Celebratio Mathematica — Burkholder

[1] D. L. Burkholder: “On a class of stochastic approximation processes,” Ann. Math. Stat. 27 : 4 (1956), pp. 1044–1059. This appears to be an abridged version of the author’s 1956 PhD thesis. MR 85653 Zbl 0075.13803 article

[2] D. L. Burkholder: On a certain class of stochastic approximation processes. Ph.D. thesis, The University of North Carolina at Chapel Hill, 1956. Advised by W. Höffding. An abridged version of this was published in Ann. Math. Stat. 27:4 (1956). MR 2938781 phdthesis

[3] D. L. Burkholder: “On the existence of a best approximation of one distribution function by another of a given type,” Ann. Math. Stat. 30 : 3 (1959), pp. 738–742. MR 106522 Zbl 0096.13102 article

[4] D. L. Burkholder: “Effect on the minimal complete class of tests of changes in the testing problem,” Ann. Math. Stat. 31 : 2 (1960), pp. 325–331. MR 117846 Zbl 0119.15305 article

A question of interest in connection with many statistical problems is the following: Does a slight change in the problem result in a different answer? Here the effect of changes in the testing problem on the minimal complete class of tests is investigated. The effects of such changes are found to be different for the two families of distributions considered: The discrete multivariate exponential family and the continuous multivariate exponential family. In Section 2, it is shown that with respect to the discrete exponential family, the minimal complete class of tests for a standard testing problem is minimal complete for a wide variety of related problems. In Section 3, an example is given showing that with respect to the continuous exponential family, on the other hand, the minimal complete class of tests for a standard problem is not necessarilty minimal complete for a slight variation of this problem. Tests that are admissible for the standard problem are not necessarily admissible for the variation.

[5] D. L. Burkholder: “Sufficiency in the undominated case,” Ann. Math. Stat. 32 : 4 (1961), pp. 1191–1200. MR 131287 Zbl 0221.62003 article

In this paper the concept of statistical sufficiency is studied within a general probability setting. It is not assumed that the family of probability measures is dominated. That is, it is not assumed that there is a \( \sigma \)-finite measure \( \mu \) such that each probability measure in the family is absolutely continuous with respect to \( \mu \). In the dominated case, the theory of sufficiency has received a thorough-going and elegant treatment by Halmos and Savage [1949], Bahadur [1954], and others. Although many families of probability measures of importance for statistical work are dominated, many others are not. Nonparametric statistical work, especially, abounds with undominated families. It seems appropriate, therefore, to see what can be learned about sufficiency in the undominated case.

[6] D. L. Burkholder and Y. S. Chow: “Iterates of conditional expectation operators,” Proc. Am. Math. Soc. 12 : 3 (1961), pp. 490–495. MR 142144 Zbl 0106.33201 article

Let \( \{T_n\} \) be a sequence of conditional expectation operators in \[ L_1 = L_1(W,F,P) \] where \( (W,F,P) \) is a probability space. Let \[ S_n = T_n\cdots T_2T_1 .\] It is known [Doob 1953, p. 331] that if \( \{T_n\} \) is monotone decreasing, that is, if the range of \( T_{n+1} \) is a subset of the range of \( T_n \) for all \( n \), then for each \( x \) in \( L_1 \) the sequence \( \{S_n x\} \) converges almost everywhere. Here, the pointwise convergence behavior of \( \{ S_n x \} \) is studied under other conditions. For example, if \( T_{2n-1} = T_1 \) and \( T_{2n} = T_2 \) for all \( n \), does \( \{S_nx\} \) converge almost everywhere? This question was first posed by J. L. Doob. It is proved here that if \( x \) is in \( L_2 \), then this is indeed the case, and, furthermore, \( \sup|S_nx| \) is in \( L_2 \). Several of the preliminary results needed, especially Theorems 1 and 2, seem to be of some interest in their own right. The linear spaces mentioned in this paper may be either real or complex. All of our results hold with either interpretation.

