Decoupled Standard Random Walk

• Decoupled Standard Random Walk is defined as a sequence of independent random variables, each matching the n-step sum distribution of a classical random walk, thereby eliminating temporal dependence.
• The model uses independent Bernoulli layers to analyze renewal counts and fluctuation behavior, revealing distinct large deviation and central limit properties compared to traditional renewal theory.
• It connects to determinantal processes and extreme value theory, offering new insights into maxima, first-passage times, and the impact of heavy-tailed increments on asymptotic behavior.

A decoupled standard random walk is a sequence $(\hat S_n)_{n\ge1}$ of independent random variables such that, for each $n\ge1$, $\hat S_n$ has the same distribution as the ordinary partial sum $S_n=\xi_1+\cdots+\xi_n$, where $(\xi_k)_{k\ge1}$ are i.i.d. copies of a nonnegative, nondegenerate random variable $\xi$. The construction preserves the one-dimensional marginals of a standard random walk while removing the temporal dependence between different indices $n$. This produces a renewal-like but non-pathwise object whose counting, fluctuation, large-deviation, and first-passage theories differ sharply from those of the usual coupled walk (Alsmeyer et al., 2024).

1. Definition and structural meaning

Let $(\xi_k)_{k\ge1}$ be i.i.d. nonnegative random variables and define the standard random walk

$S_n=\xi_1+\cdots+\xi_n,\qquad n\in\mathbb N.$

A decoupling of $(S_n)$ is any sequence $n\ge1$0 of independent random variables such that

$n\ge1$1

for every $n\ge1$2. Thus each coordinate retains the law of the $n\ge1$3-step sum of the original walk, but the sequence no longer arises from one nested trajectory (Alsmeyer et al., 2024).

This distinction is fundamental. For the genuine walk with nonnegative increments, $n\ge1$4 is monotone nondecreasing. By contrast, the decoupled sequence $n\ge1$5 is not monotone, because it is assembled from independent coordinates rather than from successive partial sums of a single increment sequence. Accordingly, $n\ge1$6 is not the position at time $n\ge1$7 of a genuine walk; it is a marginally correct but pathwise decoupled surrogate (Iksanov et al., 6 Jan 2026).

The standard auxiliary objects are the decoupled renewal counting process

$n\ge1$8

the decoupled maxima

$n\ge1$9

and the first passage time based on these maxima,

$\hat S_n$0

The original coupled walk has $\hat S_n$1 because of monotonicity, but the decoupled model does not preserve this identity. That separation between counting below a level and crossing above it is one of the central structural features of the theory (Alsmeyer et al., 2024).

2. Decoupled renewal process and its mean structure

The process $\hat S_n$2 counts how many independent marginals $\hat S_n$3 fall in $\hat S_n$4. Since the indicators $\hat S_n$5 are independent across $\hat S_n$6, $\hat S_n$7 is a sum of independent Bernoulli variables with success probabilities

$\hat S_n$8

This makes $\hat S_n$9 substantially more tractable than the ordinary renewal count built from the dependent sequence $S_n=\xi_1+\cdots+\xi_n$0 (Buraczewski et al., 7 Aug 2025).

Its mean is the classical renewal function, up to notation: $S_n=\xi_1+\cdots+\xi_n$1 One part of the literature denotes this mean by $S_n=\xi_1+\cdots+\xi_n$2, with $S_n=\xi_1+\cdots+\xi_n$3, while another uses $S_n=\xi_1+\cdots+\xi_n$4 and notes that $S_n=\xi_1+\cdots+\xi_n$5 is a renewal function (Buraczewski et al., 7 Aug 2025, Dong et al., 26 Oct 2025).

Because the dependence structure has been removed while the marginal renewal probabilities are retained, the decoupled renewal process sits between two classical objects. It is not an ordinary renewal process, since the indicators refer to independent coordinates rather than to one increasing renewal path. But it still encodes the same one-dimensional renewal information through the probabilities $S_n=\xi_1+\cdots+\xi_n$6. A plausible implication is that the model serves as a bridge between renewal theory and independent-array methods: classical renewal asymptotics continue to determine means, while fluctuations and rare events are governed by Bernoulli-sum and extreme-value mechanisms.

