## Abstract

This paper deals with different models of random walks with a reinforced memory of preferential attachment type. We consider extensions of the Elephant Random Walk introduced by Schütz and Trimper (Phys Rev E 70:044510(R), 2004) with stronger reinforcement mechanisms, where, roughly speaking, a step from the past is remembered proportional to some weight and then repeated with probability *p*. With probability \(1-p\), the random walk performs a step independent of the past. The weight of the remembered step is increased by an additive factor \(b\ge 0\), making it likelier to repeat the step again in the future. A combination of techniques from the theory of urns, branching processes and \(\alpha \)-stable processes enables us to discuss the limit behavior of reinforced versions of both the Elephant Random Walk and its \(\alpha \)-stable counterpart, the so-called Shark Random Swim introduced by Businger (J Stat Phys 172(3):701–717, 2004). We establish phase transitions, separating subcritical from supercritical regimes.

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## References

Alves, G. A., de Araújo, Cressoni, J. C., da Silva, L. R., da Silva, M. A. A., Viswanathan, G.M.: Superdiffusion driven by exponentially decaying memory. J. Stat. Mech. 2014 (2014)

Athreya, K.B., Ney, P.E.: Branching Processes. Dover Books on Mathematics (2004)

Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science

**286**, 509–512 (1999)Baur, E., Bertoin, J.: Elephant random walks and their connection to Pólya-type urns. Phys. Rev. E

**49**, 052134 (2016)Bercu, B.: A martingale approach for the elephant random walk. J. Phys. A

**51**, 015201 (2017)Bercu, B., Laulin, L.: On the multi-dimensional elephant random walk. J. Stat. Phys.

**175**(6), 1146–1163 (2019)Bertoin, J.: Noise reinforcement for Lévy processes. Preprint, arXiv:1810.08364 : To appear in Ann. Inst, Henri Poincaré B (2018)

Bertoin, J.: A version of Herbert A. Simon’s model with slowly fading memory and its connections to branching processes. J. Stat. Phys.

**176**, 679 (2019)Bertoin, J.: Universality of Noise Reinforced Brownian Motions. Preprint (2019)

Bertoin, J., Goldschmidt, C.: Dual random fragmentation and coagulation and an application to the genealogy of yule processes. Mathematics and Computer Science

**III**(2012)Bertoin, J., Uribe Bravo, G.: Supercritical percolation on large scale-free random trees. Ann. Appl. Probab.

**25–1**, 81–103 (2015)Businger, S.: The shark random swim (Lévy flight with memory). J. Stat. Phys.

**172**(3), 701–717 (2018)Coletti, C.F., Gava, R., Schütz, G.M.: Central limit theorem for the elephant random walk. J. Math. Phys.

**58**(5), 053003 (2017)Coletti, C.F., Gava, R., Schütz, G.M.: A strong invariance principle for the elephant random walk. J. Stat. Mech. Theory Exp.

**12**, 123207 (2017)Cotar, C., Thacker, D.: Edge- and vertex-reinforced random walks with super-linear reinforcement on infinite graphs. Ann. Probab.

**45**(4), 2655–2706 (2017)Diaconis, P., Rolles, S.W.W.: Bayesian analysis for reversible Markov chains. Ann. Stat.

**34**(3), 1270–1292 (2006)Gut, A., Stadtmüller, U.: Variations of the elephant random walk. Preprint, arXiv:1812.01915 (2018)

Janson, S.: Functional limit theorems for multitype branching processes and generalized Pólya urns. Stoch. Proc. Appl.

**110**(2), 177–245 (2004)Kürsten, R.: Random recursive trees and the elephant random walk. Phys. Rev. E

**93**, 032111 (2016)Mahmoud, H.: Pólya Urn Models. CRC Press, Boca Raton (2009)

Mailler, C., Marckert, J.-F.: Measure-valued Pólya processes. Electron. J. Probab.

**22**(26), 33 (2017)Mailler, C., Uribe Bravo, G.: Random walks with preferential relocations and fading memory: a study through random recursive trees. J. Stat. Mech. Theory Exp.

**9**, 093206 (2019)Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep.

**339**, 1–77 (2000)Oliveira, F.A., Ferreira, R.M.S., Lapas, L.C., Vainstein, M.H.: Anomalous diffusion: a basic mechanism for the evolution of inhomogeneous systems. Front. Phys.

**19**, 18 (2019)Paraan, F.N.C., Esguerra, J.P.: Exact moments in a continuous time random walk with complete memory of its history. Phys. Rev. E

**74**, 032101 (2006)Pemantle, R.: A survey of random processes with reinforcement. Prob. Surv.

**4**, 1–79 (2007)Sabot, C., Tarrès, P.: Edge-reinforced random walk, vertex-reinforced jump process and the supersymmetric hyperbolic sigma model. JEMS

**17**(9), 2353–2378 (2015)Sabot, C., Zeng, X.: A random Schrödinger operator associated with the vertex reinforced jump process on infinite graphs. J. Am. Math. Soc.

**32**, 311–349 (2019)Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman and Hall/CRC, Boca Raton (2000)

Schütz, G.M., Trimper, S.: Elephants can always remember: exact long-range memory effects in a non-Markovian random walk. Phys. Rev. E

**70**, 045101(R) (2004)Silver, D., Huang, A., et al.: Mastering the game of go with deep neural networks and tree search. Nature

**529**, 484–489 (2016)

## Acknowledgements

I warmly thank Silvia Businger for explaining her work and for her help, and Jean Bertoin for valuable comments. I am also grateful to two anonymous referees for their careful reading of the manuscript and for their helpful remarks.

