Step-Reinforced Random Walks

• Step-Reinforced Random Walk (SRRW) is a non-Markovian process in which each step is either a fresh i.i.d. increment or a uniformly chosen past effective increment, preserving the original step law.
• The model exhibits phase transitions at the reinforcement parameter p = 1/2, leading to diffusive, critical, and superdiffusive regimes with martingale limits and Gaussian fluctuation phenomena.
• Variants such as negative reinforcement and unified formulations connect SRRW to the elephant random walk and extend its applicability to empirical processes, group-valued settings, and geometric random walk generalizations.

A step-reinforced random walk (SRRW) is a discrete-time non-Markovian random walk with long-range memory in which each increment is either a fresh i.i.d. step or a copy of a previous effective increment chosen uniformly from the past. In its canonical positively step-reinforced form, one starts from i.i.d. increments $(X_n)_{n\ge1}$, Bernoulli reinforcement indicators, and uniform indices on the past; the resulting partial sums preserve the marginal step law while creating strong dependence across time. This mechanism contains the elephant random walk as a special case for Rademacher increments, and it has since been extended to negatively reinforced, unbalanced, amnesic, stable, and group-valued settings (Bertenghi, 2021).

1. Canonical construction and basic properties

In the standard $d$-dimensional formulation, one fixes i.i.d. $\mathbb{R}^d$-valued random vectors $\mathbf{X}_1,\mathbf{X}_2,\dots$ with common law $\mu$, finite second moment, mean $E(\mathbf{X})$, and covariance matrix $\bm{\sigma}^2$. One also fixes i.i.d. Bernoulli variables $(\varepsilon_i)_{i\ge2}$ with parameter $p\in(0,1)$, independent uniform indices $U[i-1]$ on $d$0, and defines reinforced increments recursively by

$d$1

The SRRW is then the partial-sum process

$d$2

With probability $d$3 the walk takes a new independent increment from $d$4; with probability $d$5 it reuses one of the past effective increments uniformly at random. A basic inductive property is that each reinforced step still has the same marginal law as the original increment, $d$6, even though the increments are no longer independent (Bertenghi, 2021).

In one dimension, the notation simplifies to

$d$7

If $d$8, it is standard to center the walk by subtracting $d$9; the centered process is again a step-reinforced random walk with typical step $\mathbb{R}^d$0 (Bertenghi, 2021).

The same reinforcement algorithm also admits a negatively step-reinforced version. If $\mathbb{R}^d$1 are i.i.d. real-valued increments and $\mathbb{R}^d$2, $\mathbb{R}^d$3 are as above, the negatively reinforced increments are

$\mathbb{R}^d$4

and the corresponding walk is $\mathbb{R}^d$5. The positive model is also called noise-reinforced random walk, while the negative model repeats past increments with a sign flip (Hu et al., 2023).

A central identification is with the elephant random walk (ERW). For Rademacher increments, the positively reinforced SRRW corresponds to an ERW with memory parameter $\mathbb{R}^d$6, while the negatively reinforced walk corresponds to the ERW with memory parameter $\mathbb{R}^d$7. More generally, if $\mathbb{R}^d$8 is isotropic on $\mathbb{R}^d$9, the positively reinforced SRRW corresponds to a multidimensional ERW (Bertenghi, 2021).

2. Scaling regimes and first-order asymptotics

For the positively step-reinforced walk, the reinforcement parameter induces a phase transition at $\mathbf{X}_1,\mathbf{X}_2,\dots$0. The diffusive regime is $\mathbf{X}_1,\mathbf{X}_2,\dots$1, the critical regime is $\mathbf{X}_1,\mathbf{X}_2,\dots$2, and the superdiffusive regime is $\mathbf{X}_1,\mathbf{X}_2,\dots$3. In the superdiffusive regime, the natural scaling sequence is

$\mathbf{X}_1,\mathbf{X}_2,\dots$4

The normalized process $\mathbf{X}_1,\mathbf{X}_2,\dots$5 is a martingale, and in the one-dimensional centered case $\mathbf{X}_1,\mathbf{X}_2,\dots$6 almost surely and in $\mathbf{X}_1,\mathbf{X}_2,\dots$7, where $\mathbf{X}_1,\mathbf{X}_2,\dots$8 is non-degenerate. Equivalently,

$\mathbf{X}_1,\mathbf{X}_2,\dots$9

In higher dimension,

$\mu$0

with $\mu$1 non-degenerate and non-Gaussian (Bertenghi, 2021).