[7] D. L. Burkholder: “On the order structure of the set of sufficient subfields,” Ann. Math. Stat. 33 : 2 (1962), pp. 596–599. MR 137227 Zbl 0127.34808 article

[8] D. L. Burkholder: “Semi-Gaussian subspaces,” Trans. Am. Math. Soc. 104 : 1 (1962), pp. 123–131. MR 138986 Zbl 0131.35001 article

[9] D. L. Burkholder: “Successive conditional expectations of an integrable function,” Ann. Math. Stat. 33 : 3 (1962), pp. 887–893. MR 143246 Zbl 0128.12602 article

[10] D. L. Burkholder and R. A. Wijsman: “Optimum properties and admissibility of sequential tests,” Ann. Math. Stat. 34 : 1 (1963), pp. 1–17. MR 148160 Zbl 0113.13005 article

[11] D. L. Burkholder: “Maximal inequalities as necessary conditions for almost everywhere convergence,” Z. Wahrscheinlichkeitstheor. Verw. Geb. 3 : 1 (March 1964), pp. 75–88. MR 189110 Zbl 0134.14602 article

[12] D. L. Burkholder: “Martingale transforms,” Ann. Math. Statist. 37 (1966), pp. 1494–1504. MR 0208647 Zbl 0306.60030 article

[13] D. L. Burkholder: “Independent sequences with the Stein property,” Ann. Math. Stat. 39 : 4 (1968), pp. 1282–1288. MR 228045 Zbl 0162.49404 article

Throughout this note \( Z = (Z_1,Z_2,\dots) \) is an independent sequence of complex valued random variables on a probability space \( (\Omega,\alpha,P) \).

It is convenient to say that \( Z \) has the Stein property if there is a number \( b > 0 \) such that if \( A \) is in \( \alpha \) and \( P(A) > 0 \), then there is a positive integer \( n \) such that if \( a,a_1 \), \( a_2,\dots \) are complex numbers and the series \[ \sum_{k=1}^{\infty}a_kZ_k \] converges almost everywhere, then \[ b\Bigl(\sum_{k=n}^{\infty}|a_k|^2\Bigr)^{1/2} \leq \operatorname{ess\,sup}\limits_{\omega\in A} \Bigl|a + \sum_{k=1}^{\infty}a_kZ_k(\omega)\Bigr|. \] For \( b > 0 \) and \( k \) a positive integer, let \( \pi_k(b) \) be the least upper bound of \[ P(|a + Z_k| < b) \] for \( a \) complex. Let \[ \pi(b) = \limsup_{k\to\infty}\pi_k(b) .\] The function \( \pi_k \) if essentially Lévy’s “fonction de concentration” for the random variable\( Z_k \) [Doeblin 1939; Lévy 1931, 1937].

[14] D. L. Burkholder and R. F. Gundy: “Extrapolation and interpolation of quasi-linear operators on martingales,” Acta Math. 124 (1970), pp. 249–304. MR 0440695 Zbl 0223.60021 article

[15] D. L. Burkholder, R. F. Gundy, and M. L. Silverstein: “A maximal function characterization of the class \( H^{p} \),” Trans. Amer. Math. Soc. 157 (1971), pp. 137–153. MR 0274767 Zbl 0223.30048 article

Richard Floyd Gundy	Related
Martin Louis Silverstein	Related

[16] D. L. Burkholder: “Martingale inequalities,” pp. 1–8 in Martingales (Oberwolfach, Germany, 17–23 May 1970). Lecture Notes in Mathematics 190. Springer (Berlin), 1971. MR 372985 incollection

[17] D. L. Burkholder: “Inequalities for operators on martingales,” pp. 551–557 in Actes du Congrès International des Mathématiciens (Nice, France, 1–10 September 1970), vol. 2. Gauthier-Villars (Paris), 1971. MR 420830 Zbl 0245.60041 incollection

This is a report on some recent developments in operator extrapolation theory [Burkholder et al. 1972; Burkholder and Gundy 1970]. An application to conjugate harmonic functions leading to a maximal function characterization of the Hardy class \( H^p \) is also described [Burkholder et al. 1971]. /par An operator defined on the Lebesgue class \( L^p \) (\( p\geq 1 \)) of a probability space may always be viewed as an operator on a family of martingales and it is this more flexible viewpoint that is convenient for us here.