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Communicated by Antti Knowles.

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## A Appendix

### A Appendix

*Proof of Lemma *4.1

The fact that \((Y(t),\,t\ge 0)\) is a pure birth process with the stated properties is a consequence of (17) and of the dynamics of \((T(t),\,t\ge 0)\). Standard properties of branching processes (see, e.g., [2]) show that \((\mathrm{e}^{-(b+1)t}Y(t),\,t\ge 0)\) is a square-integrable martingale, and it follows from Lemma 3 in [10] that its (almost surely and \(L^2\)-)limit is Gamma\((1/(b+1),1/(b+1))\)-distributed. \(\square \)

*Proof of Lemma *4.2

The i.i.d. property of the processes \((Y_i^{(p)}(b_i+\cdot ),\,t\ge 0)\), \(i\ge 1\), is obvious from the construction. We shall therefore prove everything for \(i=1\), in which case \(b_i=b_1=0\).

Clearly, the sum of degrees of vertices of \(T^{(p)}_1(t)\) is equal to

It now follows from the construction of the preferential attachment tree *T*(*t*) at the beginning of Sect. 4.2 (recall in particular the parameters of the exponential clocks) that \((Y_1^{(p)}(t),\,t\ge 0)\) is a pure birth process with the stated birth rate and reproduction law. It is then well-known (see again
[2]) that \((\mathrm{e}^{-(b+p)t} Y_1^{(p)}(t),\,t\ge 0)\) is a martingale, whose terminal value \(W_1\) is almost surely strictly positive. By Kolmogorov’s forward equation (see once more
[2]) we compute for \(t>0\)

This proves square-integrability of \((\mathrm{e}^{-(b+p)t} Y_1^{(p)}(t),\,t\ge 0)\), and the claim about the first and second moment of \(W_1\) follows from the last display.

It remains to show boundedness in \(L^k\) for \(k\ge 3\), that is, we have to show that there exists a constant \(c_k<\infty \) such that

In order to prove this, we adapt [8, Proof of Lemma 3] to our situation. First, we note that the generator \(\mathfrak {G}\) of \((Y_1^{(p)}(t),\,t\ge 0)\) is given for any smooth function \(f:(0,\infty )\rightarrow \mathbb {R}\) by

Specifying to \(f(x)=x^\ell \) for some integer \(\ell \ge 3,\)

We prove now by induction that (47) holds for all \(k\in \mathbb {N}\). We already know it for \(k=1\) and \(k=2\), so let us assume that for some \(\ell \ge 3\), (47) holds for all \(k=1,\ldots ,\ell -1\). Kolmogorov’s forward equation reads

In combination with (48), and using (47) for \(k=1,\ldots ,\ell -1\), we deduce that for some \(\gamma >0\) depending on *b*, we have

By Jensen’s inequality,

so that

Going back to (49) and integrating, we obtain

Thus, (47) does hold for \(k=\ell \) as well, as wanted. \(\square \)

*Proof of Lemma *4.3

We fix a small \(\varepsilon >0\) and a sequence \((x_n,\,n\in \mathbb {N})\) of positive integers with \(\lim _{n\rightarrow \infty }x_n=\infty \) and \(x_n\le n\). Recalling Lemma 4.1 and the notation from there, we define for each \(k\in \mathbb {N}\) the event

Lemma 4.1 ensures that \(\lim _{n\rightarrow \infty }\mathbb {P}\left( \bigcap _{k=x_n}^\infty E^1_k\right) =1\). On \(E^1_k\), it holds for *k* sufficiently large that

Writing *D*(*k*) for the number of subtrees present at time \(\tau _k\), i.e.,

we deduce from the construction of \(T^{(p)}(t)\) that *D*(*k*) has the same law as \(1+\sum _{i=1}^{k-1}\epsilon _{i,1-p}\), where \(\epsilon _{i,1-p}\), \(i\ge 1\), are i.i.d. Bernoulli random variables with success probability \(1-p\). Consequently, an application of the law of large numbers shows that if we define

then \(\lim _{n\rightarrow \infty }\mathbb {P}\left( \bigcap _{k=x_n}^\infty E_k^2\right) =1\). On \(E_k^2\) it holds by construction that

Using (50), we find that on the event \(E^1_k\cap E^2_k\), for *k* large enough and provided \(\varepsilon \) is sufficiently small,

Letting

we have by the properties of \(E^1_k\) and \(E^2_k\) that \(\lim _{n\rightarrow \infty }\mathbb {P}\left( E_n\right) =1\).

On the event \(E_n\), it holds by construction that for all *n* large and *i* with \(x_n\le i\le n\),

Entirely similar, one sees that on \(E_n\)

Now notice that

if \(\varepsilon \downarrow 0\). Since \(\varepsilon >0\) can be chosen arbitrarily small, we can clearly construct a sequence \((\varepsilon _n)\) with \(\varepsilon _n\downarrow 0\) such that on \(E_n\), the stated bounds hold. \(\square \)

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### Cite this article

Baur, E. On a Class of Random Walks with Reinforced Memory.
*J Stat Phys* **181**, 772–802 (2020). https://doi.org/10.1007/s10955-020-02602-3

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DOI: https://doi.org/10.1007/s10955-020-02602-3

### Keywords

- Reinforced random walks
- Preferential attachment
- Memory
- Stable processes
- Branching processes
- Pólya urns

### Mathematics Subject Classification

- 60G50
- 60G52
- 60K35
- 05C85