3. Fluctuation theory for $S_n=\xi_1+\cdots+\xi_n$7

Under the assumption that the law of $S_n=\xi_1+\cdots+\xi_n$8 belongs to the domain of attraction of a stable law with index $S_n=\xi_1+\cdots+\xi_n$9, a functional limit theorem holds for the centered decoupled renewal process after proper scaling, centering, and normalization. In finite-dimensional distributions,

$(\xi_k)_{k\ge1}$0

where $(\xi_k)_{k\ge1}$1 is a centered stationary Gaussian process with explicit covariance; if $(\xi_k)_{k\ge1}$2 is Lipschitz, the convergence strengthens to $(\xi_k)_{k\ge1}$3 with the $(\xi_k)_{k\ge1}$4-topology (Alsmeyer et al., 2024).

A later development proves a functional central limit theorem in $(\xi_k)_{k\ge1}$5 with the $(\xi_k)_{k\ge1}$6-topology under the heavy-tail assumption

$(\xi_k)_{k\ge1}$7

after a logarithmic time change $(\xi_k)_{k\ge1}$8 satisfying $(\xi_k)_{k\ge1}$9 with $\xi$0. The centered process

$\xi$1

converges to a centered Gaussian process $\xi$2; for $\xi$3, the transformed process $\xi$4 is stationary Gaussian (Dong et al., 26 Oct 2025).

The same paper establishes laws of the iterated or single logarithm for $\xi$5 in four regimes. The almost sure fluctuation order is:

• Finite variance: $\xi$6.
• Infinite variance, normal domain of attraction: $\xi$7.
• Regularly varying tail, $\xi$8: $\xi$9.
• Regularly varying tail, $n$0: $n$1.

The corresponding lower limits are the negatives of the upper-limit constants (Dong et al., 26 Oct 2025).

These results show that decoupling preserves the renewal-scale centering but alters the fluctuation field. In the ordinary renewal setting, fluctuations are driven by the dependence structure of one partial-sum path. In the decoupled setting, they arise from a superposition of independent Bernoulli layers indexed by $n$2, and the limiting Gaussian processes are correspondingly different.

4. Local large deviations and determinantal-process connections

A central problem is the asymptotic behavior of the local probabilities

$n$3

For heavy-tailed increments with infinite mean,

$n$4

the renewal function satisfies

$n$5

and for every fixed $n$6, $n$7,

$n$8

where $n$9 is the Legendre transform of a convex function $(\xi_k)_{k\ge1}$0 built from the inverse $(\xi_k)_{k\ge1}$1-stable subordinator. A local central limit theorem also yields the case $(\xi_k)_{k\ge1}$2 at the CLT scale (Buraczewski et al., 7 Aug 2025).

In the finite-mean case $(\xi_k)_{k\ge1}$3, the logarithmic asymptotics split according to tail heaviness and according to whether $(\xi_k)_{k\ge1}$4 or $(\xi_k)_{k\ge1}$5. For $(\xi_k)_{k\ge1}$6, the right tail of $(\xi_k)_{k\ge1}$7 controls the event; for $(\xi_k)_{k\ge1}$8, the left tail controls it. The principal scales are summarized below.

Regime	Assumption	Main logarithmic scale
Infinite mean	$(\xi_k)_{k\ge1}$9, $S_n=\xi_1+\cdots+\xi_n,\qquad n\in\mathbb N.$0	$S_n=\xi_1+\cdots+\xi_n,\qquad n\in\mathbb N.$1
Very heavy tails, $S_n=\xi_1+\cdots+\xi_n,\qquad n\in\mathbb N.$2	$S_n=\xi_1+\cdots+\xi_n,\qquad n\in\mathbb N.$3, $S_n=\xi_1+\cdots+\xi_n,\qquad n\in\mathbb N.$4	$S_n=\xi_1+\cdots+\xi_n,\qquad n\in\mathbb N.$5
Semi-heavy tails, $S_n=\xi_1+\cdots+\xi_n,\qquad n\in\mathbb N.$6	$S_n=\xi_1+\cdots+\xi_n,\qquad n\in\mathbb N.$7, $S_n=\xi_1+\cdots+\xi_n,\qquad n\in\mathbb N.$8	$S_n=\xi_1+\cdots+\xi_n,\qquad n\in\mathbb N.$9
Light tails, $(S_n)$0	$(S_n)$1 for some $(S_n)$2	$(S_n)$3
Light tails, $(S_n)$4	finite mean and nontrivial left tail	$(S_n)$5

The proofs use the representation of $(S_n)$6 as a sum of independent Bernoulli variables, together with exponential tilting in the infinite-mean case and a product approximation in the finite-mean case. In the latter regime, for $(S_n)$7,

$(S_n)$8

which reduces the local deviation problem to sharp large-deviation estimates for $(S_n)$9 (Buraczewski et al., 7 Aug 2025).