The question central to the work under survey is: Suppose that \( \mathfrak{M} \) is a family of martingales on a probability space \( (\Omega,\alpha,P) \), and \( U \) and \( V \) are operators on \( \mathfrak{M} \) with values in the set of nonnegative \( \alpha \)-measurable functions. If \( \Phi \) is a nonnegative function on \( [0,\infty] \), under what conditions does \[ E\Phi(Vf)\leq cE\Phi(Uf),\qquad f\in\mathfrak{M},\] follow from some more easily proved inequality, perhaps \[ \|Vf\|_2 \leq c \|Uf\|_2,\qquad f\in\mathfrak{M} \,? \] Here \( E \) denotes expectation, integration over \( \Omega \) with respect to \( P \), and the letter \( c \) denotes a positive real number, not necessarily the same number from line to line.

[18] D. L. Burkholder and R. F. Gundy: “Extrapolation and interpolation of quasi-linear operators on martingales,” Matematika 15 : 3 (1971), pp. 91–141. Russian translation of article published in Acta Math. 124 (1970). Zbl 0223.60022 article

[19] D. L. Burkholder and R. F. Gundy: “Distribution function inequalities for the area integral,” pp. 527–544 in Collection of articles honoring the completion by Antoni Zygmund of 50 years of scientific activity, VI, published as Studia Math. 44 : 6. Państwowe Wydawnictwo Naukowe (Warsaw), 1972. MR 340557 Zbl 0219.31009 incollection

Antoni Zygmund	Related	Volume
Richard Floyd Gundy	Related

[20] D. L. Burkholder, B. J. Davis, and R. F. Gundy: “Integral inequalities for convex functions of operators on martingales,” pp. 223–240 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. Le Cam, J. Neyman, and E. L. Scott. University of California Press (Berkeley and Los Angeles, CA), 1972. MR 400380 Zbl 0253.60056 incollection

Richard Floyd Gundy	Related
Elizabeth Leonard Scott	Related
Burgess James Davis	Related
Jerzy Neyman	Related
Lucien Marie Le Cam	Related

[21] D. L. Burkholder and R. F. Gundy: “Boundary behaviour of harmonic functions in a half-space and Brownian motion,” Ann. Inst. Fourier (Grenoble) 23 : 4 (1973), pp. 195–212. MR 365691 Zbl 0253.31010 article

Let \( u \) be harmonic in the half-space \( \mathbb{R}^{n+1}_+ \), \( n\geq 2 \). We show that \( u \) can have a fine limit at almost every point of the unit cubes in \[ \mathbb{R}^n=\partial \mathbb{R}^{n+1}_+ \] but fail to have a nontangential limit at any point of the cube. The method is probabilistic and utilizes the equivalence between conditional Brownian motion limits and fine limits at the boundary. In \( \mathbb{R}^2_+ \) it is known that the Hardy classes \( \mathbf{H}^p \), \( 0 < p < \infty \), may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms of the integrability of a Brownian motion maximal function. This result is shown to hold in \( \mathbb{R}^{n+1}_+ \), for \( n\geq 2 \).

[22] D. L. Burkholder: “Distribution function inequalities for martingales,” Ann. Probab. 1 : 1 (1973), pp. 19–42. MR 365692 Zbl 0301.60035 article

[23] D. L. Burkholder: “\( H^p \) spaces and exit times of Brownian motion,” Bull. Am. Math. Soc. 81 : 3 (May 1975), pp. 556–558. MR 362519 Zbl 0321.60059 article

[24] D. L. Burkholder: “One-sided maximal functions and \( H^p \),” J. Funct. Anal. 18 : 4 (April 1975), pp. 429–454. MR 365693 Zbl 0294.31005 article

Let \( N \) be the nontangential maximal function of a function \( u \) harmonic in the Euclidean half-space \[ \mathbb{R}^n\times (0,\infty) \] and let \( N^- \) be the nontangential maximal function of its negative part. If \[ u(0,y) = o(y^{-n}) \] as \( y\to\infty \), then \[ \|N\|_p \leq c_p\|N^-\|_p ,\] \( 0 < p < 1 \), and more. The basic inequality of the paper can be used not only to derive such global results but also may be used to study the behavior of \( u \) near the boundary. Similar results hold for martingales with continuous sample functions. In addition, Theorem 1.3 contains information about the zeros of \( u \). For example, if \( u \) belongs to \( H^p \) for some \( 0 < p < 1 \), then every thick cone in the half-space must contain a zero of \( u \).

