arxiv: 2605.08983 · v1 · submitted 2026-05-09 · 🧮 math.PR

Recognition: 2 theorem links

· Lean Theorem

A functional limit theorem for self-normalized linear processes with random coefficients and i.i.d. heavy-tailed innovations

Danijel Krizmanic

Pith reviewed 2026-05-12 01:57 UTC · model grok-4.3

classification 🧮 math.PR

keywords functional limit theoremself-normalized sumslinear processesheavy-tailed innovationsrandom coefficientsSkorokhod M2 topologycadlag functions

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The pith

Self-normalized partial sums of linear processes with random coefficients and heavy-tailed innovations converge in the Skorokhod M2 topology under a bounded partial sums condition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper establishes a functional limit theorem for the self-normalized partial sum process of a strictly stationary linear process that has random coefficients and is driven by i.i.d. heavy-tailed innovations. The proof requires that every partial sum of the coefficient series stays almost surely between zero and the sum of the whole series. The limiting object lives in the space of cadlag functions on the unit interval, and the topology used is the Skorokhod M2 topology. Readers interested in time series with infinite variance would see this as a tool for describing the asymptotic shape of accumulated sums when ordinary central limit theorems fail.

Core claim

Under the condition that all partial sums of the series of coefficients are almost surely bounded between zero and the sum of the series, the self-normalized partial sum process of the strictly stationary linear process converges in distribution in the space of cadlag functions on [0,1] equipped with the Skorokhod M2 topology.

What carries the argument

The self-normalized partial sum process of the linear process, with normalization taken from the process itself, under the almost-sure boundedness condition on the partial sums of the random coefficients.

If this is right

The theorem covers linear processes whose coefficients vary randomly from realization to realization while preserving strict stationarity.
Self-normalization removes the need to estimate a separate scale parameter when the innovations have infinite variance.
Convergence in the M2 topology permits sample paths with jumps, which is natural for heavy-tailed driving noise.
The result supplies a functional version that can be used to derive limit theorems for statistics built from the entire path, not just the endpoint.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The boundedness condition on coefficient partial sums might be verified directly for many parametric families of random coefficients used in practice.
Because M2 is weaker than the J1 topology, the result may continue to hold in situations where the limiting path has more discontinuities than standard stable Levy processes allow under J1.
The same normalization and topology could be tested for linear processes whose innovations are only stationary rather than i.i.d., though that extension lies outside the present argument.

Load-bearing premise

All partial sums of the series of coefficients are almost surely bounded between zero and the sum of the series.

What would settle it

A concrete counterexample process in which the partial sums of the coefficients are not a.s. bounded between zero and the total sum, yet the self-normalized sums still converge in the Skorokhod M2 topology, would show the condition is not necessary.

read the original abstract

In this article we derive a self-normalized functional limit theorem for strictly stationary linear processes with i.i.d. heavy-tailed innovations and random coefficients under the condition that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series. The convergence takes part in the space of c\`{a}dl\`{a}g functions on $[0,1]$ with the Skorokhod $M_{2}$ topology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper derives a self-normalized functional limit theorem for strictly stationary linear processes with i.i.d. heavy-tailed innovations and random coefficients. The result holds under the explicit condition that all partial sums of the coefficient series are almost surely bounded between zero and the total sum of the series, with convergence established in the space of càdlàg functions on [0,1] under the Skorokhod M2 topology.

Significance. If the derivation is valid, the result extends existing functional limit theorems to self-normalized partial-sum processes with random coefficients and infinite-variance innovations. This is relevant for modeling dependent heavy-tailed time series where standard normalization fails, and the M2 topology choice accommodates the jump structure induced by heavy tails.

major comments (1)

[Abstract / main assumption] The central theorem is conditioned on the partial-sum boundedness assumption for the random coefficients (stated in the abstract and presumably used in the tightness and convergence arguments). The manuscript provides no verification, examples, or sufficient conditions under which this a.s. boundedness holds for standard random-coefficient models, leaving open whether the result applies beyond specially constructed cases.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their thorough review and insightful comments. Below, we provide a point-by-point response to the major comment and outline the revisions we plan to make.

read point-by-point responses

Referee: [Abstract / main assumption] The central theorem is conditioned on the partial-sum boundedness assumption for the random coefficients (stated in the abstract and presumably used in the tightness and convergence arguments). The manuscript provides no verification, examples, or sufficient conditions under which this a.s. boundedness holds for standard random-coefficient models, leaving open whether the result applies beyond specially constructed cases.

Authors: We acknowledge that the manuscript does not currently include explicit verification or examples for the partial-sum boundedness assumption. This is a valid point, and we will revise the paper to address it. In the revised manuscript, we will insert a new Remark following the statement of the main assumption, providing sufficient conditions under which the boundedness holds. Specifically, a sufficient condition is that the random coefficients are non-negative almost surely. In this case, the partial sums are monotonically non-decreasing and hence bounded above by the infinite sum (which is assumed finite in the context of the linear process). We will also provide a concrete example, such as a random coefficient AR(1) model where the coefficient is uniform on [0, 0.5], ensuring the condition holds a.s. This will demonstrate that the result applies to standard models beyond specially constructed cases. We believe this addition will clarify the scope of the theorem without altering the main results. revision: yes

Circularity Check

0 steps flagged

No significant circularity; standard limit theorem derivation

full rationale

The paper derives a conditional functional limit theorem for self-normalized partial sums of a linear process. The key assumption (partial sums of coefficients a.s. bounded between 0 and the total sum) is stated explicitly as a sufficient condition for tightness and convergence in the Skorokhod M2 topology. All steps are standard probabilistic arguments (characteristic functions, truncation, weak convergence criteria) applied to the given stationary process with i.i.d. heavy-tailed innovations; none reduce by construction to the target statement, fitted parameters, or self-citation chains. The result is a genuine theorem under the stated regime rather than a tautology or renaming of inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard assumptions of probability theory for linear processes, heavy-tailed distributions, and convergence in Skorokhod space; no free parameters or invented entities are introduced in the abstract.

axioms (2)

standard math Properties of the Skorokhod M2 topology on the space of càdlàg functions
Invoked to state the mode of convergence.
domain assumption Strict stationarity of the linear process
Required for the functional limit to hold uniformly.

pith-pipeline@v0.9.0 · 5369 in / 1239 out tokens · 44462 ms · 2026-05-12T01:57:38.021494+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
self-normalized functional limit theorem ... in the space of càdlàg functions on [0,1] with the Skorokhod M2 topology
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
under the condition that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series

Reference graph

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