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

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## 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

### Proof

*n*

*n*

*n*

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.

*n*and a small positive \(\epsilon \). In the proof of Theorem 4.13 in [1], we should take into account that for large

*n*

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