[25] D. L. Burkholder: “Harmonic analysis and probability,” pp. 136–149 in Studies in harmonic analysis (Chicago, 1974). Edited by J. M. Ash. MAA Studies in Mathematics 13. Mathematical Association of America (Washington, DC), 1976. MR 463788 Zbl 0339.42011 incollection

[26] D. L. Burkholder: “Brownian motion and classical analysis,” pp. 5–14 in Probability (Urbana, IL, March 1976). Edited by J. L. Doob. Proceedings of Symposia in Pure Mathematics 31. American Mathematical Society (Providence, RI), 1977. MR 474524 Zbl 0365.60075 incollection

[27] D. L. Burkholder: “Exit times of Brownian motion, harmonic majorization, and Hardy spaces,” Adv. Math. 26 : 2 (November 1977), pp. 182–205. MR 474525 Zbl 0372.60112 article

Let \( R \) be an open, connected subset of \( \mathbb{R}^n \) (\( n \geq 2 \)), \( X \) a Brownian motion in \( \mathbb{R}^n \) starting at a point \( x \) in \( R \), and \( \tau \) the first time \( X \) leaves \( R \): \[ \tau(\omega) = \inf\{t > 0: X_t(\omega)\notin R\}. \] Several basic methods are available for the study of such exit times. For example, if the region \( R \) has a Green’s function \( G(x,y) \) and this is appropriately normalized, then the expectation of \( \tau \) as a function of the starting point \( x \) is given by \[ E_x\tau = \int_R G(x, y) dy. \] (See [Hunt 1956, p. 309].) Or from a slightly different point of view, \[ v(x) = E_x\tau \] is, under certain conditions, the unique solution of Poisson’s equation \[ \Delta v = -2 \] in \( R \) vanishing at the boundary. (See, for example, [Dynkin and Yushkevich 1969, p. 68].)

However, in many problems of interest, the region \( R \) is so large that \( E_x\tau \) is infinite. Even if finite, information about other moments \( E_x\tau^p \) may be needed. For such problems, the classical methods are not always satisfactory.

Here we present a new approach to the study of exit times that yields information about \( E_x\tau^p \) for all positive real \( p \).

[28] D. L. Burkholder and T. Shintani: “Approximation of \( L^1 \)-bounded martingales by martingales of bounded variation,” Proc. Am. Math. Soc. 72 : 1 (October 1978), pp. 166–169. MR 494472 Zbl 0387.60048 article

[29] D. L. Burkholder: “Boundary value estimation of the range of an analytic function,” Mich. Math. J. 25 : 2 (1978), pp. 197–211. MR 501568 Zbl 0372.30026 article

[30] D. L. Burkholder: “A sharp inequality for martingale transforms,” Ann. Probab. 7 : 5 (1979), pp. 858–863. MR 542135 Zbl 0416.60047 article

[31] D. L. Burkholder: “Martingale theory and harmonic analysis in Euclidean spaces,” pp. 283–301 in Harmonic analysis in Euclidean spaces (Williamstown, MA, 10–28 July 1978), Part 2. Edited by G. Weiss and S. Wainger. Proceedings of Symposia in Pure Mathematics 35. American Mathematical Society (Providence, RI), 1979. MR 545315 Zbl 0417.60055 incollection

Stephen Wainger	Related
Guido Leopold Weiss	Related

[32] D. L. Burkholder: “Weak inequalities for exit times and analytic functions,” pp. 27–34 in Probability theory (Warsaw, 6 February–11 June 1976). Edited by Z. Ciesielski. Banach Center Publications 5. PWN (Warsaw), 1979. MR 561465 Zbl 0416.60048 incollection

