations, and show that every system of such eqiiations is equivalent to a finite set of irreducible systems. It will possibly strike one as curious that so general a result should be obtainable at a time when existence theorems for non-linear difference equations are almost entirely lacking.

Although our proof resembles greatly that for differential equations, there are also essential differences. These arise out of the circumstance that the derivative of a polynomial in several functions involves the derivatives of the functions linearly, while no corresponding result holds for the operation of differencing.

[11] J. L. Doob: “Note on probability,” Bull. Am. Math. Soc. 40 : 11 (1934), pp. 798. Research announcment for an article published in Ann. Math. 37:2 (1936). JFM 60.0479.08 article

[12] J. L. Doob: “Probability distributions and statistics,” Bull. Am. Math. Soc. 40 : 3 (1934), pp. 221. JFM 60.0479.09 article

[13] J. L. Doob: “Probability and statistics,” Trans. Am. Math. Soc. 36 : 4 (1934), pp. 759–775. MR 1501765 JFM 60.0467.02 Zbl 0010.17303 article

The theory of probability has made much progress recently in the direction of completely mathematical formulations of its methods and results. The purpose of this paper is to make a further contribution in this direction. In order to analyze the results of repeated trials of an experiment, a certain space of infinitely many dimensions is the proper tool. This space is discussed in the first section of the paper. In the second section, the results of the first are applied to obtain for the first time a complete proof of the validity of the method of maximum likelihood of R. A. Fisher, which is used in statistics to estimate the true probability distribution when the results of a repeated experiment are known.

[14] J. L. Doob and B. O. Koopman: “On analytic functions with positive imaginary parts,” Bull. Am. Math. Soc. 40 : 8 (1934), pp. 601–605. MR 1562915 JFM 60.0254.03 Zbl 0010.17101 article

[15] J. L. Doob: “Stochastic processes and statistics,” Proc. Natl. Acad. Sci. USA 20 : 6 (June 1934), pp. 376–379. JFM 60.0467.01 Zbl 0009.22101 article

[16] J. L. Doob: “The theory of statistical estimation,” Bull. Am. Math. Soc. 41 : 1 (1935), pp. 27. JFM 61.0583.10 article

[17] J. L. Doob: “The ranges of analytic functions,” Ann. Math. (2) 36 : 1 (January 1935), pp. 117–126. A research announcement for this was published in Bull. Am. Math. Soc. 39:3 (1933). MR 1503212 JFM 61.0355.05 Zbl 0011.16803 article

Let \( f(z) \) be a function analytic for \( |z| < 1 \). The set of values assumed by \( f(z) \) in \( |z| < 1 \), i.e. its range, has been the subject of numerous investigations. These investigations have been concerned with families of functions, which, because of certain restrictions, (such as \( f(O) = 0 \), \( |f^{\prime}(0)| \geqq 1 \)), have no limiting functions, (in the sense of uniform convergence in every closed subregion of \( |z| < 1 \)), which are identically constant. Montel [1929] investigated the general case of families of non-constant functions having no constant limiting functions, obtaining results which contain results obtained by Bloch, Bohr, Fekete, Koebe, Landau, Valiron. In this paper, certain more inclusive families of functions will be considered which may have constant limiting functions.

[18] J. L. Doob: “The limiting distributions of certain statistics,” Ann. Math. Stat. 6 : 3 (September 1935), pp. 160–169. JFM 61.1296.01 Zbl 0012.26801 article

[19] J. L. Doob: “Stochastic processes depending on a continuously varying parameter,” Bull. Am. Math. Soc. 42 : 7 (1936), pp. 490–491. Research announcement for the article published in Trans. Am. Math. Soc. 42:1 (1937). JFM 62.0634.09 article

[20] J. L. Doob: “Statistical estimation,” Trans. Am. Math. Soc. 39 : 3 (1936), pp. 410–421. MR 1501855 JFM 62.0611.01 Zbl 0014.16901 article

[21] J. L. Doob: “Note on probability,” Ann. Math. (2) 37 : 2 (April 1936), pp. 363–367. A research announcement for this was published in Bull. Am. Math. Soc. 40:11 (1934). MR 1503284 JFM 62.0589.02 Zbl 0013.40802 article

[22] J. L. Doob: “Book review: V. Volterra, ‘Leçons sur la théorie mathématique de la lutte pour la vie’,” Bull. Am. Math. Soc. 42 : 5 (1936), pp. 304–305. Book by V. Volterra (Gauthier-Villars, 1931). MR 1563293 article

[23] J. L. Doob: “Stochastic processes depending on a continuous parameter,” Trans. Am. Math. Soc. 42 : 1 (1937), pp. 107–140. A research announcement for this article was published in Bull. Am. Math. Soc. 42:7 (1936). MR 1501916 JFM 63.1075.01 Zbl 0017.02701 article

[24] J. L. Doob: “Book review: R. von Mises, ‘Wahrscheinlichkeit statistik und wahrheit’,” Bull. Am. Math. Soc. 43 : 5 (1937), pp. 316–317. Book by R. von Mises (Springer, 1936). MR 1563534 article

[25] J. L. Doob: “Book review: E. Tornier, ‘Wahrscheinlichkeitsrechnung und allgemeine integrationstheorie’,” Bull. Am. Math. Soc. 43 : 5 (1937), pp. 317–318. Book by E. Tornier (Teubner, 1936). MR 1563535 article

[26] J. L. Doob: “Stochastic processes with an integral-valued parameter,” Trans. Am. Math. Soc. 44 : 1 (July 1938), pp. 87–150. MR 1501964 JFM 64.0532.03 Zbl 0019.12701 article

[27] J. L. Doob: “One-parameter families of transformations,” Duke Math. J. 4 : 4 (1938), pp. 752–774. MR 1546095 JFM 64.0192.04 Zbl 0020.10902 article

