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Alpha-Skew Random Walk Fourth-Moment Tightness and Tauberian Return Convolution

Mathematical Methods AIMech:AlphaSkewRandomWalkFourthMomentTightnessAndTauberianReturnConvolution DRAFT
Aliases: Seol Fourth-Order Moment Tightness Proof for Alpha-Skew Random Walks, Alpha-Skew Random Walk Tauberian Return Convolution Formulation
A stochastic-process convergence and fourth-order moment tightness formulation for the linearly interpolated rescaled alpha-skew random walk X_n^{(alpha)}(t) = n^{-1/2}(S_{[nt]}^{(alpha)} + (nt - [nt])(S_{[nt]+1}^{(alpha)} - S_{[nt]}^{(alpha)})) on Z (where S^{(alpha)} steps from 0 to +1 with probability alpha and to -1 with probability 1-alpha, and steps +/-1 with probability 1/2 away from 0) that bounds third- and fourth-order dependent cross-increment expectations via the origin-return sequence g(2i) = comb(2i, i) 2^{-2i} with generating function g_tilde(t) = (1 - t^2)^{-1/2} and its 2-fold and 4-fold convolutions (g*g)(j) and (g*g*g*g)(j), establishing uniform tightness E|X_n^{(alpha)}(t) - X_n^{(alpha)}(s)|^4 <= C_alpha |s - t|^2 in C(R_+, R).
Identity Notes & Concept Scope No AIO class represents the alpha-skew random walk or its fourth-moment tightness construction for skew Brownian motion; MarkovChain (AIO:MarkovChain) is the closest broad match covering discrete-time birth-death Markov chains and random-walk scaling limits.

Claims & Grounded Evidence

4 Claims
Mechanism #reflected-transition-law-and-origin-supported-conditional-increment-drift
For any alpha in (0, 1), |S^{(alpha)}| has the exact distribution of a simple symmetric random walk |S| reflected at 0, -S^{(alpha)} has the distribution of S^{(1-alpha)}, P(S_k^{(alpha)} = m) = alpha P(|S_k| = m) for m > 0, (1 - alpha) P(|S_k| = -m) for m < 0, and P(S_k^{(alpha)} = 0) = g(k) = comb(2i, i) 2^{-2i} for k = 2i (Propositions 2.1-2.2), which induces the state-dependent conditional drift E(S_{j+1}^{(alpha)} - S_j^{(alpha)} | S_j^{(alpha)}) = (2*alpha - 1) 1_{S_j^{(alpha)} = 0} and unit conditional second moment E[(S_{i+1}^{(alpha)} - S_i^{(alpha)})^2 | S_i^{(alpha)}] = 1 (Proposition 2.3).
Mechanism #tauberian-return-convolutions-and-fourth-moment-kolmogorov-chentsov-tightness
Because the origin-return generating function satisfies g_tilde(t) = sum_{k=0}^infty g(k) t^k = (1 - t^2)^{-1/2}, the two-fold convolution mu = g * g has generating function (1 - t^2)^{-1} with sum_{j=0}^m mu(j) ~ m and the four-fold convolution nu = g * g * g * g has generating function (1 - t^2)^{-2} with sum_{j=0}^m nu(j) ~ m^2 by Karamata's Tauberian theorem (Propositions 2.4-2.5), bounding the triple and quadruple cross-increment sums in Lemma 2.6 by C |k - j|^2 and proving E|X_n^{(alpha)}(t) - X_n^{(alpha)}(s)|^4 <= C_alpha |s - t|^2 for all s, t > 0 and n in N (Theorem 1.1 and Corollary 1.2).
Application #skew-brownian-motion-scaling-limit-and-heterogeneous-porous-media-tracking-application
Uniform tightness in C(R_+, R) establishes alpha-skew Brownian motion as the functional central limit scaling limit of the alpha-skew random walk on the integer lattice, providing the convergence foundation for Lagrangian random-walk particle tracking across discontinuous permeability interfaces in heterogeneous porous media and biological membranes.
Limitation #non-independent-increments-and-second-moment-tightness-insufficiency-limitation
Whenever alpha != 1/2, the increments S_{k+1}^{(alpha)} - S_k^{(alpha)} of the alpha-skew random walk are statistically dependent through origin visits, and second-order moment bounds E|X_n^{(alpha)}(t) - X_n^{(alpha)}(s)|^2 <= C |t - s| (exponent 1 on |t - s|) are insufficient to establish tightness in C(R_+, R) via the Kolmogorov-Chentsov criterion even in the symmetric case alpha = 1/2, necessitating fourth-order moment control (Section 1, p. 1; Lemma 2.6, p. 4).

⚠️ Curated Limitations & Negative Results

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes
AIO Target Class Mapping Relation Mapping Rationale
AIO:MarkovChain
Markov Chain
 broad_match The alpha-skew random walk S^{(alpha)} is a discrete-time birth-death Markov chain on Z with permeable skew barrier at the origin.

Evidence Sources & Primary Literature

3 Sources
Source ID Publication / Document Type / Access Reference & Checksum
r838-primary-pdf
Youngsoo Seol
primary paper
full_text
ID: https://arxiv.org/abs/1106.5042
sha256:d3cf8decaac1…
r838-arxiv-abs
Youngsoo Seol
primary paper
abstract_only
ID: https://arxiv.org/abs/1106.5042
sha256:16dfb2333fed…
r838-metadata-cr
Youngsoo Seol
primary paper
abstract_only
ID: https://arxiv.org/abs/1106.5042
sha256:d2d7b907c7e3…

Ontology & Grounding Topology

Open in interactive network graph →
CONCEPT KIND Mathematical Methods AIMech:AlphaSkewRandomWalkFourthMomentTightnessAndTauberianReturnConvolution Alpha-Skew Random Walk... 4 claims · 3 sources AIO GROUNDING AIO:MarkovChain (1)

Curation Provenance

Curator: Antigravity
Method: ai_assisted
Model: Gemini
Created: 2026-10-19T15:21:00Z
Updated: 2026-10-19T15:21:00Z
Source File: data/curated/AlphaSkewRandomWalkFourthMomentTightnessAndTauberianReturnConvolution.yaml · Schema: curated_concept.yaml

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