Correction to: Asymptotic moments of spatial branching processes (Probability Theory and Related Fields, (2022), 184, 3-4, (805-858), 10.1007/s00440-022-01131-2)

In this short note, we remark that there were erroneous limits used in the non-critical cases for the moment convergence and occupation moments in [2]. The source of the error was a misuse of the sense in which uniformity held in various convergence arguments. This affects the nature of the constants that appear in the limits. The results for the critical cases, which were the principal results, are correct with one minor adjustment in the statement which is that (Formula presented.) should read (Formula presented.) , for any (Formula presented.). In what follows we assume the notation and hypotheses of [2] and provide the correct statements and brief corrections of the proofs for the non-critical setting. The theorem numbers correspond to the same theorem numbers in [2]. (Supercritical, (Formula presented.)) Suppose that (H1) holds along with (H2) for some (Formula presented.) and (Formula presented.). Redefine (Formula presented.) where (Formula presented.) and we define iteratively for (Formula presented.) (Formula presented.) with (Formula presented.) is the set of all non-negative N-tuples (Formula presented.) such that (Formula presented.) and at least two of the (Formula presented.) are strictly positive1 if (Formula presented.) is a branching Markov process. Alternatively, if (Formula presented.) is a superprocess, define iteratively for (Formula presented.) with (Formula presented.) and (Formula presented.) (Formula presented.) where (Formula presented.) and (Formula presented.) Here the sums run over the set (Formula presented.) of positive integers such that (Formula presented.). Then, for all (Formula presented.) (Formula presented.) (Subcritical, (Formula presented.)) Suppose that (H1) holds along with (H2) for some (Formula presented.) and (Formula presented.). Redefine (Formula presented.) where we define iteratively (Formula presented.) and for (Formula presented.) , (Formula presented.) if (Formula presented.) is a branching Markov process. Alternatively, if (Formula presented.) is a superprocess, we still have (Formula presented.) but (Formula presented.) for (Formula presented.) , with (Formula presented.) Here the sums run over the set (Formula presented.) of non-negative integers such that (Formula presented.). Then, for all (Formula presented.) (Formula presented.) (Supercritical, (Formula presented.)) Let (Formula presented.) be either a branching Markov process or a superprocess. Suppose that (H1) holds along with (H2) for some (Formula presented.) and (Formula presented.). Redefine (Formula presented.) where (Formula presented.) was defined in Theorem 2 (both for branching Markov processes and superprocesses), albeit that (Formula presented.). Then, for all (Formula presented.) (Formula presented.) (Subcritical, (Formula presented.)) Suppose that (H1) holds along with (H2) for some (Formula presented.) and (Formula presented.). Redefine (Formula presented.) where (Formula presented.) and for (Formula presented.) , the (Formula presented.) are defined recursively via (Formula presented.) if X is a branching Markov process. Alternatively, if X is a superprocess, (Formula presented.) where (Formula presented.) and (Formula presented.) Then, for all (Formula presented.) (Formula presented.) We give a brief proof of Theorems 2 and 3 to give a sense of the corrected reasoning. The proofs of Theorems 5 and 6 are left out given that the corrected reasoning uses the same logic. The reader may also consult [1] for more details. Suppose for induction that the result is true for all (Formula presented.) -th integer moments with (Formula presented.). From the evolution equation in Proposition 1 of [2], noting that (Formula presented.) , when the limit exists, we have (Formula presented.) where (Formula presented.) It is easy to see that, pointwise in (Formula presented.) and (Formula presented.) , using the induction hypothesis and (H2), (Formula presented.) where we have again used the fact that the (Formula presented.) s sum to k to extract the (Formula presented.) term. Using the expressions for H[f](x, u, t) and H[f](x) together with the definition of (Formula presented.) , we have, for any (Formula presented.) , as (Formula presented.) , (Formula presented.) where (Formula presented.) is an upper estimate for (Formula presented.) Note, convergence to zero as (Formula presented.) in (8) follows thanks to the induction hypothesis (ensuring that (Formula presented.) is uniformly bounded), (H2) and the uniform boundedness of (Formula presented.). The induction hypothesis, (H2) and dominated convergence again ensure that (Formula presented.) As such, in (7), we can split