[28] J. L. Doob: “Book review: P. Lévy, ‘Théorie de l’addition des variables aléatoires’,” Bull. Am. Math. Soc. 44 : 1 (1938), pp. 19–20. Book by P. Lévy (Gauthier-Villars, 1937). MR 1563667 article

[29] J. L. Doob: “Book review: L. Bachelier, ‘Les lois des grands nombres du calcul des probabilités’,” Bull. Am. Math. Soc. 44 : 1 (1938), pp. 20. Book by L. Bachelier (Gauthier-Villars, 1937). MR 1563668 article

[30] J. L. Doob: “Book review: É. Borel, ‘Valeur pratique et philosophie des probabilités’,” Bull. Am. Math. Soc. 45 : 9 (1939), pp. 651–652. Book by É. Borel (Gauthiers-Villars, 1939). MR 1564040 article

[31] J. L. Doob: “Book review: J. Ville, ‘Étude critique de la notion de collectif’,” Bull. Am. Math. Soc. 45 : 11 (1939), pp. 824. Book by J. Ville (Gauthier-Villars, 1939). MR 1564064 article

[32] J. L. Doob: “Regularity properties of certain families of chance variables,” Trans. Am. Math. Soc. 47 : 3 (1940), pp. 455–486. MR 2052 JFM 66.0609.04 Zbl 0023.24101 article

[33] J. L. Doob: “The law of large numbers for continuous stochastic processes,” Duke Math. J. 6 : 2 (1940), pp. 290–306. MR 2053 JFM 66.0621.01 Zbl 0023.24401 article

[34] J. L. Doob and W. Ambrose: “On two formulations of the theory of stochastic processes depending upon a continuous parameter,” Ann. Math. (2) 41 : 4 (October 1940), pp. 737–745. MR 2735 JFM 66.0620.01 Zbl 0025.19804 article

The theory of stochastic processes depending upon a continuous parameter is the theory of measure (probability) relations on a collection of functions, \( \{x(t)\} \), \( t \) ranging over the real numbers. There is some difficulty however in finding an appropriate collection of functions on which to consider the measure relations. On the one hand it is desirable to consider as large a class of functions as possible while on the other hand it is desirable that the functions considered have enough regularity properties, both individually and as a class, that one may systematically make use of known theorems from the theory of functions in investigating these measure relations. One way of choosing the collection of functions to be considered has been given by Doob [1937]; he considers first a measure defined on the space of all real-valued functions, \( x(t) \), and carries this measure over to certain subspaces. Then he shows that in certain cases these subspaces will have desirable regularity properties. Another approach was given by Wiener [1938; 1922–1923], who takes a function, \( f(t,x) \), subject to certain regularity conditions, and then considers the collection of \( t \)-functions obtained from \( f(t,x) \) by fixing \( x \) and allowing \( t \) to vary. Wiener defines a measure on this space of \( t \)-functions in terms of a measure on \( x \)-space. The principal result of the present paper is the establishing of some relations between these two approaches to the theory of stochastic processes depending upon a continuous parameter. In section 1 we give the precise formulations of these two kinds of stochastic processes, in section 2 we show their equivalence, and in section 3 we obtain some further theorems relating the two.

[35] J. L. Doob: “Probability as measure,” Ann. Math. Stat. 12 : 2 (1941), pp. 206–214. MR 4387 JFM 67.0453.02 Zbl 0025.34501 article

The following pages outline a treatment of probability suitable for statisticians and for mathematiciasn working in that field. No attempt will be made to develop a thoery of probability which does not use numbers for probabilities. The theory will be developed in such a such a way that the classical proofs of probability theorems will need no change, although the reasoning used may have a sounder mathematical basis. It will be seen that this mathematical basis is hightly technical, but that, as applied to simple problems, it becomes the set-up used by every statistician. The formal and empirical aspects of probability will be kept carefully separate. In this way, we hope to avoid the airy flights of fancy which distinguish many probability discussions and which are irrelevant to the problems actually encountered by either mathematician or statistician.

[36] J. L. Doob: “A minimum problem in the theory of analytic functions,” Duke Math. J. 8 : 3 (1941), pp. 413–424. MR 4885 JFM 67.1021.01 Zbl 0063.01144 article

[37] R. von Mises and J. L. Doob: “Discussion of papers on probability theory,” Ann. Math. Stat. 12 : 2 (1941), pp. 215–217. MR 4388 Zbl 0025.34502 article

[38] J. L. Doob: “Topics in the theory of Markoff chains,” Trans. Am. Math. Soc. 52 : 1 (1942), pp. 37–64. MR 6633 Zbl 0063.09001 article

Let \( P(t) \): \( (p_{ij}(t)) \) be a matrix (finite- or infinite-dimensional), depending on \( t > 0 \), whose elements satisfy the following conditions \begin{align*} & p_{ij}(t) \geq 0, \qquad \sum_j p_{ij}(t) = 1, \\ & P(s)P(t) = P(t)P(s) = P(s+t). \end{align*} Then \( p_{ij}(t) \) can be considered a transition probability of a Markoff chain: A system is supposed which can assume various numbered states, and \( p_{ij}(t) \) is the probability that the system is in the \( j \)th state at the end of a time interval of length \( t \), if it was in the ith state at the beginning of the interval. The present paper will be divided into two parts. In the first, the regularity properties of \( P(t) \), and its asymptotic properties as \( t\to 0 \), \( t\to\infty \) are studied. These problems have been solved in the finite-dimensional case by Doeblin [1938, 1939]. In the infinite-dimensional case new situations can arise, and the results are somewhat different. The method of approach is new, depending on two theorems concerning matrices whose elements are non-negative, and which have row sums less than or equal to 1. The method of approach can also be applied to the study of the asymptotic properties of the powers of a matrix of non-negative elements, with row sums 1. In the second part of the paper, the actual transitions connected with Markoff chains are investigated: That is, the properties of the function \( \xi(t) \), the number of the state which the given system assumes at time \( t \), are investigated. The continuity properties of \( \xi(t) \) are analyzed, and related to the regularity properties of the \( p_{ij}(t) \).

