Random walks are determined by their trace on the positive half-line

We prove that the law of a random walk $X_n$ is determined by the one-dimensional distributions of $\max(X_n, 0)$ for $n = 1, 2, \ldots$, as conjectured recently by Lo\"ic 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.


Introduction and main result
main result was previously stated without proof in a more general form in [OU90], and an erroneous proof was given in [Ula92].
A random walk X n is said to be non-degenerate if P(X n > 0) = 0. Similarly, a finite signed Borel measure µ on R is said to be non-degenerate if the restriction of µ to (0, ∞) is a non-zero measure.
Theorem 1.1. -If X n and Y n are non-degenerate random walks such that max(X n , 0) and max(Y n , 0) are equal in distribution for all n = 1, 2, . . . , then X n and Y n are equal in distribution for n = 1, 2, . . .
Following [CD20], we remark that various reformulations of the above result are possible. A non-degenerate random walk X n is determined by any of the following objects: • The law of the ascending ladder process (T k , S k ); here S k = X T k is the k th running maximum of the random walk. • The upward space-time Wiener-Hopf factor Φ + (q, ξ), that is, the characteristic function of (T 1 , S 1 ). • The distributions of the running maxima max(0, X 1 , X 2 , . . . , X n ) for all n = 1, 2, . . . Theorem 1.1 clearly implies that a non-degenerate Lévy process X t is determined by any of the following objects: • The distributions of max(X t , 0) for all t > 0 (or even for t = 1, 2, . . .).
• The law of the ascending ladder process (T t , S t ).
• The upward space-time Wiener-Hopf factor κ + (q, ξ), that is, the characteristic exponent of (T t , S t ). • The distributions of the running suprema sup{X s : s ∈ [0, t]} for all t > 0. For further discussion, we again refer to [CD20], where Theorem 1.1 was proved under various relatively mild additional conditions. For related research, see [CD20,LMS76,Ost85,OU90,Ula90,Ula92] and the references therein.
Theorem 1.1 was given without proof in [OU90] in a more general form: Theorem 4 therein claims that µ = ν if µ and ν are non-degenerate finite Borel measures on R and the restrictions of µ * n k and ν * n k to (0, ∞) are equal for k = 1, 2, . . . , where n 1 = 1 and n 2 − 1, n 3 − 1, . . . are distinct and have no common divisor other than 1. Noteworthy, this result is stated for measures on the Euclidean space of arbitrary dimension, and their restrictions to the half-space. A proof is given in [Ula92] under the additional condition n 2 = 2, and only in dimension one. However, the argument in [Ula92] contains a gap, that we describe at the end of this article.

Proof
All measures considered below are finite, signed Borel measures. For a measure µ on R, we denote the restrictions of µ to (0, ∞) and (−∞, 0] by µ + = 1 (0,∞) µ and µ − = 1 (−∞,0] µ. This should not be confused with the Hahn decomposition of µ into the positive and negative part. By µ * n we denote the n-fold convolution of µ, and we define µ * 0 to be the Dirac measure δ 0 . For brevity, we write µ * n ± = (µ ± ) * n , as opposed to (µ * n ) ± . We record the following elementary identities: We denote the characteristic function of a measure µ by µ: for z ∈ R, and also for those z ∈ C for which the integral converges. We recall that µ + is a bounded holomorphic function in the upper complex half-plane C + = {z ∈ C : Im z > 0}, continuous on the boundary. Similarly, µ − is a bounded holomorphic function on the lower complex half-plane C − = {z ∈ C : Im z < 0}.
A holomorphic function f on C − is said to be of bounded type (or belong to the Nevanlinna class) if log |f (x)| has a harmonic majorant on C − . Equivalently, f is of bounded type if it is a ratio of two bounded holomorphic functions on C − . We recall the following fundamental factorisation theorem for holomorphic functions on C − which are bounded or of bounded type, and we refer to [Gar07,Mas09] for further details.
The function f o is an outer function, a holomorphic function determined uniquely up to multiplication by a constant of modulus 1 by the formula: Finally, the function f s is a singular inner function, a holomorphic function determined uniquely up to multiplication by a constant of modulus 1 by the expression:

