Random walks are determined by their trace on the positive half-line
Annales Henri Lebesgue, Volume 3 (2020) , pp. 1389-1397.


KeywordsRandom walk, Lévy process, Wiener–Hopf factorisation, Nevanlinna class


We prove that the law of a random walk X n is determined by the one-dimensional distributions of max(X n ,0) for n=1,2,..., as conjectured recently by Loïc Chaumont and Ron Doney. Equivalently, the law of X n is determined by its upward space-time Wiener–Hopf factor. Our methods are complex-analytic.


[CD20] Chaumont, Loïc; Doney, Ron On distributions determined by their upward, space-time Wiener–Hopf factor, J. Theor. Probab., Volume 33 (2020) no. 2, pp. 1011-1033 | Article | MR 4091570 | Zbl 07202139

[Con73] Conway, John B. Functions of one complex variable, Graduate Texts in Mathematics, Volume 11, Springer, 1973 | MR 447532 | Zbl 0277.30001

[Gar07] Garnett, John Brady Bounded Analytic Functions, Graduate Texts in Mathematics, Volume 236, Springer, 2007 | MR 2261424

[LMS76] Letac, Gérard G.; Mazet, Pierre; Schiffman, Gérard A note on left-continuous random walks, J. Appl. Probab., Volume 13 (1976) no. 4, pp. 814-817 | Article | MR 428462 | Zbl 0356.60007

[LO77] Linnik, Yuriĭ V.; Ostrovskiĭ, Iosif V. Decomposition of random variables and vectors, Translations of Mathematical Monographs, Volume 48, American Mathematical Society, 1977 | MR 428382

[Mas09] Mashreghi, Javad Representation theorems in Hardy spaces, London Mathematical Society Student Texts, Cambridge University Press, 2009 | Zbl 1169.22001

[Ost85] Ostrovskiĭ, Iosif V. Generalization of the Titchmarsh convolution theorem and the complex-valued measures uniquely determined by their restrictions to a half-line, Stability Problems for Stochastic Models (Uzhgorod, 1984) (Lecture Notes in Mathematics) Volume 1155, Springer, 1985, pp. 256-283 | Article | MR 825330 | Zbl 0574.30032

[OU90] Ostrovskiĭ, Iosif V.; Ulanovskiĭ, Alexander M. Probability distributions and Borel measures uniquely determined by their restrictions to a half-space, Probability theory and mathematical statistics. Vol. II (Vilnius, 1989) (Grigelionos, B.; Prohorov, Yu V.; Sazonov, V. V.; Statulevičius, V., eds.), Mokslas, 1990, pp. 278-287

[Ula90] Ulanovskiĭ, Alexander M. Unique determination of the convolutions of measures in R m , m2, by their restriction to a set, J. Math. Sci., Volume 49 (1990) no. 6, pp. 1298-1301 | Article

[Ula92] Ulanovskiĭ, Alexander M. On the determination of a measure by the restriction of its n-fold convolutions to a massive set, Teor. Funkts. Funkts. Anal. Prilozh., Volume 57 (1992), pp. 102-109 (English translation in: J. Math. Sci. 77(1) (1995): 2997–3002)

[Vig01] Vigon, Vincent Simplifiez vos Lévy en titillant la factorisation de Wiener–Hopf (2001) (Ph. D. Thesis)

[You15] Younsi, M. On removable sets for holomorphic functions, EMS Surv. Math. Sci., Volume 2 (2015) no. 2, pp. 219-254 | Article | MR 3429163 | Zbl 1331.30002