[39] J. L. Doob: “The Brownian movement and stochastic equations,” Ann. Math. (2) 43 : 2 (April 1942), pp. 351–369. Reprinted in Selected papers on noise and stochastic processes (1954). MR 6634 Zbl 0063.01145 article

[40] J. L. Doob: “What is a stochastic process?,” Am. Math. Mon. 49 : 10 (December 1942), pp. 648–653. MR 7210 Zbl 0060.28906 article

[41] J. L. Doob and R. A. Leibler: “On the spectral analysis of a certain transformation,” Am. J. Math. 65 : 2 (April 1943), pp. 263–272. MR 7944 Zbl 0061.24903 article

[42] J. L. Doob: “The elementary Gaussian processes,” Ann. Math. Stat. 15 : 3 (1944), pp. 229–282. MR 10931 Zbl 0060.28907 article

[43] J. L. Doob: “Markoff chains: Denumerable case,” Trans. Am. Math. Soc. 58 : 3 (November 1945), pp. 455–473. MR 13857 Zbl 0063.01146 article

[44] J. L. Doob: “Probability in function space,” Bull. Am. Math. Soc. 53 : 1 (1947), pp. 15–30. MR 19858 Zbl 0032.03401 article

[45] J. L. Doob: “Asymptotic properties of Markoff transition probabilities,” Trans. Am. Math. Soc. 63 : 3 (May 1948), pp. 393–421. MR 25097 Zbl 0041.45406 article

[46] J. L. Doob: “Renewal theory from the point of view of the theory of probability,” Trans. Am. Math. Soc. 63 : 3 (1948), pp. 422–438. MR 25098 Zbl 0041.45405 article

[47] J. L. Doob: “On a problem of Marczewski,” Colloq. Math. 1 (1948), pp. 216–217. MR 27314 Zbl 0038.03703 article

[48] J. L. Doob: “Time series and harmonic analysis,” pp. 303–343 in Proceedings of the Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 13–18 August 1945 and 27–29 January 1946). Edited by J. Neyman. University of California Press (Berkeley and Los Angeles), 1949. MR 27979 Zbl 0038.29902 incollection

Although many articles on the present subject have appeared in the mathematical, statistical, and physical literature, there still seems to be some justification for one more. The statisticians have applied only small parts of the theory; the physicists have gone deeper, but write like physicists; the mathematicians have gone furthest, but write like mathematicians, only for posterity. Their work is frequently not understood, and is in general either ignored or applied in simplified forms which often are formally more formidable than the original rigorous one. The present paper attempts to give a compact outline of the harmonic analysis of stochastic processes, with applications to physical problems.

[49] J. L. Doob: “Heuristic approach to the Kolmogorov–Smirnov theorems,” Ann. Math. Stat. 20 : 3 (1949), pp. 393–403. MR 30732 Zbl 0035.08901 article

Asymptotic theorems on the difference between the (empirical) distribution function calculated from a sample and the true distribution function governing the sampling process are well known. Simple proofs of an elementary nature have been obtained for the basic theorems of Komogorov [1933] and Smirnov [1939] by Feller [1948], but even these proofs conceal to some extent, in their emphasis on elementary methodology, the naturalness of the results (qualitatively at least), and their mutual relations. Feller suggested that the author publish his own approach (which had also been used by Kac), which does not have these disadvantages, although rather deep analysis would be necessary for its rigorous justification. The approach is therefore presented (at one critical point) as heuristic reasoning which leads to results in investigations of this kind, even though the easiest proofs may use entirely different methods.

No calculations are required to obtain the qualitative results, that is the existence of limiting distributions for large samples of various measures of the discrepancy between empirical and true distribution functions. The numerical evaluation of these limiting distributions requires certain results concerning the Brownian movement stochastic process and its relation to other Gaussian processes which will be derived in the Appendix.

[50] J. L. Doob: “Application of the theory of martingales,” pp. 23–27 in Le calcul des probabilités et ses applications [The calculus of probabilities and its applications] (Lyon, France, 28 June–3 July 1948). CNRS International Colloquia 13. Centre National de la Recherche Scientifique (Paris), 1949. MR 33460 Zbl 0041.45101 incollection

[51] J. L. Doob: “Book review: N. Arley, ‘On the theory of stochastic processes and their application to the theory of cosmic radiation’,” Am. Math. Mon. 57 : 7 (August–September 1950), pp. 497–498. Book by N. Arley (Wiley, 1948). MR 1527645 article

[52] J. L. Doob: “Continuous parameter martingales,” pp. 269–277 in Proceedings of the second Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 31 July–12 August 1950). Edited by J. Neyman. University of California Press (Berkeley and Los Angeles), 1951. MR 44770 Zbl 0044.33802 incollection

[53] J. L. Doob: “Book review: M. Fréchet, ‘Généralités sur les probabilités. Éléments aléatoires’,” Bull. Am. Math. Soc. 57 : 3 (1951), pp. 206–207. Book by M. Freéchet (Gauthier-Villars, 1950). MR 1565307 article

[54] J. L. Doob: “The measure-theoretic setting of probability theory,” Ann. Soc. Polon. Math. 25 (1952), pp. 199–209. MR 58897 Zbl 0049.09601 article

[55]J. L. Doob: Stochastic processes. John Wiley & Sons (New York), 1953. MR 0058896 Zbl 0053.26802

[56] J. L. Doob: “The Brownian movement and stochastic equations,” pp. 319–337 in Selected papers on noise and stochastic processes. Edited by N. Wax. Dover Publications (New York), 1954. Reprinted from Ann. Math. 43:2 (1942). incollection

