On a Class of Random Walks with Reinforced Memory

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.

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

$$\begin{aligned} \left( 2(|T_1^{(p)}(t)|-1)+H^{(p)}_1(t)\right) . \end{aligned}$$

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 $t>0$

$$\begin{aligned} \mathbb {E}\left[ Y_1^{(p)}(t)\right] =\mathrm{e}^{(b+p)t}\,,\quad \quad \mathbb {E}\left[ Y_1^{(p)}(t)^2\right] =\frac{(b+1)(b+2p)}{b+p}\left( \mathrm{e}^{2(b+p)t}-\mathrm{e}^{(b+p)(b_i+t)}\right) . \end{aligned}$$

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

$$\begin{aligned} \mathbb {E}\left[ Y_1^{(p)}(t)^k\right] \le c_k \mathrm{e}^{k(b+p)t}\quad for all t\ge 0. \end{aligned}$$

(47)

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

$$\begin{aligned} \mathfrak {G}f(x)=x(1-p)\left( f(x+b)-f(x)\right) +xp\left( f(x+b+1)-f(x)\right) . \end{aligned}$$

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

$$\begin{aligned} \mathfrak {G}f(x)&=x(1-p)\sum _{j=0}^{\ell -1}{\ell \atopwithdelims ()j}x^jb^{\ell -j}+xp\sum _{j=0}^{\ell -1} {\ell \atopwithdelims ()j}x^j(b+1)^{\ell -j}\nonumber \\&=\ell (b+p)x^\ell +(1-p)\sum _{j=0}^{\ell -2}{\ell \atopwithdelims ()j}x^{j+1}b^{\ell -j}+p\sum _{j=0}^{\ell -2}{\ell \atopwithdelims ()j}x^{j+1}(b+1)^{\ell -j}. \end{aligned}$$

(48)

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

$$\begin{aligned} \frac{d }{d t}\mathbb {E}\left[ f(Y_1^{(p)}(t))\right] =\mathbb {E}\left[ \mathfrak {G}f(Y_1^{(p)}(t))\right] . \end{aligned}$$

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

$$\begin{aligned} \frac{d }{d t}\ln \mathbb {E}\left[ Y_1^{(p)}(t)^\ell \right] \le (b+p)\ell +\gamma \frac{\mathrm{e}^{(\ell -1)(b+p)t}}{\mathbb {E}\left[ Y_1^{(p)}(t)^\ell \right] }. \end{aligned}$$

(49)

By Jensen’s inequality,

$$\begin{aligned} \mathbb {E}\left[ Y_1^{(p)}(t)^\ell \right] \ge \mathrm{e}^{\ell (b+p)t}\quad for all t\ge 0\,, \end{aligned}$$

so that

$$\begin{aligned} \int _0^\infty \frac{\mathrm{e}^{(\ell -1)(b+p)t}}{\mathbb {E}\left[ Y_1^{(p)}(t)^\ell \right] }d t\le \int _0^\infty \mathrm{e}^{-(b+p)t}d t=\frac{1}{b+p}. \end{aligned}$$

Going back to (49) and integrating, we obtain

$$\begin{aligned} \mathbb {E}\left[ Y_1^{(p)}(t)^\ell \right] \le \mathrm{e}^{\gamma \frac{1}{b+p}}\mathrm{e}^{\ell (b+p)t}\quad for all t\ge 0. \end{aligned}$$

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

$$\begin{aligned} E^1_k:=\left\{ W(1-\varepsilon )\le \mathrm{e}^{-(b+1)\tau _k}\left( (b+1)k-b\right) \le W(1+\varepsilon )\right\} . \end{aligned}$$

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

$$\begin{aligned} \tau _k&\le \frac{1}{b+1}\left( \ln k - \ln W +\ln (b+1)-\ln \left( 1-\varepsilon \right) \right) \,,\nonumber \\ \tau _k&\ge \frac{1}{b+1}\left( \ln k - \ln W +\ln (b+1)-2\ln \left( 1+\varepsilon \right) \right) . \end{aligned}$$

(50)

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

$$\begin{aligned} D(k)=\max \left\{ i\ge 1: T_i^{(p)}(\tau _k)\ne \emptyset \right\} \,, \end{aligned}$$

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

$$\begin{aligned} E^2_k:=\left\{ k(1-p)(1-\varepsilon )\le D(k) \le k(1-p)(1+\varepsilon )\right\} \,, \end{aligned}$$

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

$$\begin{aligned} b_{\lceil k(1-p)(1+\varepsilon )\rceil }\ge \tau _k. \end{aligned}$$

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,

$$\begin{aligned} b_k \ge \tau _{\lfloor \frac{k}{(1-p)(1+\varepsilon )}\rfloor }\ge \frac{1}{b+1}\left( \ln (k-1)-\ln W+\ln (b+1)-\ln (1-p)-3\ln (1+\varepsilon )\right) . \end{aligned}$$

Letting

$$\begin{aligned} E_n:=\bigcap _{k=x_n}^\infty \left( E^1_k\cap E^2_k\right) \,, \end{aligned}$$

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 $x_n\le i\le n$,

$$\begin{aligned} \tau _n-b_i\le \frac{1}{b+1}\left( \ln n-\ln (i-1)+\ln (1-p)+3\ln (1+\varepsilon )-\ln (1-\varepsilon )\right) . \end{aligned}$$

Entirely similar, one sees that on $E_n$

$$\begin{aligned} \tau _n-b_i\ge \frac{1}{b+1}\left( \ln n-\ln (i+1)+\ln (1-p)+2\ln (1-\varepsilon )-2\ln (1+\varepsilon )\right) . \end{aligned}$$

Now notice that

$$\begin{aligned} \max \left\{ 3\ln (1+\varepsilon )-\ln (1-\varepsilon ),\,2\ln (1+\varepsilon )-2\ln (1-\varepsilon )\right\} \downarrow 0 \end{aligned}$$

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. $\square$