The strong-law behavior of positive and negative reinforcement differs. Under $\mu$2, the positively reinforced walk satisfies

$\mu$3

where $\mu$4, whereas the negatively reinforced walk satisfies

$\mu$5

Thus positive reinforcement preserves the LLN drift, while negative reinforcement reduces it by the factor $\mu$6 (Hu et al., 2023).

A separate but closely related line of work establishes a spatial phase transition for genuinely $\mu$7-dimensional SRRWs on $\mu$8. Under a $\mu$9 moment condition, the walk undergoes a phase transition between recurrence and transience in dimensions $E(\mathbf{X})$0, and it is transient for all reinforcement parameters in dimensions $E(\mathbf{X})$1. More precisely, in dimension $E(\mathbf{X})$2 recurrence holds for $E(\mathbf{X})$3 and transience for $E(\mathbf{X})$4; in dimension $E(\mathbf{X})$5 recurrence holds for $E(\mathbf{X})$6 and transience for $E(\mathbf{X})$7; and in dimensions $E(\mathbf{X})$8 the walk is transient for all $E(\mathbf{X})$9 (Qin, 2024).

3. Limit theorems and quantitative asymptotics

The principal second-order theorem for the superdiffusive positive model states that after subtracting the random leading term $\bm{\sigma}^2$0, the remaining fluctuations are Gaussian on the $\bm{\sigma}^2$1 scale. If $\bm{\sigma}^2$2, $\bm{\sigma}^2$3, and $\bm{\sigma}^2$4 is the covariance matrix of $\bm{\sigma}^2$5, then

$\bm{\sigma}^2$6

In one dimension this becomes

$\bm{\sigma}^2$7

This extends the Kubota–Takei result for the elephant random walk from Rademacher steps to arbitrary $\bm{\sigma}^2$8 step distributions and to higher dimensions (Bertenghi, 2021).

For the positively reinforced walk in the subcritical and critical regimes, and for the negatively reinforced walk on its full parameter range, strong invariance principles are available. Under $\bm{\sigma}^2$9, the positive walk with $(\varepsilon_i)_{i\ge2}$0 admits a Brownian coupling after centering by $(\varepsilon_i)_{i\ge2}$1, and the critical case $(\varepsilon_i)_{i\ge2}$2 has the $(\varepsilon_i)_{i\ge2}$3 normalization. For the negatively reinforced walk, the fluctuation scale is $(\varepsilon_i)_{i\ge2}$4 and a strong invariance principle holds for every $(\varepsilon_i)_{i\ge2}$5. As by-products, the law of the iterated logarithm and the functional central limit theorem are obtained for both signs of reinforcement (Hu et al., 2023).

Recent quantitative work goes beyond weak convergence and gives Berry–Esseen bounds. Under a finite third absolute moment, both positively and negatively step-reinforced random walks admit rates of convergence to normality. The proofs use randomly weighted sum representations, Berry–Esseen bounds for functionals of independent random variables, and comparison arguments between mixed normal laws and fixed Gaussian laws (Hu, 3 Apr 2025).

The critical parameter $(\varepsilon_i)_{i\ge2}$6 also governs the asymptotic geometry of the two-dimensional positive model. At $(\varepsilon_i)_{i\ge2}$7, under a fourth moment assumption,

$(\varepsilon_i)_{i\ge2}$8

and

$(\varepsilon_i)_{i\ge2}$9

This identifies the critical two-dimensional walk as transient but directionally non-convergent (Qin, 2024).

4. Variants, unifying formulations, and alternative memory schemes

The positive and negative models can be unified by allowing a reinforced step either to keep or to flip its sign. One recent formulation introduces parameters $p\in(0,1)$0 and $p\in(0,1)$1, takes a fresh step with probability $p\in(0,1)$2, and under reinforcement reuses a uniformly chosen past step with probability $p\in(0,1)$3 or flips it with probability $p\in(0,1)$4. The effective parameter is

$p\in(0,1)$5

This framework contains the elephant random walk ($p\in(0,1)$6), the positively step-reinforced random walk ($p\in(0,1)$7), and the negatively step-reinforced random walk ($p\in(0,1)$8). In the subcritical regime $p\in(0,1)$9, it yields unified normal and stable central limit theorems for randomly weighted sums representing the walk (Hu et al., 13 Oct 2025).

A closely related two-parameter class writes the same effective quantity in the form

$U[i-1]$0

where $U[i-1]$1 is the probability of reinforcement and $U[i-1]$2 is the probability of preserving the chosen sign. In that formulation, the law of large numbers is

$U[i-1]$3

and the model exhibits the same diffusive, critical, and superdiffusive trichotomy according to whether $U[i-1]$4, $U[i-1]$5, or $U[i-1]$6. This gives a unified treatment of ERW, positive SRRW, and negative SRRW in a single martingale framework (Aguech et al., 20 Apr 2025).