[57] J. L. Doob: “Semimartingales and subharmonic functions,” Trans. Am. Math. Soc. 77 : 1 (1954), pp. 86–121. MR 64347 Zbl 0059.12205 article

[58] J. L. Doob: “Martingales and one-dimensional diffusion,” Trans. Am. Math. Soc. 78 : 1 (1955), pp. 168–208. MR 70885 Zbl 0068.11301 article

[59] J. L. Doob: “A probability approach to the heat equation,” Trans. Am. Math. Soc. 80 : 1 (1955), pp. 216–280. MR 79376 Zbl 0068.32705 article

[60] J. L. Doob: “Probability methods applied to the first boundary value problem,” pp. 49–80 in Proceedings of the third Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 26–31 December 1954 and July–August 1955), vol. 2: Contributions to probability theory. Edited by J. Neyman. University of California Press (Berkeley and Los Angeles), 1956. MR 84886 Zbl 0074.09101 incollection

[61] J. L. Doob: “Present state and future prospects of stochastic process theory,” pp. 348–355 in Proceedings of the International Congress of Mathematicians, 1954 (Amsterdam, 2–9 September 1954), vol. 3. Edited by J. C. H. Gerretsen and J. de Groot. Erven P. Noordhoff N.V. (Groningen), 1956. MR 84895 Zbl 0074.33801 incollection

Johan Cornelis Hendrik Gerretsen	Related
Johannes de Groot	Related

[62] D. L. Dub: Stochastic processes (Moscow). Izdatelstvo Inostrannoy Literatury, 1956. Russian translation of 1953 English original. MR 85654 book

[63] J. L. Doob: “Interrelations between Brownian motion and potential theory,” pp. 202–204 in Proceedings of the International Congress of Mathematicians, 1954 (Amsterdam, 2–9 September 1954), vol. 3. Edited by J. C. H. Gerretsen and J. de Groot. Erven P. Noordhoff N.V. (Groningen), 1956. MR 95542 Zbl 0072.43102 incollection

Johan Cornelis Hendrik Gerretsen	Related
Johannes de Groot	Related

[64] J. L. Doob: “Book review: M. Loève, “Probability theory: Foundations. Random sequences,” Bull. Am. Math. Soc. 62 : 1 (1956), pp. 62–65. Book by M. Loève (Van Nostrand, 1955). MR 1565753 article

[65] J. L. Doob: “Brownian motion on a Green space,” Theory Probab. Appl. 2 : 1 (1957), pp. 1–30. A version of this was originally published in Teor. Veroyatnost. i Primenen. 2:1 (1957). article

The space \( R \), locally isometric to an open \( N \)-dimensional sphere, is called a Green space of dimensionality \( N \). A Markov process in this space which is locally a Brownian motion of dimensionality \( N \) we shall call a Brownian motion in \( R \). It is shown that for each value of the variance parameter there exists synonymously a transition probability for the Brownian motion process in \( R \). This transition probability from the constant \( \xi \) to the constant \( \eta \), at time \( t \) has a density \( p(t,\xi,\eta) \). It is shown that for a reasonable choice of this density the function \( p \), when it is equal to zero for \( t\leqq 0 \), defines for fixed \( \eta \) a superparabolic function on \( (t,\xi) \) which is parabolic everywhere except the point \( (0,\eta) \); the singularity at this point is the same as for the transition probability density of the corresponding \( N \)-dimensional Brownian motion. In addition, \[ p(t,\xi,\eta) = p(t,\eta,\xi) .\]

[66] J. L. Doob: “A new look at the first boundary-value problem,” pp. 21–33 in Applied probability: Proceedings of symposia in applied mathematics. Edited by L. A. MacColl. Proceedings of Symposia in Applied Mathematics 7. McGraw-Hill (New York), 1957. MR 92278 Zbl 0079.31202 incollection

[67] J. L. Doob: “Brownian motion on a Green space,” Teor. Veroyatnost. i Primenen. 2 : 1 (1957), pp. 3–33. With Russian summary. A version of this was republished in Theory Probab. Appl. 2:1 (1957). MR 106509 Zbl 0078.32505 article

The purpose of this paper is to extend the basic results of Brownian motion theory, as already developed for spaces which are subsets of Euclidean \( N \)-space, to Green spaces. These spaces are the natural ones for Brownian motion. A significant advantage in going from Euclidean spaces to Green spaces is that, in much of the discussion, the character of the boundary does not enter, so that one can always deal with an abstract Green space instead of with its open subsets and their relative boundaries. For example, in discussing the Dirichlet problem, or Green’s functions, on the space as a whole rather than on its subsets, the non-probabilistic treatment requires that the boundary points be defined individually even to formulate the problem to be attacked. We shall see that this is not necessary in the probabilistic approach.

[68] J. L. Doob: “La théorie des probabilités et le premier problème des fonctions frontières” [Probability and the first boundary function problem], Publ. Inst. Statist. Univ. Paris 6 (1957), pp. 289–290. MR 106510 Zbl 0089.34002 article

[69]J. L. Doob: “Conditional Brownian motion and the boundary limits of harmonic functions,” Bull. Soc. Math. France 85 (1957), pp. 431–458. MR 0109961 Zbl 0097.34004

[70] J. L. Doob: “Probability theory and the first boundary value problem,” Ill. J. Math. 2 : 1 (1958), pp. 19–36. Dedicated to Paul Lévy on the occasion of his seventieth birthday. MR 106511 Zbl 0086.08403 article

[71] J. L. Doob: “Boundary limit theorems for a half-space,” J. Math. Pures Appl. (9) 37 (1958), pp. 385–392. MR 109962 Zbl 0097.34101 article

[72] J. L. Doob: “Corrections to ‘Discrete potential theory and boundaries’,” J. Math. Mech. 8 : 6 (1959), pp. 993. Corrections to an article published in J. Math. Mech. 8:3 (1959). article

