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Finite-Time Extremal Index, Cluster Size, and Sojourn Time in Non-Markovian Regenerative Processes

Mathematical Methods AIMech:FiniteTimeExtremalIndexClusterSizeAndSojournTimeInRegenerativeProcesses DRAFT

Aliases: Finite-Time Extremal Index Reciprocity and Sojourn Identity in Regenerative Models, Abadi-Freitas Non-Markovian Regenerative Extremal Clustering Formulation

A probabilistic extreme-value and quantitative-recurrence formulation for stationary discrete-time non-Markovian regenerative processes (X_n)_{n in Z} generated by i.i.d. symbol draws Z_n ~ (p_a) and run-length laws q_a(k) with finite mean regeneration time nu = sum_a p_a E(q_a) that distinguishes the cluster-entrance mean cluster size E_{E_a}(N_a) (conditioned on E_a = {X_{-p(U_a)} <= a, X_0 > a}) from the unanchored expected sojourn time E_{U_a}(N_a) (conditioned on U_a = {X_0 > a}), proving exact finite-time reciprocity E_{E_a}(N_a) = 1 / theta_{p(U_a)}(a), the stationary second-moment sojourn identity E_{U_a}(N_a) = (theta_1(a)/2)*(E_{E_a}(N_a^2) + E_{E_a}(N_a)), and the O'Brien horizon bound |1 - theta_q / theta_1| <= q P(U_a | R_0).

Identity Notes & Concept Scope No AIO class represents finite-time extremal index and sojourn-time clustering identities in non-Markovian regenerative processes; MarkovChain (AIO:MarkovChain) is the closest broad match covering discrete-time stochastic processes and transition/recurrence analysis.

Claims & Grounded Evidence

4 Claims

Mechanism #finite-time-extremal-index-cluster-size-reciprocity-and-sojourn-second-moment-identity

For any stationary discrete-time regenerative process with minimum observable period p(U_a) = 1, the finite-time escape probability theta_1(a) = P(X_0 > a, X_1 <= a | X_0 > a) equals the exact reciprocal of the mean cluster size E_{E_a}(N_a) = 1 / theta_1(a) conditioned on cluster entrance E_a = {X_{-1} <= a, X_0 > a} across both Smith's model (theta_1(a) = (1 - e_a)/2, E_{E_a}(N_a) = 2/(1 - e_a)) and deterministic block models (theta_1(a) = e_a(1 - e_a)/g_a, E_{E_a}(N_a) = g_a/[e_a(1 - e_a)]), whereas the unanchored expected sojourn time obeys the exact second-moment identity E_{U_a}(N_a) = (theta_1(a)/2)*(E_{E_a}(N_a^2) + E_{E_a}(N_a)) (Lemma 7, Equation 18), converging to 3/2 != 2 in Smith's model as a -> infty.

supports Source: Clustering indices and decay of correlations in non-Markovian models · Locator: Section 4.1-4.2, Definitions 1-3 and Equations 9-16, pp. 7-14; Section 6, Lemma 7, Equations 18-20, p. 15

Mechanism #obrien-finite-horizon-approximation-bound-and-morse-code-correlation-decay

By factorizing the regeneration event at the second exceedance time j in {2, ..., q}, Lemma 4 proves the sharp finite-horizon O'Brien bound |1 - theta_q(a)/theta_1(a)| <= q P(U_a | R_0) (Equation 17), ensuring theta_q / theta_1 -> 1 whenever q = o(P(U_a | R_0)^{-1}), while Section 3 establishes exponential psi-mixing correlation decay c_n = P(R_{n+1} | R_0) = 1/(2 - p_1) + [(1 - p_1)/(2 - p_1)](p_1 - 1)^n on finite alphabets (Equation 8).

supports Source: Clustering indices and decay of correlations in non-Markovian models · Locator: Section 3, Equation 8, pp. 6-7; Section 5, Lemma 4 and Remark 5, Equation 17, pp. 14-15