These asymptotics have a direct application to determinantal point processes. For the infinite Ginibre ensemble $n\ge1$00 with kernel

$n\ge1$01

Kostlan-type identities give

$n\ge1$02

for the exponential choice of $n\ge1$03. More generally, for the determinantal process $n\ge1$04 with Mittag-Leffler kernel

$n\ge1$05

one has

$n\ge1$06

where $n\ge1$07 has the gamma law with parameters $n\ge1$08 and $n\ge1$09. This link is one of the main motivations for the decoupled model (Buraczewski et al., 7 Aug 2025).

5. Maxima, first passage, and the loss of monotonic equivalence

The first strong-law-type results show that the asymptotics of the decoupled maxima

$n\ge1$10

and the first passage time

$n\ge1$11

can differ substantially from the corresponding quantities for the coupled walk. If $n\ge1$12, then

$n\ge1$13

If $n\ge1$14 but $n\ge1$15, then in general

$n\ge1$16

and under an additional tail condition,

$n\ge1$17

If $n\ge1$18, then

$n\ge1$19

These statements show that decoupling may radically alter first-passage asymptotics relative to the classical law $n\ge1$20 (Alsmeyer et al., 2024).

A subsequent functional theory analyzes the running maxima and first passage times in the Skorokhod space with the $n\ge1$21-topology and distinguishes five regimes determined by the right tail of $n\ge1$22 and the borderline $n\ge1$23 case (Iksanov et al., 6 Jan 2026).

Regime	Tail condition	Limit type
1	$n\ge1$24, or $n\ge1$25 with $n\ge1$26	Fréchet-type extremal process
2	Same tail scale, first passage version	Inverse of regime 1 extremal process
3	$n\ge1$27, or $n\ge1$28 with $n\ge1$29	Shifted extremal process
4	Corresponding first passage version of regime 3	Inverse extremal-like process
5	$n\ge1$30 with $n\ge1$31	Hybrid of linear drift and extremal component

In regime 1, extremes dominate and no centering is needed for the maxima. In regime 3, centering by the mean $n\ge1$32 becomes necessary, and the limit is driven by a Poisson random measure on $n\ge1$33. In the boundary regime, Gaussian fluctuations and rare large jumps both contribute, producing a hybrid limit (Iksanov et al., 6 Jan 2026).

The first passage processes converge to generalized inverses of the limiting maxima. This gives inverse extremal-like limit processes rather than the Gaussian-type limits familiar from ordinary renewal theory. The papers emphasize the contrast: for ordinary random walks with positive mean and finite variance, the number of visits and the first passage time have the same Brownian limit; for decoupled walks, $n\ge1$34 has stationary Gaussian limits, whereas $n\ge1$35 has inverse extremal-like limits (Iksanov et al., 6 Jan 2026). A common misconception is therefore that decoupling is a minor perturbation of renewal theory. The available results show the opposite: once dependence across times is removed, counting below a threshold and crossing above it become fundamentally different asymptotic problems.

6. Relation to the decoupled continuous-time random walk

The expression “decoupled” also appears in the continuous-time random-walk literature, but it denotes a different construction. There the process is

$n\ge1$36

where $n\ge1$37 are jump lengths, $n\ge1$38 are waiting times, $n\ge1$39 is the number of jumps up to time $n\ge1$40, and decoupling means that the jump lengths and waiting times are statistically independent, with joint density $n\ge1$41 (Denisov et al., 2010).

For superheavy-tailed waiting times satisfying

$n\ge1$42

with $n\ge1$43 slowly varying, the exceedance probability

$n\ge1$44

is the key control parameter. If the jump distribution has finite first and second moments, then the long-time position density has a simple exponential scaling limit: a symmetric two-sided exponential in the unbiased case $n\ge1$45, and an asymmetric one-sided exponential in the biased case $n\ge1$46. All moments grow more slowly than any positive power of time, so the model is a generic framework for superslow diffusion (Denisov et al., 2011).

This continuous-time theory is related by terminology rather than by direct identity of models. In the decoupled standard random walk, decoupling removes dependence across the index $n\ge1$47 while keeping $n\ge1$48. In the decoupled continuous-time random walk, decoupling separates temporal waiting times from spatial jumps. The shared term therefore masks two different mechanisms: independent marginals across discrete times in one case, and factorized space-time dynamics in the other (Denisov et al., 2010, Denisov et al., 2011).