[73] J. L. Doob: “A relativized Fatou theorem,” Proc. Nat. Acad. Sci. U.S.A. 45 : 2 (February 1959), pp. 215–222. MR 107095 Zbl 0106.07801 article

[74] J. L. Doob: “Discrete potential theory and boundaries,” J. Math. Mech. 8 : 3 (1959), pp. 433–458. Corrections were published in J. Math. Mech. 8:6 (1959). MR 107098 Zbl 0101.11503 article

In [1957], Hunt has studied a general potential theory based on continuous parameter Markov processes. The purpose of this paper is to outline the corresponding theory based on discrete parameter processes, and to apply the theory to obtain the Martin exit and entrance boundaries in the special case of Markov chains (countable state space). The discret parameter theory is, of course, essentially simpler than that treated by Hunt (who did not treat boundary problems), and has the advantage that it involves topological considerations only at an advanced stage. We shall not discuss the analouges of all the results of Hunt, but only give enough to ensure understanding of the issues involved, and to cover the application to boundaries.

[75] J. L. Doob: “A Markov chain theorem,” pp. 50–57 in Probability and statistics: The Harald Cramér volume. Edited by U. Grenander. Wiley Publications in Statistics. John Wiley & Sons (New York), 1959. MR 110123 Zbl 0095.12701 incollection

Harald Cramér	Related
Ulf Grenander	Related

[76] J. L. Doob: “A non-probabilistic proof of the relative Fatou theorem,” Ann. Inst. Fourier. Grenoble 9 (1959), pp. 293–300. MR 117454 Zbl 0095.08203 article

Let \( R \) be a Green space, as defined by Brelot and Choquet, with Martin boundary \( R^{\prime} \). Naïm [1957] has extended the Cartan fine topology on \( R \) to \( R\cup R^{\prime} \). Limits involving this topology will be called “fine limits”.

Let \( h \) be a strictly positive superharmonic function. Then \( h \) has a canonical integral representation [Brelot 1956], going back to Martin if \( R \) is an open subset of a Euclidean space, involving a uniquely determined measure \( \mu^h \) on \( R\cup R^{\prime} \). It has been shown [Doob 1957] using probabilistic methods that, if \( u \) is a positive superharmonic function on \( R \), then \( u/h \) has a finite fine limit at \( \mu^h \)-almost every point of \( R\cup R^{\prime} \). The purpose of this note is to prove this theorem non-probabilistically. Note that, if \( h \) is harmonic, the theorem is a boundary limit theorem, because then \( \mu^h \) is a measure of subsets of \( R^{\prime} \). In particular, if \( h \) is a constant function, the theorem states that \( u \) has a finite fine limit at \( \mu^h \)-almost every point of \( R^{\prime} \). This is the justification for calling the theorem the relative Fatou theorem.

[77] J. L. Doob: “Some problems concerning the consistency of mathematical models,” pp. 27–33 in Information and decision processes. Edited by R. E. Machol. McGraw-Hill (New York), 1960. MR 117140 incollection

[78] J. L. Doob, J. L. Snell, and R. E. Williamson: “Application of boundary theory to sums of independent random variables,” pp. 182–197 in Contributions to probability and statistics: Essays in honor of H. Hotelling. Edited by I. Olkin. Stanford Studies in Mathematics and Statistics 2. Stanford University Press (Stanford, CA), 1960. MR 120667 Zbl 0094.32202 incollection

James Laurie Snell	Related
Ingram Olkin	Related
Ralph E. Williamson	Related
Harold Hotelling	Related

[79] J. L. Doob: “Relative limit theorems in analysis,” J. Analyse Math. 8 : 1 (December 1960), pp. 289–306. MR 125974 Zbl 0115.26803 article

[80] J. L. Doob: “Appreciation of Khinchin,” pp. 17–20 in Proceedings of the fourth Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 20 June–30 July 1960), vol. 2. Edited by J. Neyman. University of California Press (Berkeley, CA), 1961. MR 131342 incollection

Jerzy Neyman	Related
Aleksandr Yakovlevich Khinchin	Related

[81] J. L. Doob: “A relative limit theorem for parabolic functions,” pp. 61–70 in Transactions of the second Prague conference on information theory, statistical decision functions, random processes (Liblice, Czech Republic, 1–6 June 1959). Edited by J. Kožešnik. Nakladatelství Československé Akademie Věd (Prague), 1961. MR 132923 Zbl 0101.11302 incollection

[82] J. L. Doob: “Notes on martingale theory,” pp. 95–102 in Proceedings of the fourth Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 20 June–30 July 1960), vol. 2. Edited by J. Neyman. University of California Press (Berkeley, CA), 1961. MR 133182 Zbl 0126.14003 incollection

Although several writers, for example Bernstein, Lévy, and Ville, had used what would now be identified as martingale concepts, the first systematic studies appeared in [Doob 1940, 1953]. Since then, martingale theory has been applied extensively, but little progress has been made in the theory itself. The purpose of hte present paper is to point out how much spade work remains to be done in the theory, by deriving new theorems without the use of deep technical apparatus.

Throughout this paper, the more appropriate nomenclature “submartingale,”, “supermartingale” is used, rather than the “semimartingale” found in [1953]. The unifying thread in the following work will be the fact that certain simple operations on submartingales transform them into submartingales. This leads to a new submartingale convergence theorem, to a sharpening of the upcrossing inequality, and thereby into an examination of apparently hitherto unnoticed interrelations between martingale and potential theory.