[33] D. L. Burkholder: “Brownian motion and the Hardy spaces \( H^p \),” pp. 97–118 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 623466 Zbl 0497.30028 incollection

David Alexander Brannan	Related
James Gourlay Clunie	Related

[34] D. L. Burkholder: “Some new inequalities in complex analysis (and what they do),” Real Anal. Exch. 5 : 1 (1980), pp. 124–127. Zbl 0423.30023 article

[35] D. L. Burkholder: “A geometrical characterization of Banach spaces in which martingale difference sequences are unconditional,” Ann. Probab. 9 : 6 (1981), pp. 997–1011. MR 632972 Zbl 0474.60036 article

We study Banach-space-valued martingale transforms and, in particular, characterize those Banach spaces for which the classical theorems of the real-valued case carry over. For example, if \( B \) is a Banach space and \( 1 < p < \infty \), then there exists a positive real number \( c_p \) such that \[ \|\epsilon_1 d_1 + \cdots \epsilon_n d_n\|_p \leq c_p\|d_1 + \cdots + d_n\|_p \] for all \( B \)-valued martingale difference sequences \( d = (d_1 \), \( d_2,\dots) \) and all numbers \( \epsilon_1 \), \( \epsilon_2,\dots \) in \( \{-1,1\} \) if and only if there is a symmetric biconvex function \( \zeta \) on \( B\times B \) satisfying \( \zeta(0,0) > 0 \) and \[ \zeta(x,y)\leq |x+y| \] if \( |x|\leq 1\leq |y| \).

[36] D. L. Burkholder: “Martingale transforms and the geometry of Banach spaces,” pp. 35–50 in Probability in Banach spaces, III (Medford, MA, 4–16 August 1980). Edited by A. Beck. Lecture Notes in Mathematics 860. Springer (Berlin), 1981. MR 647954 Zbl 0471.60012 incollection

This paper is a continuation of [1980], which contains a geometrical characterization of the class of Banach spaces having the unconditionality property for martingale differences. Important information about this class, which we denote by UMD, is contained in the work of Maurey, Pisier, Aldous, and others. For example, if \( B \in \) UMD, then \( B \) is superreflexive [Maurey 1975] but there do exist superreflexive spaces that are not in UMD [Pisier 1975]; if \( 1 < p < \infty \), and the Lebesgue–Bochner space has an unconditional basis, then \( B \in \) UMD [Aldous 1979]. One of the main objects of study in this paper and [1980] is the class MT of Banach spaces \( B \) for which \( B \)-valued martingale transforms are well-behaved. The geometrical condition introduced in [1980] also characterizes the class MT, information about which is of value in the study of \( B \)-valued stochastic integrals and \( B \)-valued singular integrals. Before recalling the probability background, we shall describe this geometrical condition.

[37] D. L. Burkholder: “A nonlinear partial differential equation and the unconditional constant of the Haar system in \( L^p \),” Bull. Am. Math. Soc. (N.S.) 7 : 3 (1982), pp. 591–595. Research announcement. MR 670133 Zbl 0504.46022 article

[38] D. L. Burkholder: “A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions,” pp. 270–286 in Conference on harmonic analysis in honor of Antoni Zygmund (Chicago, 23–28 March 1981), vol. 1. Edited by W. Beckner, A. P. Calderón, R. Fefferman, and P. W. Jones. Wadsworth Mathematics Series. Wadsworth (Belmont, CA), 1983. MR 730072 incollection

Alberto Pedro Calderón	Related	Volume
William Beckner	Related
Antoni Zygmund	Related	Volume
Robert Allen Fefferman	Related
Peter Wilcox Jones	Related

[39] D. L. Burkholder: “Boundary value problems and sharp inequalities for martingale transforms,” Ann. Probab. 12 : 3 (1984), pp. 647–702. MR 744226 Zbl 0556.60021 article

