## 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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## 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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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