[83] J. L. Doob: “Conformally invariant cluster value theory,” Ill. J. Math. 5 : 4 (1961), pp. 521–549. MR 186821 Zbl 0196.42201 article

[84] J. L. Doob: “Book review: M. Loève, ‘Probability theory’,” Bull. Am. Math. Soc. 67 : 5 (1961), pp. 446–447. Book by Michel Loève (Van Nostrand, 1960). MR 1566140 article

[85] J. L. Doob: “Boundary properties for functions with finite Dirichlet integrals,” Ann. Inst. Fourier (Grenoble) 12 (1962), pp. 573–621. MR 173783 Zbl 0121.08604 article

[86] J. L. Doob: “A ratio operator limit theorem,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 1 : 3 (January 1963), pp. 288–294. MR 163333 Zbl 0122.36302 article

[87] J. L. Doob: “One-sided cluster-value theorems,” Proc. London Math. Soc. (3) 13 (1963), pp. 461–470. MR 166365 Zbl 0116.05301 article

[88] M. Brelot and J. L. Doob: “Limites angulaires et limites fines” [Angular limits and fine limits], Ann. Inst. Fourier (Grenoble) 13 : 2 (1963), pp. 395–415. MR 196107 Zbl 0132.33902 article

[89] J. L. Doob: “Cluster values of sequences of analytic functions,” Sankhyā Ser. A 25 : 2 (July 1963), pp. 137–148. MR 213567 Zbl 0145.31101 article

[90] J. L. Doob: “Some classical function theory theorems and their modern versions,” Ann. Inst. Fourier (Grenoble) 15 : 1 (1965), pp. 113–136. Errata to this article were published in Ann. Inst. Fourier 17:1 (1967). MR 203065 Zbl 0154.07503 article

[91]K. L. Chung and J. L. Doob: “Fields, optionality and measurability,” Amer. J. Math. 87 : 2 (April 1965), pp. 397–424. MR 0214121 Zbl 0192.24604 article

[92] J. L. Doob: “Wiener’s work in probability theory,” pp. 69–72 in Norbert Wiener, 1894–1964, published as Bull. Am. Math. Soc. 72 : 1, part 2. Issue edited by F. Browder, E. H. Spanier, and M. Gerstenhaber. American Mathematical Society (Providence, RI), January 1966. MR 184259 Zbl 0131.00512 incollection

Edwin H. Spanier	Related
Murray Gerstenhaber	Related
Norbert Wiener	Related
Felix E. Browder	Related

[93] J. L. Doob: “Applications to analysis of a topological definition of smallness of a set,” Bull. Am. Math. Soc. 72 : 4 (1966), pp. 579–600. MR 203665 Zbl 0142.09001 article

[94] J. L. Doob: “Remarks on the boundary limits of harmonic functions,” SIAM J. Numer. Anal. 3 : 2 (June 1966), pp. 229–235. MR 393529 Zbl 0141.30404 article

The purpose of this paper is to extend to arbitrary domains two theorems on complex-valued harmonic functions defined on balls or halfspaces.

The first of the two theorems is the famous one of F. and M. Riesz that a function on the unit circle which is of bounded variation and of power series type (that is, has Fourier series of the form \( \sum_0^{\infty}a_ne^{ni\theta} \)) is absolutely continuous.

The second theorem states, roughly, that if a real harmonic function \( u \) defined on a halfspace has a nontangential limit at every point of a subset \( B \) of the bounding hyperplane then any one of various functions whose partial derivatives are dominated suitably by those of \( u \) have nontangential limits almost everywhere (Lebesgue measure) on \( B \). Stein [1961] has carried such theorems farthest.

[95] J. L. Doob: “Errata: ‘Some classical function theory theorems and their modern versions’,” Ann. Inst. Fourier (Grenoble) 17 : 1 (1967), pp. 469. Errata to an article published in Ann. Inst. Fourier 15:1 (1965). MR 220961 article

[96] J. L. Doob: “Applications of a generalized F. Riesz inequality,” Rev. Roumaine Math. Pures Appl. 12 (1967), pp. 1185–1191. MR 233947 Zbl 0156.12202 article

[97] J. L. Doob: “Generalized sweeping-out and probability,” J. Funct. Anal. 2 : 2 (May 1968), pp. 207–225. MR 222959 Zbl 0186.50403 article

In this paper, in further confirmation of the close relation between potential theory and probability, generalized sweeping-out (balayage) will be investigated from a probabilistic point of view. The key concept is that of a supermartingale with values in a compact space \( K \). A specified reference family \( S \) of functions from \( K \) to the reals determines a partial ordering of measures on \( K \); a measure following a measure is a balayage of the latter. It is shown that under appropriate hypotheses on \( S \) an ordered (totally ordered) family of measures is the family of marginal distributions of a supermartingale with state space \( K \). If the measure family is maximal in the order, if \( K \) is metrizable and if the supermartingale is chosen properly, the supermartingale sample paths are right-continuous paths to the Choquet boundary of \( K \) relative to \( S \). The functions in \( S \) are generalized superharmonic functions and the supermartingale paths to the Choquet boundary of \( K \) are analogs of Brownian paths to the Martin boundary of a Green space in the classical potential theoretic context.

[98] J. L. Doob: “Compactification of the discrete state spaces of a Markov process,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 10 : 3 (September 1968), pp. 236–251. MR 234525 Zbl 0164.19202 article

[99] J. L. Doob: “An application of stochastic process separability,” Enseignement Math. (2) 15 (1969), pp. 101–105. MR 250366 Zbl 0179.47604 article

[100] J. L. Doob: “State spaces for Markov chains,” Trans. Am. Math. Soc. 149 : 1 (1970), pp. 279–305. MR 258131 Zbl 0231.60048 article

If \( p(t,i,j) \) is the transition probability (from \( i \) to \( j \) in time \( t \)) of a continuous parameter Markov chain, with \[ p(0+,i,i) = 1 ,\] entrance and exit spaces for \( p \) are defined. If \( L \) [or \( L^* \)] is an entrance [or exit] space, the function \( p(\,\cdot\,,\,\cdot\,,j) \) [or \( p(\,\cdot\,,i,\,\cdot\,)/h(\,\cdot\,) \)] has a continuous extension to \( (0,\infty)\times L \) [or \( (0,\infty)\times L^* \), for a certain norming function \( h \) on \( L^* \)]. It is shown that there is always a space which is both an entrance and exit space. On this space one can define right continuous strong Markov processes, for the parameter interval \( [0,b] \), with the given transition function as conditioned by specification of the sample function limits at 0 and \( b \).