Let \( p^* \) be the maximum of \( p \) and \( q \) where \( 1 < p < \infty \) and \( \frac1p+\frac1q=1 \). If \( d=(d_1 \), \( d_2,\dots) \) is a martingale difference sequence in real \( L^p(0,1) \), \( \varepsilon=(\varepsilon_1 \), \( \varepsilon_2,\dots) \) is a sequence of numbers in \( \{-1,1\} \), and \( n \) is a positive integer, then \[ \Bigl\| \sum_{k=1}^n \varepsilon_k d_k \Bigr\|_p \leq (p^*-1) \Bigl\| \sum_{k=1}^n d_k \Bigr\|_p \] and the constant \( p^*-1 \) is best possible. Furthermore, strict inequality holds if and only if \( p\neq 2 \) and \[ \Bigl\| \sum_{k=1}^n d_k \Bigr\|_p > 0 .\] This improves an earlier inequality of the author by giving the best constant and conditions for equality. The inequality holds with the same constant if \( \epsilon \) is replaced by a real-valued predictable sequence uniformly bounded in absolute value by 1, thus yielding a similar inequality for stochastic integrals. The underlying method rests on finding an upper or a lower solution to a novel boundary value problem, a problem with no solution (the upper is not equal to the lower solution) except in the special case \( p=2 \). The inequality above, in combination with the work of Ando, Dor, Maurey, Odell, Olevskii, Pelczynski, and Rosenthal, implies that the unconditional constant of a monotone basis of \( L^p(0,1) \) is \( p^*-1 \). The paper also contains a number of other sharp inequalities for martingale transforms and stochastic integrals. Along with other applications, these provide answers to some questions that arise naturally in the study of the optimal control of martingales.

[40] D. L. Burkholder: “An elementary proof of an inequality of R. E. A. C. Paley,” Bull. London Math. Soc. 17 : 5 (September 1985), pp. 474–478. MR 806015 Zbl 0566.46014 article

[41] D. L. Burkholder: “An extension of a classical martingale inequality,” pp. 21–30 in Probability theory and harmonic analysis (New York). Edited by J.-A. Chao and W. Woyczyński. Pure and Applied Mathematics 98. Dekker, 1986. MR 830228 Zbl 0594.60020 incollection

Wojbor A. Woyczynski	Related
Jia-Arng Chao	Related

[42] D. L. Burkholder: “Martingales and Fourier analysis in Banach spaces,” pp. 61–108 in Probability and analysis (Varenna, Italy, 31 May–8 June 1985). Edited by G. Letta and M. Pratelli. Lecture Notes in Mathematics 1206. Springer (Berlin), 1986. MR 864712 Zbl 0605.60049 incollection

The power of martingale theory in the study of the Fourier analysis of scalarvalued functions is now widely appreciated. Our aim here is to describe some new martingale methods and their application to the Fourier analysis of functions having values in a Banach space.

One of the themes of this work is that the new methods developed for B-valued martingales can be used to obtain new information even in the real case, for example, the best constants in some basic inequalities. These sharp inequalities for real-valued martingales lead, in turn, to new information about some of the classical Banach spaces, for example, the unconditional constant of any monotone basis of \( L^p(O,1) \) where \( 1 < p < \infty \).

Giorgio Letta	Related
Maurizio Pratelli	Related

[43] D. L. Burkholder: “A sharp and strict \( L^p \)-inequality for stochastic integrals,” Ann. Probab. 15 : 1 (1987), pp. 268–273. MR 877602 Zbl 0617.60042 article

[44] D. L. Burkholder: “A proof of Pełczyński’s conjecture for the Haar system,” Studia Math. 91 : 1 (1988), pp. 79–83. MR 957287 Zbl 0652.42012 article

Let \( H \) be a real or complex Hilbert space with norm \( |\cdot| \). Let \( 1 < p < \infty \) and \( p^* = \max\{p \), \( p/(p-1)\} \). Suppose that \( f \) and \( g \) belong to the Lebesgue–Bochner space \( L_H^p[0,1) \) and \( (h_n)_{n\geq 0} \) is the sequence of Haar functions on \( [0,1) \). Let \[ f = \sum_{k=0}^{\infty}a_kh_k,\qquad g = \sum_{k=0}^{\infty}b_kh_k \] where \( a_k,b_k\in H \) and the two series converge in \( L_H^p[0,1) \). The main result of the paper is: If \( |b_k|\leq |a_k| \) for all \( k\geq 0 \), then \[ \|g\|_p \leq (p^* - 1)\|f\|_p \] and the constant \( p^*-1 \) is best possible. Strict inequality holds if \( p\neq 2 \) and \( \|f\|_p > 0 \).

