Journal of Theoretical Probability

, Volume 32, Issue 1, pp 541–543

# Correction to: Conservative and Semiconservative Random Walks: Recurrence and Transience

• Vyacheslav M. Abramov
Correction

## 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 $$n>n_0$$
\begin{aligned} \frac{\lambda _n}{\mu _n}\ge 1+\frac{1}{n}+\frac{c}{n\ln n}, \end{aligned}
(1)
and is recurrent if there exists a value $$n_0$$ such that for all $$n>n_0$$
\begin{aligned} \frac{\lambda _n}{\mu _n}\le 1+\frac{1}{n}+\frac{1}{n\ln n}. \end{aligned}
(2)

### Proof

Following [2], a birth-and-death process is recurrent if and only if
\begin{aligned} \sum _{n=1}^\infty \prod _{k=1}^{n}\frac{\mu _k}{\lambda _k}=\infty . \end{aligned}
Write
\begin{aligned} \sum _{n=1}^\infty \prod _{k=1}^{n}\frac{\mu _k}{\lambda _k}=\sum _{n=1}^\infty \exp \left( \sum _{k=1}^{n}\ln \left( \frac{\mu _k}{\lambda _k}\right) \right) . \end{aligned}
(3)
Now, suppose that (1) holds. Then, for sufficiently large n
\begin{aligned} \frac{\mu _n}{\lambda _n}\le 1-\frac{1}{n}-\frac{c}{n\ln n}+O\left( \frac{1}{n^2}\right) , \end{aligned}
and since the function $$x\mapsto \ln x$$ is increasing on $$(0,\infty )$$, then
\begin{aligned} \begin{aligned} \ln \left( \frac{\mu _n}{\lambda _n}\right)&\le \ln \left( 1-\frac{1}{n}-\frac{c}{n\ln n}+O\left( \frac{1}{n^2}\right) \right) \\&=-\frac{1}{n}-\frac{c}{n\ln n}+O\left( \frac{1}{n^2}\right) . \end{aligned} \end{aligned}
Hence, for sufficiently large n
\begin{aligned} \sum _{k=1}^n\ln \left( \frac{\mu _k}{\lambda _k}\right) \le -\ln n-c\ln \ln n+O(1), \end{aligned}
and thus, by (3), for some constant $$C<\infty$$,
\begin{aligned} \sum _{n=1}^{\infty }\prod _{k=1}^{n}\frac{\mu _k}{\lambda _k}\le C\sum _{n=1}^{\infty }\frac{1}{n(\ln n)^c}<\infty , \end{aligned}
provided that $$c>1$$. The transience follows.
On the other hand, suppose that (2) holds. Then, for sufficiently large n
\begin{aligned} \frac{\mu _n}{\lambda _n}\ge 1-\frac{1}{n}-\frac{1}{n\ln n}+O\left( \frac{1}{n^2}\right) , \end{aligned}
and, consequently,
\begin{aligned} \ln \left( \frac{\mu _n}{\lambda _n}\right) \ge \ln \left( 1-\frac{1}{n}-\frac{1}{n\ln n}+O\left( \frac{1}{n^2}\right) \right) . \end{aligned}
Similarly to that was provided before, for some constant $$C^\prime$$,
\begin{aligned} \sum _{n=1}^{\infty }\prod _{k=1}^{n}\frac{\mu _k}{\lambda _k}\ge C^\prime \sum _{n=1}^{\infty }\frac{1}{n\ln n}=\infty . \end{aligned}
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:
\begin{aligned} \frac{L_n}{M_n}\le 1+\frac{2-d}{n}+\frac{1-\epsilon }{n\ln n}, \end{aligned}
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
\begin{aligned} \frac{\lambda _n(1,d)}{\mu _n(1,d)}=1+\frac{d-1}{n}+O\left( \frac{1}{n^2}\right) \end{aligned}
is satisfied (see the proof of Lemma 4.2), and hence,
\begin{aligned} \frac{p_n}{1-p_n}\asymp \left[ \frac{\lambda _n(1,d)}{\mu _n(1,d)}\cdot \frac{L_n}{M_n}\right] \le 1+\frac{1}{n}+\frac{1-\epsilon }{n\ln n}+\frac{C}{n^2}, \end{aligned}
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.

## Notes

### Acknowledgements

The help of the reviewer is highly appreciated.

## References

1. 1.
Abramov, V.M.: Conservative and semiconservative random walks: recurrence and transience. J. Theor. Probab. 31(3), 1900–1922 (2018)
2. 2.
Karlin, S., McGregor, J.: The classification of the birth-and-death processes. Trans. Am. Math. Soc. 86(2), 366–400 (1957)
3. 3.
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)Google Scholar