[101] J. L. Doob: “Stochastic processes determined by families of continuous functions,” pp. 378–382 in Proceedings of the international conference on functional analysis and related topics (Tokyo, 1–8 April 1969). University of Tokyo Press, 1970. MR 266301 Zbl 0203.50102 incollection

[102] J. L. Doob: “Separability and measurable processes,” J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970), pp. 297–304. MR 279841 Zbl 0206.18503 article

[103] J. L. Doob: “Book review: K. L. Chung, ‘Markov processes with stationary transition probabilities’,” Bull. Am. Math. Soc. 76 : 4 (1970), pp. 688–690. Book by Kai Lai Chung (Springer, 1967), actual title being “Markov chains with stationary transition probabilities”. MR 1566555 article

[104] J. L. Doob: “Martingale theory: Potential theory,” pp. 203–206 in Potential theory (Stresa, Italy, 2–10 July 1969). Edited by M. Brelot. C.I.M.E. Sessions. Cremonese (Rome), 1970. Zbl 0214.17301 incollection

[105] J. L. Doob: “What is a martingale?,” Am. Math. Mon. 78 : 5 (May 1971), pp. 451–463. MR 283864 Zbl 0215.25801 article

[106] J. L. Doob: “State space for Markov processes,” pp. 13–14 in Martingales (Oberwolfach, Germany, 17–23 May 1970). Edited by H. Dinges. Lecture Notes in Mathematics 190. Springer (Berlin), 1971. Meeting report. MR 362475 incollection

[107] J. L. Doob: “Obituary: Paul Lévy,” J. Appl. Probability 9 : 4 (December 1972), pp. 870–872. MR 342336 Zbl 0245.01020 article

[108] J. L. Doob: “William Feller and twentieth century probability,” pp. xv–xx in Proceedings of the sixth Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 21 June–18 July 1970), vol. 2: Probability theory. Edited by L. M. Le Cam, J. Neyman, and E. L. Scott. University of California Press (Berkeley, CA), 1972. MR 386946 Zbl 0251.01009 incollection

William Feller	Related
Elizabeth Leonard Scott	Related
Jerzy Neyman	Related
Lucien Marie Le Cam	Related

[109] J. L. Doob: “The structure of a Markov chain,” pp. 131–141 in Proceedings of the sixth Berkeley symposium on mathematical statistics and probability (Berkeley, CA, 21 June–18 July 1970), vol. 3: Probability theory. Edited by L. M. Le Cam, J. Neyman, and E. L. Scott. University of California Press (Berkeley, CA), 1972. MR 405603 Zbl 0255.60042 incollection

Lucien Marie Le Cam	Related
Elizabeth Leonard Scott	Related
Jerzy Neyman	Related

[110] J. L. Doob: “An inequality useful in martingale theory,” Sankhyā Ser. A 35 : 1 (March 1973), pp. 1–4. MR 356216 Zbl 0272.60039 article

[111] J. L. Doob: “Boundary approach filters for analytic functions,” Ann. Inst. Fourier (Grenoble) 23 : 3 (1973), pp. 187–213. MR 367206 Zbl 0251.30034 article

Let \( D \) be the unit disk of the complex plane, with boundary \( C \). Let \( H^{\infty} \) be the class of bounded holomorphic functions on \( D \). The purpose of this paper is to discuss the filters along which the members of \( H^{\infty} \) have limits at \( C \). If \( A \) is a subset of \( D \) with accumulation point 1, discussing the limits of members of \( H^{\infty} \) at the point 1 along \( A \) is equivalent to discussing the limits of these functions along the filter generated by the traces on \( A \) of neighborhoods of 1. It is essential to treat filters with limit 1 which are not of this type, however, for example the filter \( \Gamma_{\!A} \) corresponding to nontangential approach to 1 and the filter \( \Gamma_{\!F} \) corresponding to approach to 1 in the fine topology relative to \( D \).

[112] J. L. Doob: “What are martingales?,” Wiadom. Mat. (2) 16 (1973), pp. 37–50. MR 458574 Zbl 0285.60032 article

[113] J. L. Doob: “Analytic sets and stochastic processes,” pp. 1–12 in Stochastic processes and related topics (Bloomington, IN, 31 July–9 August 1974), vol. 1. Edited by M. L. Puri. Academic Press (New York), 1975. Book dedicated to Jerzy Neyman. MR 380959 Zbl 0322.60006 incollection

Madan Lal Puri	Related
Jerzy Neyman	Related

[114] J. L. Doob: “Stochastic process measurability conditions,” Ann. Inst. Fourier (Grenoble) 25 : 3–4 (1975), pp. 163–176. Dedicated to Marcel Brelot on the occasion of his 70th birthday. MR 420805 Zbl 0287.60032 article

[115] Probability (Urbana, IL, March 1976). Edited by J. L. Doob. Proceedings of Symposia in Pure Mathematics 31. American Mathematical Society (Providence, RI), 1977. Zbl 0342.00013 book

[116] J. L. Doob: “Classical potential theory and Brownian motion,” pp. 147–179 in Aspects of contemporary complex analysis (Durham, UK, 1–20 July 1979). Edited by D. A. Brannan and J. Clunie. Academic Press (London and New York), 1980. MR 623469 Zbl 0513.31004 incollection

