An alternative formulation of the discrete-time fractional Poisson process

Referee: [Introduction and §2] Introduction and §2: The choice of the discrete Mittag-Leffler distribution (defined via the generalized binomial coefficient or fractional difference operator) as the waiting-time law is presented without explicit justification as the canonical discrete analogue to the continuous Mittag-Leffler law. The paper should compare this choice to other possible discretizations that also converge to the continuous fractional Poisson process (e.g., via binomial thinning or alternative fractional operators) and explain why alternatives would not be equally natural; without this, the non-equivalence result, while valid for the chosen law, has reduced interpretive force for the broader claim of an 'alternative formulation'.

Authors: We appreciate the referee's observation that additional justification for selecting the discrete Mittag-Leffler distribution would strengthen the paper. In the revised introduction and Section 2 we will insert a short comparative discussion. We will note that this law is obtained by applying the fractional difference operator directly to the geometric distribution, thereby preserving the same formal relationship to the continuous Mittag-Leffler law that the fractional derivative preserves for the exponential distribution. While other discretizations (for example, binomial thinning of a Poisson process or alternative fractional operators) can be defined and may converge to the continuous fractional Poisson process under suitable scaling, they do not arise from the same fractional-difference construction and therefore do not yield an exact renewal process with the same closed-form waiting-time PGF. We will briefly contrast the binomial-thinning approach to illustrate this distinction, thereby clarifying why the chosen law provides a natural discrete analogue and why the demonstrated non-equivalence is specific to this construction. revision: partial

Referee: [§4] §4 (equivalence discussion): The non-equivalence is asserted by comparing the derived count distribution to the Sibuya-subordinated process, but the manuscript does not provide an explicit side-by-side expression for the two count PMFs or a numerical check for small n and fractional orders to illustrate the difference; this makes it difficult to assess whether the divergence is substantive or merely formal.

Authors: We agree that an explicit side-by-side presentation would make the non-equivalence easier to verify. In the revised Section 4 we will display the probability mass function of the renewal-based count process next to the corresponding PMF of the Sibuya-subordinated process. In addition, we will add a small numerical table that reports the probabilities for N(n) = k under both constructions for n = 5 and n = 10 and for fractional orders α = 0.5 and α = 0.75. These concrete values will demonstrate that the two distributions differ already at small n, confirming that the divergence is substantive rather than merely formal. revision: yes