This result yields Pełczyński’s conjecture: The classical inequality of Paley and Marcinkiewicz for the Haar system holds with the same constant if the multiplier sequence of signs \( \pm 1 \) is replaced by a sequence of unimodular complex numbers.

[45] D. L. Burkholder: “Sharp inequalities for martingales and stochastic integrals,” pp. 75–94 in Colloque Paul Lévy sur les processus stochastiques [Paul Lévy colloquium on stochastic processes] (Palaiseau, France, 22–26 June 1987). Edited by L. Schwartz. Astérisque 157–158. Sociét’e Mathématique de France (Paris), 1988. MR 976214 Zbl 0656.60055 incollection

Laurent Schwartz	Related
Paul Lévy	Related

[46] D. L. Burkholder: “Differential subordination of harmonic functions and martingales,” pp. 1–23 in Harmonic analysis and partial differential equations (El Escorial, Spain, 9–13 June 1987). Edited by J. García-Cuerva. Lecture Notes in Mathematics 1384. Springer (Berlin), 1989. MR 1013814 Zbl 0675.31003 incollection

[47] D. L. Burkholder: “On the number of escapes of a martingale and its geometrical significance,” pp. 159–178 in Almost everywhere convergence: Proceedings of the international conference on almost everywhere convergence in probability and ergodic theory (Columbus, OH, 11–14 June 1988). Edited by G. A. Edgar and L. Sucheston. Academic Press (Boston, MA), 1989. MR 1035244 Zbl 0685.60049 incollection

Gerald Arthur Edgar	Related
Louis Sucheston	Related

[48] D. L. Burkholder, É. Pardoux, and A. Sznitman: École d’été de probabilités de Saint-Flour XIX–1989 [12th Saint-Flour probability summer school–1989] (Saint-Flour, France, 16 August–2 September 1989). Edited by P. L. Hennequin. Lecture Notes in Mathematics 1464. Springer (Berlin), 1991. MR 1108182 Zbl 0722.00029 book

Paul-Louis Hennequin	Related
Alain-Sol Sznitman	Related
Étienne Pardoux	Related

[49] D. L. Burkholder: “Explorations in martingale theory and its applications,” pp. 1–66 in D. L. Burkholder, E. Pardoux, and A.-S. Sznitman: École d’été de probabilités de Saint-Flour XIX–1989 [12th Saint-Flour probability summer school–1989] (Saint-Flour, France, 16 August–2 September 1989). Edited by P. L. Hennequin. Lecture Notes in Mathematics 1464. Springer (Berlin), 1991. MR 1108183 Zbl 0771.60033 incollection

Paul-Louis Hennequin	Related
Alain-Sol Sznitman	Related
Étienne Pardoux	Related

[50] D. L. Burkholder: “Book review: G. Pisier, ‘The volume of convex bodies and Banach space geometry’,” Bull. Am. Math. Soc. (N.S.) 25 : 1 (1991), pp. 140–145. MR 1567936 article

[51] “59 are chosen for National Academy of Sciences,” NYT (10 May 1992), pp. 1001024. article

[52] D. L. Burkholder: “Strong differential subordination and stochastic integration,” Ann. Probab. 22 : 2 (1994), pp. 995–1025. MR 1288140 Zbl 0816.60046 article

[53] D. L. Burkholder: “Sharp norm comparison of martingale maximal functions and stochastic integrals,” pp. 343–358 in Proceedings of the Norbert Wiener Centenary Congress, 1994 (East Lansing, MI, 27 November–3 December 1994). Edited by V. Mandrekar and P. R. Masani. Proceedings of Symposia in Applied Mathematics 52. American Mathematical Society (Providence, RI), 1997. MR 1440921 Zbl 0899.60040 incollection

Pesi Rustom Masani	Related
Norbert Wiener	Related
Vidyadhar Shantaram Mandrekar	Related