ANNALES HENRI LEBESGUE
where a ∈ R is a constant and σ is a signed measure, singular with respect to the Lebesgue measure. Furthermore, for almost all x ∈ R with respect to both the Lebesgue measure and the measure σ, the limit f (x) of f (x + iy) as y → 0 − exists. This boundary limit f (x) is non-zero almost everywhere with respect to the Lebesgue measure and zero almost everywhere with respect to σ. The symbol f (x) used in the definition of the outer function f o refers precisely to this boundary limit. Additionally, we have and any parameters α j , z j , a, σ and boundary values |f (x)|, x ∈ R, which satisfy these conditions, correspond to some function f of bounded type. Finally, f is a bounded holomorphic function in the lower complex half-plane if and only if a 0, σ is a non-negative measure and the boundary values |f (x)| are bounded for x ∈ R. We note basic properties of A and B. By continuity of g, A and A∪B are closed sets, and D is an open set. Since g is holomorphic on C − (and not identically zero), B is a countable (possibly finite) set with no accumulation points on C − . By Theorem 2.2, A has zero Lebesgue measure (as a subset of R). In particular, D is connected. Indeed: the sets D ∩ C + = C + and D ∩ C − = C − \ B are clearly path-connected, the set D ∩ R = R \ A is non-empty, and since D is open, each point of D ∩ R is path-connected with points from both D ∩ C + and D ∩ C − .
We define a function ϕ on D by the formula By definition, ϕ is holomorphic both on C + and on C − \ B, as well as meromorphic on C − . Furthermore, ϕ is continuous at each point z ∈ R \ A, because both f (defined on C + ∪ R) and h/g (defined on (C − \ B) ∪ (R \ A)) are continuous at z and f (z) = h(z)/g(z). By a standard application of Morera's theorem (see [Con73,Theorem IV.5.10 and Exercise IV.5.9], or [Gar07, Exercise II.12]), ϕ is holomorphic in D. It remains to note that ϕ(z)g(z) = h(z) for z ∈ C − \ B.
Proof. -Let µ be such a measure, and suppose that both µ + and µ − are non-zero measures . Let ϕ, f, g, h, A, B, D be as in the proof of Lemma 2.3. Clearly, ϕ n is the holomorphic extension of f n , the characteristic function of µ * n + . An application of Lemma 2.3 to the measure µ * n + + µ − implies that for all n = 1, 2, . . . , the function ϕ n g extends from C − \ B to a function h n which is bounded and holomorphic on C − and continuous on C − ∪ R, namely, h n is the characteristic function of µ * n + * µ − . Consider the factorisations g = g b g o g s and h n = h n,b h n,o h n,s given in Theorem 2.2, and let σ g , a g and σ h,n , a h,n denote the corresponding non-negative measures σ and constants a for g and h n , respectively. Note that Theorem 2.2 applies both to g and to h n = ϕ n g, as these functions are not identically zero: f and g are characteristic functions of non-zero measures µ + and µ − , while h n is the product of g and the holomorphic extension of f n .
Recall that By definition, ϕ n,o and ϕ n,s have no zeros in C − . This means that if z 0 ∈ C − is a pole of ϕ of order α 0 , then z 0 is a pole of ϕ n,b = h n,b /g b of order nα 0 , and therefore g b has a zero at z 0 of multiplicity at least nα 0 for all n = 1, 2, . . . Since all zeroes of g b have finite multiplicity, ϕ has no poles in C − . In particular, ϕ extends to a holomorphic function on C \ A, which will be denoted again by ϕ, and ϕ n,b = h n,b /g b has no poles in C − . Therefore, the zeros of h n,b must cancel the zeros of g b , and ϕ n,b is a Blaschke product.
Since h n (x)/g(x) = (f (x)) n for x ∈ R \ A and A has Lebesgue measure zero, we have In particular, ϕ n,o is a bounded outer function, namely, the outer function in the factorisation of the bounded holomorphic function (f (z)) n on the lower complex half-plane. Finally ϕ n,s is the ratio of two singular inner functions, and hence a singular inner function. If we denote a ϕ,n = a h,n − a g and σ ϕ,n = σ h,n − σ g , then |ϕ n,s (z)| = exp −a ϕ,n Im z − 1 π R − Im z |z − x| 2 σ ϕ,n (dx) .
The above properties imply that ϕ n is of bounded type, and therefore the factors ϕ n,b , ϕ n,o , ϕ n,s , the signed measure σ ϕ,n and the constant a ϕ,n ∈ R are uniquely determined (up to multiplication by a constant of modulus 1 in case of ϕ n,o and ϕ n,s ).
Since ϕ = f on C + and f is a bounded holomorphic function on C + , we have proved that ϕ is a bounded holomorphic function on C \ A. However, A has zero Lebesgue measure (as a subset of R). By Painlevé's theorem (see [You15, Theorem 2.7]), ϕ extends to a bounded holomorphic function on C. This, in turn, implies that ϕ is constant, and so µ + is constant, contradicting the assumption that µ + is a non-zero measure on (0, ∞).
The author of the present article was not able to correct the error in [Ula92]. The proof given above uses a related, but essentially different idea.