1 Correction to: J Theor Probab (2018) 31:1900–1922 https://doi.org/10.1007/s10959-017-0747-3
The aim of this note is to correct the errors in the formulation and proof of Lemma 4.1 in [1] and some claims that are based on that lemma. The correct formulation of the aforementioned lemma should be as follows.
Lemma 4.1
Let the birth-and-death rates of a birth-and-death process be \(\lambda _n\) and \(\mu _n\) all belonging to \((0,\infty )\). Then, the birth-and-death process is transient if there exist \(c>1\) and a value \(n_0\) such that for all \(n>n_0\)
and is recurrent if there exists a value \(n_0\) such that for all \(n>n_0\)
Proof
Following [2], a birth-and-death process is recurrent if and only if
Write
Now, suppose that (1) holds. Then, for sufficiently large n
and since the function \(x\mapsto \ln x\) is increasing on \((0,\infty )\), then
Hence, for sufficiently large n
and thus, by (3), for some constant \(C<\infty \),
provided that \(c>1\). The transience follows.
On the other hand, suppose that (2) holds. Then, for sufficiently large n
and, consequently,
Similarly to that was provided before, for some constant \(C^\prime \),
The recurrence follows. \(\square \)
As \(n\rightarrow \infty \), asymptotic expansion (4.5) obtained in the proof of Lemma 4.2 in [1] guarantees its correctness. However, the corrected version of Lemma 4.1 requires more delicate arguments in the proofs of Lemma 4.2 and Theorem 4.13 in [1]. Specifically, in the proof of Lemma 4.2 instead of limit relation (4.6) we should study the cases \(d=2\) and \(d\ge 3\) separately in terms of the present formulation of Lemma 4.1.
In the formulation of Theorem 4.13 in [1], assumption (4.12) must be replaced by the stronger one:
for all large n and a small positive \(\epsilon \). In the proof of Theorem 4.13 in [1], we should take into account that for large n
is satisfied (see the proof of Lemma 4.2), and hence,
for a fixed constant C and large n. So, according to Lemma 4.1 the process is recurrent.
Note that the statements of Lemma 4.1 are closely related to those of Theorem 3 in [3] that prove recurrence and transience for the model studied there.
References
Abramov, V.M.: Conservative and semiconservative random walks: recurrence and transience. J. Theor. Probab. 31(3), 1900–1922 (2018)
Karlin, S., McGregor, J.: The classification of the birth-and-death processes. Trans. Am. Math. Soc. 86(2), 366–400 (1957)
Menshikov, M.V., Asymont, I.M., Iasnogorodskii, R.: Markov processes with asymptotically zero drifts. Probl. Inf. Transm. 31, 248–261 (1995), translated from Problemy Peredachi Informatsii 31, 60–75 (in Russian)
Acknowledgements
The help of the reviewer is highly appreciated.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abramov, V.M. Correction to: Conservative and Semiconservative Random Walks: Recurrence and Transience. J Theor Probab 32, 541–543 (2019). https://doi.org/10.1007/s10959-018-0871-8
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10959-018-0871-8