David Alexander Brannan	Related
James Gourlay Clunie	Related

[117] J. L. Doob: “A potential theoretic approach to martingale theory,” pp. 227–231 in Statistics and probability: Essays in honor of C. R. Rao. Edited by G. Kallianpur, P. R. Krishnaiah, and J. K. Ghosh. North-Holland (Amsterdam and New York), 1982. MR 659475 Zbl 0483.60069 incollection

Gopinath Kallianpur	Related
Calyampudi Radhakrishna Rao	Related
Paruchuri R. Krishnaiah	Related
Jayanta Kumar Ghosh	Related

[118] J. L. Doob: Classical potential theory and its probabilistic counterpart. Grundlehren der Mathematischen Wissenschaften 262. Springer (New York), 1984. Reprinted in 2001. MR 731258 Zbl 0549.31001 book

[119] J. L. Doob: “Commentary on probability,” pp. 353–354 in A century of mathematics in America, Part 2. Edited by P. L. Duren, R. Askey, and U. C. Merzbach. History of Mathematics 2. American Mathematical Society (Providence, RI), 1989. MR 1003142 incollection

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Richard Allen Askey	Related	Volume

[120] J. L. Doob: “Kolmogorov’s early work on convergence theory and foundations,” Ann. Probab. 17 : 3 (1989), pp. 815–821. MR 1009435 Zbl 0687.01008 article

[121] J. L. Doob: Stochastic processes, Reprint edition. Wiley Classics Library 24. John Wiley & Sons (New York), 1990. Reprint of 1953 original. MR 1038526 Zbl 0696.60003 book

[122] J. L. Doob: “Probability vs. measure,” pp. 189–193 in Paul Halmos: Celebrating 50 years of mathematics. Edited by J. H. Ewing and F. W. Gehring. Springer, 1991. MR 1113273 Zbl 0791.60001 incollection

Paul Richard Halmos	Related
Frederick W. Gehring	Related
John H. Ewing	Related

[123] J. L. Doob: “Book review: P. R. Masani, ‘Norbert Wiener 1894–1964’,” Bull. Am. Math. Soc. (N.S.) 27 : 2 (1992), pp. 304–305. Book by P. R. Masani (Birhäuser, 1990). MR 1567992 article

Norbert Wiener	Related
Pesi Rustom Masani	Related

[124] J. L. Doob: Measure theory. Graduate Texts in Mathematics 143. Springer (New York), 1994. MR 1253752 Zbl 0791.28001 book

[125] J. L. Doob: “The development of rigor in mathematical probability (1900–1950),” pp. 157–170 in Development of mathematics 1900–1950 (Bourglinster, Luxembourg, June 1992). Edited by J.-P. Pier. Birkhäuser (Basel), 1994. Republished in Am. Math. Mon. 103:7 (1996). MR 1298633 Zbl 0806.01014 incollection

[126] J. L. Doob: “The development of rigor in mathematical probability (1900–1950),” Am. Math. Mon. 103 : 7 (August–September 1996), pp. 586–595. Originally published in Development of mathematics 1900–1950 (1994). MR 1404084 Zbl 0865.01011 article

[127] J. L. Snell: “A conversation with Joe Doob,” Statist. Sci. 12 : 4 (November 1997), pp. 301–311. MR 1619190 Zbl 0955.01556 article

[128] J. L. Doob: Classical potential theory and its probabilistic counterpart, Reprint edition. Classics in Mathematics. Springer (Berlin), 2001. Originally published in 1984. MR 1814344 Zbl 0990.31001 book

[129] American mathematician who was regarded as the grand old man of probability, 2004. misc

[130] Joseph Doob, a math pioneer, dies at 94, 2004. Webpage on University of Illinois at Urbana-Champaign Department of Mathematics. misc

[131] Record of the celebration of the life of Joseph Leo Doob, 2005. Webpage on University of Illinois at Urbana-Champaign Department of Mathematics. Account of an event held in October 2004. misc

[132] J. L. Snell: “Obituary: Joseph Leonard Doob,” J. Appl. Probab. 42 : 1 (March 2005), pp. 247–256. MR 2144907 Zbl 1072.01550 article

[133] D. Burkholder and P. Protter: “Joseph Leo Doob, 1910–2004,” Stochastic Process. Appl. 115 : 7 (July 2005), pp. 1061–1072. MR 2147241 Zbl 1073.01516 article

Donald L. Burkholder	Related	Volume
Philip Elliott Protter	Related

[134] D. Burkholder: “Foreword,” pp. v in Joseph Doob: A collection of mathematical articles in his memory, published as Ill. J. Math. 50 : 1–4. Duke University Press (Durham, NC), 2006. MR 2247820 incollection

[135] D. Burkholder: “Joseph Leo Doob (1910–2004),” pp. vii–viii in Joseph Doob: A collection of mathematical articles in his memory, published as Ill. J. Math. 50 : 1–4. Issue edited by D. Burkholder. Duke University Press (Durham, NC), 2006. MR 2247821 Zbl 1130.01303 incollection

[136] M. Yor: “Joseph Leo Doob (27 février 1910–7 juin 2004)” [Joseph Leo Doob (27 Feburary 1910–7 June 2004)], Gaz. Math. 114 (2007), pp. 33–39. MR 2361707 Zbl 1221.01089 article

[137] Joseph Doob: A collection of mathematical articles in his memory, published as Ill. J. Math. 50 : 1–4. Issue edited by D. Burkholder. Duke University Press (Durham, NC), 2007. Zbl 1120.60002 book

[138] J. L. Doob: “Letter from Joseph L. Doob, dated June 6, 1985,” J. Électron. Hist. Probab. Stat. 5 : 1 (June 2009). Two page letter to Pierre Crépel. MR 2520680 Zbl 1170.01385 article

[139] J. Yang and L. Pan: “J. L. Doob and stochastic processes,” Math. Pract. Theory 45 : 9 (2015), pp. 281–288. With an English summary. Zbl 1349.01046 article

Jing Yang	Related
Liyun Pan	Related