Application #poincare-recurrence-cylinder-hitting-time-exponential-law-application

Applied to Poincare recurrence and cylinder hitting times tau_n = inf{t >= 1 | X_t^{t+n-1} = a_0^{n-1}}, replacing the memory-scale horizon q(a, alpha) with the short observable periodicity p(a_0^{n-1}) yields the exponential hitting-time scaling parameter theta_{p(U_a)} directly from characteristic-polynomial root asymptotics r_1 in Smith's model (Equations 10-14) or Euclidean block remainders s_n in the deterministic block model (Equations 15-16).

supports Source: Clustering indices and decay of correlations in non-Markovian models · Locator: Section 1, Application: Hitting times, p. 3; Section 4.2.1-4.2.2, Equations 10-16, pp. 10-14

Limitation #infinite-alphabet-non-psi-mixing-mass-escape-and-cyclic-nonconvergence-limitation

Over an infinite countable alphabet N with non-zero symbol probabilities p_a > 0, the regenerative process fails to be psi-mixing (Section 3, p. 7), the cluster-size distribution in the deterministic block model suffers escape of mass P_E(N_a <= k) -> 0 so the limiting cluster-size expectation does not exist (Section 4.2.2, p. 12), and for n-cylinder hitting in the block model the Euclidean remainder s_n = ceil(n/a)*a - n cycles periodically through {0, ..., a-1} so neither lim_{n->infty} theta_1(n) nor lim_{n->infty} E_E(N_n) exists (Section 4.2.2, p. 13).

supports Source: Clustering indices and decay of correlations in non-Markovian models · Locator: Section 3, The Infinite case, p. 7; Section 4.2.2, pp. 12-13

⚠️ Curated Limitations & Negative Results

Over an infinite countable alphabet N with non-zero symbol probabilities p_a > 0, the regenerative process fails to be psi-mixing (Section 3, p. 7), the cluster-size distribution in the deterministic block model suffers escape of mass P_E(N_a <= k) -> 0 so the limiting cluster-size expectation does not exist (Section 4.2.2, p. 12), and for n-cylinder hitting in the block model the Euclidean remainder s_n = ceil(n/a)*a - n cycles periodically through {0, ..., a-1} so neither lim_{n->infty} theta_1(n) nor lim_{n->infty} E_E(N_n) exists (Section 4.2.2, p. 13).

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:MarkovChain` Markov Chain	broad_match	The formulation characterizes recurrence times, correlation decay, and extreme-event clustering in discrete-time regenerative stochastic processes extending Markov chains.

Evidence Sources & Primary Literature

3 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`r837-primary-pdf`	Clustering indices and decay of correlations in non-Markovian models Miguel Abadi, Ana Cristina Moreira Freitas, Jorge Milhazes Freitas	primary paper full_text	ID: https://arxiv.org/abs/1810.03216 sha256:0011d680154a…
`r837-arxiv-abs`	Clustering indices and decay of correlations in non-Markovian models (arXiv abstract) Miguel Abadi, Ana Cristina Moreira Freitas, Jorge Milhazes Freitas	primary paper abstract_only	ID: https://arxiv.org/abs/1810.03216 sha256:079766c8ff44…
`r837-metadata-cr`	Clustering indices and decay of correlations in non-Markovian models (Crossref metadata) Miguel Abadi, Ana Cristina Moreira Freitas, Jorge Milhazes Freitas	primary paper abstract_only	ID: https://arxiv.org/abs/1810.03216 sha256:24b696f974ce…

Ontology & Grounding Topology

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Curation Provenance

Curator: Antigravity

Method: ai_assisted

Model: Gemini

Created: 2026-10-19T15:20:00Z

Updated: 2026-10-19T15:20:00Z

Source File: data/curated/FiniteTimeExtremalIndexClusterSizeAndSojournTimeInRegenerativeProcesses.yaml · Schema: curated_concept.yaml

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