Random walks are determined by their trace on the positive half-line. (Les marches aléatoires sont déterminées par leur trace sur la demi-droite positive.) (English. French summary) Zbl 1454.60059
Summary: We prove that the law of a random walk \(X_n\) is determined by the one-dimensional distributions of \(\max (X_n, 0)\) for \(n = 1, 2, \dots\), as conjectured recently by L. Chaumont and R. Doney [J. Theor. Probab. 33, No. 2, 1011–1033 (2020; Zbl 1455.60031)]. Equivalently, the law of \(X_n\) is determined by its upward space-time Wiener-Hopf factor. Our methods are complex-analytic.
MSC:
| 60G50 | Sums of independent random variables; random walks |
| 60G51 | Processes with independent increments; Lévy processes |
| 45E10 | Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) |
| 30H15 | Nevanlinna spaces and Smirnov spaces |
Citations:
Zbl 1455.60031References:
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