The standard uniform-memory SRRW has also been generalized to amnesic reinforcement. In that model, the choice of a past step is governed by a positive memory sequence $U[i-1]$7 with partial sums $U[i-1]$8, and the selection probability is proportional to $U[i-1]$9. Under regular variation of $d$00, the diffusive regime becomes

$d$01

where $d$02 is the regular variation index of $d$03. The corresponding functional invariance principle shows that the limit process is always the sum of a noise-reinforced Brownian motion and a Brownian motion, with the two Gaussian components not independent (Bertenghi et al., 2024).

The term “step-reinforced random walk” is also used more broadly for models where the law of the next increment depends on recent-step averages rather than on exact copying of a past increment. One such class consists of random walks reinforced by their recent history through threshold-based switching among increment distributions. This suggests that the expression “SRRW” can denote either the canonical uniform-copying model or a broader family of increment-based self-interacting walks, depending on context (Pinsky, 2017).

5. Functional extensions, empirical processes, and geometric generalizations

A notable application of the superdiffusive central limit theorem concerns reinforced empirical processes. If $d$04 is a reinforced sample on $d$05 obtained by the same reinforcement algorithm, the reinforced empirical process

$d$06

admits a refined Donsker-type expansion. Bertoin had shown that for $d$07,

$d$08

where $d$09 has exchangeable increments. The second-order theorem then yields

$d$10

with $d$11 a standard Brownian bridge. The finite-dimensional covariance coincides with that of the Brownian bridge, and Kallenberg’s bridge theory upgrades finite-dimensional convergence to process convergence in $d$12 (Bertenghi, 2021).

Geometric extensions now place SRRWs on groups. A generalized SRRW on a group allows a fresh step sampled from $d$13 or a transformed past step $d$14, unifying several previously studied models. Upper bounds on transition probabilities have been established for every reinforcement parameter $d$15, with the decay rate linked to the geometry of the underlying group. On Euclidean space, the walk is transient in all dimensions $d$16 for any $d$17. On finitely generated groups, isoperimetric-profile bounds imply polynomial, stretched-exponential, or exponential decay according to the group geometry, and on nonamenable Cayley graphs one obtains exponential decay of $d$18. In particular, this resolves the open problem of exponential decay for the elephant random walk on Cayley trees (Peres et al., 8 Apr 2026).

These functional and geometric results indicate that SRRW theory is not limited to one-dimensional martingale asymptotics. It also interfaces with empirical-process theory, exchangeable-increment bridges, recursive-tree representations, and heat-kernel-type estimates on groups. A plausible implication is that step reinforcement preserves a large amount of Gaussian second-order structure even when first-order behavior is non-Gaussian or geometry-dependent.

6. Terminology, related models, and open problems

Step-reinforced random walks belong to the broader class of reinforced processes in which transition rules depend on the past trajectory. The surrounding literature connects SRRWs to ERW and MERW, Pólya urns, preferential attachment, shark random swim, noise-reinforced Lévy processes, reinforced empirical processes, and other self-interacting models. One line of work studies preferential-attachment-type reinforcement where the probability of reusing a past step is itself reinforced by an additive weight $d$19, producing “memory-reinforced” and “strongly memory-reinforced” variants with additional phase transitions (Baur, 2019).

A common misconception is terminological rather than probabilistic: in some recent MCMC and stochastic-optimization papers, “SRRW” denotes self-repellent random walk, not step-reinforced random walk. Those models are nonlinear Markov chains that discourage revisiting heavily sampled states and are analytically unrelated to the copy-a-past-step mechanism, even though the acronym coincides (Doshi et al., 2023).

Several open directions are explicitly identified in the SRRW literature. For superdiffusive positive reinforcement, the explicit law of the martingale limit $d$20 is not known in general, and extensions to non-uniform choice of past steps or more general kernels remain natural problems. Strong invariance principles in the superdiffusive positive regime, multidimensional strong results for the models treated only in one dimension, and heavy-tailed extensions beyond finite variance are also stated as open directions. In the unbalanced setting, the critical and supercritical heavy-tailed regimes remain to be developed further (Bertenghi, 2021, Hu et al., 2023, Hu et al., 13 Oct 2025).

Taken together, these developments show that SRRW is best understood not as a single theorem or a single scaling law, but as a family of long-memory increment processes organized around a robust structural idea: repeated recycling of past steps, sometimes with sign changes or generalized memory kernels, generates martingale normalizations, recursive-tree representations, and phase transitions whose critical value is often one-half, while still leaving room for Gaussian second-order limits and refined invariance principles across a surprisingly broad range of models.