[54] D. L. Burkholder: “Some extremal problems in martingale theory and harmonic analysis,” pp. 99–115 in Harmonic analysis and partial differential equations: Essays in honor of Alberto P. Calderón (Chicago, February 1996). Edited by M. Christ, C. E. Kenig, and C. Sadosky. Chicago Lectures in Mathematics. University of Chicago Press, 1999. Essays in honor of Calderón’s 75th birthday. MR 1743857 Zbl 0954.60034 incollection

Alberto Pedro Calderón	Related	Volume
Cora Sadosky	Related
Francis Michael Christ	Related
Carlos Eduardo Kenig	Related

[55] D. L. Burkholder: “Martingales and singular integrals in Banach spaces,” Chapter 6, pp. 233–269 in Handbook of the geometry of Banach spaces, vol. 1. Edited by W. B. Johnson and J. Lindenstrauss. Elsevier (Amsterdam), 2001. MR 1863694 Zbl 1029.46007 incollection

[56] D. L. Burkholder: “The best constant in the Davis inequality for the expectation of the martingale square function,” Trans. Am. Math. Soc. 354 : 1 (2002), pp. 91–105. MR 1859027 Zbl 0984.60041 article

A method is introduced for the simultaneous study of the square function and the maximal function of a martingale that can yield sharp norm inequalities between the two. One application is that the expectation of the square function of a martingale is not greater than \( \sqrt{3} \) times the expectation of the maximal function. This gives the best constant for one side of the Davis two-sided inequality. The martingale may take its values in any real or complex Hilbert space. The elementary discrete-time case leads quickly to the analogous results for local martingales \( M \) indexed by \( [0,\infty) \). Some earlier inequalities are also improved and, closely related, the Lévy martingale is embedded in a large family of submartingales.

[57] 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

Joseph L. Doob	Related	Volume
Philip Elliott Protter	Related

[58] 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

[59] 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

[60] 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

[61] D. L. Burkholder: “Comments on (5), (14), (22), (30), and (31),” pp. 553–558 in Selected papers of Alberto P. Calderón with commentary. Edited by A. Bellow, C. E. Kenig, and P. Malliavin. AMS Collected Works 21. American Mathematial Society (Providence, RI), 2008. MR 2435333 incollection

Paul Malliavin	Related
Alberto Pedro Calderón	Related	Volume
Alexandra Bellow	Related
Carlos Eduardo Kenig	Related

[62] A collection of articles in honor of Don Burkholder, published as Ill. J. Math. 54 : 4. Issue edited by R. Bauer, R. Song, and R. Sowers. University of Illinois, Department of Mathematics (Urbana, IL), 2010. Zbl 1328.01047 book

Renming Song	Related
Robert Otto Bauer	Related
Richard Bucher Sowers	Related

[63] L. Ahlberg: Illinois mathematician elected fellow of AAAS, 11 January 2011. Illinois News Bureau online article. misc

[64]D. L. Burkholder: Selected works of Donald L. Burkholder. Edited by B. Davis and R. Song. Selected Works in Probability and Statistics. Springer (Berlin), 2011. MR 2850314 Zbl 1271.01046 book

Renming Song	Related
Burgess James Davis	Related

[65] “Obituary: Donald Burkholder,” The News-Gazette (17 April 2013). article

[66] Memorial issue for Donald Burkholder (1927–2013), published as Ann. Probab. 45 : 1. Issue edited by R. Bañuelos and B. Davis. Institute of Mathematical Statistics (Beachwood, OH), 2017. book

Rodrigo Bañuelos	Related
Burgess James Davis	Related

[67] R. Bañuelos and B. Davis: “Introduction to memorial issue for Donald Burkholder (1927–2013),” pp. 1–3 in Memorial issue for Donald Burkholder (1927–2013), published as Ann. Probab. 45 : 1. Issue edited by R. Bañuelos and B. Davis. Institute of Mathematical Statistics (Beachwood, OH), 2017. MR 3601643 Zbl 1359.01025 incollection

Rodrigo Bañuelos	Related
Burgess James Davis	Related

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