the integral on the right-hand side over (Formula presented.) and (Formula presented.) , for (Formula presented.). Using (9), we can ensure that, for any arbitrarily small (Formula presented.) , making use of the boundedness in (H1), there is a global constant (Formula presented.) such that, for all t sufficiently large, (Formula presented.) On the other hand, we can also control the integral over (Formula presented.) , again appealing to (H1) and (H2) to ensure that (Formula presented.) We can again work with a (different) global constant (Formula presented.) such that (Formula presented.) In conclusion, using (10) and (11), we can take limits as (Formula presented.) in (7) and the statement of the theorem follows for branching Markov processes. The proof in the superprocess setting starts the same way as in [2] up to equation (90) therein, noting that the term (Formula presented.) in the moment evolution equation (Formula presented.) from equation (77) of [2] can be compensated in the limit using (Formula presented.) defined in (1) above. The remainder of the proof deals with the compensation of the integral term in (12). We have (Formula presented.) where (Formula presented.) as (Formula presented.) that is, (Formula presented.) The induction hypothesis and (H1) allow us to get (Formula presented.) Using the same arguments used above from (10) onwards, we get the desiblack result. (Formula presented.) First note that since we only compensate by (Formula presented.) , the term (Formula presented.) that appears in equation (41) of [2] does not vanish after the normalisation. Due to assumption (H1), we have (Formula presented.) Next we turn to the integral term in (41) of [2]. Define (Formula presented.) , for (Formula presented.) to be the set of tuples (Formula presented.) with exactly n positive terms and whose sum is equal to k. Similar calculations to those given above yield (Formula presented.) Now suppose for induction that the result holds for all (Formula presented.) -th integer moments with (Formula presented.). Roughly speaking the argument can be completed by noting that the integral in the definition of (Formula presented.) can be written as (Formula presented.) which is convergent by appealing to (H2), the fact that (Formula presented.) and the induction hypothesis. As a convergent integral, it can be truncated at (Formula presented.) and the residual of the integral over (Formula presented.) can be made arbitrarily small by taking t sufficiently large. By changing variables in (14) when the integral is truncated at arbitrarily large t, so it is of a similar form to that of (13), we can subtract it from (13) to get (Formula presented.) where (Formula presented.) One proceeds to split the integral of the difference over [0, 1] into two integrals, one over (Formula presented.) and one over (Formula presented.). For the aforesaid integral over (Formula presented.) , we can control the behaviour of (Formula presented.) as (Formula presented.) , making it arbitrarily small, by appealing to uniform dominated control of its argument in square brackets thanks to (H1). The integral over (Formula presented.) can thus be bounded, as (Formula presented.) , by (Formula presented.). For the integral over (Formula presented.) , we can appeal to the uniformity in (H1) and (H2) to control the entire term (Formula presented.) (over time and its argument in the square brackets) by a global constant. Up to a multiplicative constant, the magnitude of the integral is thus of order (Formula presented.) which tends to zero as (Formula presented.). In the superprocess setting, as in the original proof, the exponential scaling kills the term (Formula presented.) in (12). For the integral term in (12), define (Formula presented.) by (Formula presented.) and noting that (Formula presented.) can be written as (Formula presented.) which is also convergent by appealing to (H2). The rest of the proof follows similar arguments to that of the particle system. That is, one splits the integral (15) at t and uses the integral over [0, t] to compensate the integral component of (12), changing variable so that it becomes an integral over [0, 1] and handling things as with the particle system. The remaining integral from (Formula presented.) can be argued away as arbitrarily small because of the convergence of (15). (Formula presented.) Fundamentally, the corrected results offer the same rates of convergence and simply the constants take a different iterative structure. The corrections also remove the discrepancy between when there is dependency on x or not in the constants in the case of branching Markov processes and superprocesses. Hence, the original attempt at explaining the discrepancy in the dependency on the limiting constants is no longer necessary nor valid.