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REVIEW 3 major objections 4 minor 66 references

Retained hidden excess generates memory in price-limited markets

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read When a stock closes at its daily price limit, the unobserved part of the move carries over to the next day and creates memory even though the shocks driving prices each day are independent.

desk verdict The theoretical core—clipping-induced retained excess as a memory mechanism—is new, plausible, and well-supported by simulation; the empirical section does not independently test it. read the letter →

arxiv 2608.08625 v1 pith:73QMPOPE submitted 2026-08-09 cond-mat.stat-mech q-fin.TR

Debraj Das This is my paper · ORCID
classification cond-mat.stat-mechq-fin.TR
keywords pricelimitshiddenexcesstemporalmemoryheavy-tailedshocksregularvariationsingle-big-jumpprinciplepersistence-reversalasymmetrystochasticrecursion
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that the unobserved overshoot created when a daily price move is clipped at an exchange-imposed limit can itself be the source of apparent memory. The author introduces a minimal recursion in which the next day's latent return is an independent shock plus a retained fraction of yesterday's hidden excess. For heavy-tailed shocks and a wide price band, a close at the limit is typically caused by one dominant shock, whose age and overshoot determine the following day. The paper derives a same-sign next-day drift proportional to the band width, a finite probability of closing at the same limit again, and a power-law-suppressed chance of hitting the opposite limit, and reports qualitative agreement with exchange data.

What carries the argument

The engine is the single-dominant-shock decomposition of a limit-close history. An upper-limit close on day $t$ is ascribed to a shock of age $j$ (with $j=0$ fresh and $j\ge1$ inherited through retained excess). To survive $j$ rounds of clipping and retention, that shock must exceed $CB_j$, where $B_j=\sum_{m=0}^{j}\lambda^{-m}$; the threshold-exceedance ratio $V$ converges to a Pareto($\nu$) variable. The age distribution weights histories by $B_j^{-\nu}/Z(\lambda,\nu)$, and the retained fraction $A_j=\sum_{m=0}^{j}\lambda^{m}$ converts the overshoot into next-day drift and persistence. This machinery turns the intractable clipped recursion into a tractable mixture of single-shock histories.

What would settle it

Simulate or observe a heavy-tailed price-limited market at two large band widths $C_1<C_2$; the model predicts $m_+(C_2)/m_+(C_1)\approx C_2/C_1$ and a same-limit persistence probability that tends to a fixed positive constant as $C$ grows. If the mean next-day response flattens with $C$, or if persistence tends to zero, the single-dominant-shock retained-excess mechanism is falsified.

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Extended reading notes

Core claim

The central claim is that clipping alone does not create memory; clipping plus partial retention does. In the model $X_{t+1}=\epsilon_{t+1}+\lambda(X_t-\mathrm{clip}(X_t,-C,C))$, the hidden excess $L_t=X_t-\mathrm{clip}(X_t,-C,C)$ is the only coupling between days, so with $\lambda=0$ returns are independent. For symmetric shocks with regularly varying tails, the stationary law of $X_t$ inherits the tail index $\nu$ but has its tail amplitude multiplied by $(1-\lambda^\nu)^{-1}$. In the wide-band limit, conditioning on an upper-limit close yields $m_+(C)/C\to M(\lambda,\nu)>0$ for any $\lambda>0$, with $M\sim\lambda/(\nu-1)$ as $\lambda\to0$; the same-limit persistence probability $P_{++}(C)$ approaches the finite value $1-\pi_0$; and the opposite-limit reversal probability satisfies $P_{+-}(C)\sim k C^{-\nu}\Psi(\lambda,\nu)$. The same statements mirror for lower-limit closes.

Load-bearing premise

The wide-band predictions all assume that a limit close comes from one dominant shock, with all other shocks negligible and the latent state on that shock's day equal to the shock itself; that assumption is only asymptotically justified and is not accurate for narrow bands.

Editorial extensions

If this is right

  • After a limit close, the mean next-day return has the same sign as the close and grows linearly with the band width $C$, for any nonzero retention $\lambda$.
  • The probability of closing at the same limit the next day approaches the finite value $1-\pi_0$ in the wide-band limit, while hitting the opposite limit is suppressed as $C^{-\nu}$.
  • Without retention ($\lambda=0$) there is no memory and the normalized response vanishes; any positive retention, however small, gives a positive response.
  • The stationary latent return inherits the tail index of the driving shocks but with an amplitude enlarged by $(1-\lambda^\nu)^{-1}$.
  • Exchange data from price-limited stocks show the predicted same-sign next-day response and its increase across wider bands, though the quantitative fit is incomplete.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The persistence-reversal asymmetry is a general signature: any clipped observable whose hidden part feeds back with a delay should show the same pattern, so the predictions could be tested in order-flow, queue, or neural saturation settings.
  • The empirical asymmetry between upper and lower limits suggests a natural extension with direction-dependent retention; panic-driven declines would then produce stronger memory than news-driven rallies.
  • Because the wide-band formulas are parameter-free once $(\lambda,\nu)$ are fixed, a clean test is to calibrate $\lambda$ at the widest band and check whether it predicts the response slope at all narrower bands; the author's own data show this fails at $C=2\%$, pointing to finite-band corrections.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes a minimal latent-state model in which clipping of daily returns at an exchange-imposed band [−C,C] creates a hidden excess L_t = X_t − R_t, and a fraction λ of that excess is retained in the next day's latent return. Even with independent driving shocks, the retention couples successive days and generates memory. For symmetric shocks with regularly varying tails, the stationary latent return preserves the tail index with an enhanced amplitude (1−λ^ν)^{-1}. In the wide-band limit C≫s_ε, limit closes are argued to admit a single-dominant-shock description; from this the paper derives the next-day mean response, the same-limit persistence probability, and the opposite-limit reversal probability, all as functions of λ and the tail index ν. Simulations with Student-t shocks support the asymptotic formulas. The empirical section uses NSE limit-close data to compare the predicted mean response with data for C=2%,5%,10%,20%. The paper concludes that the data are qualitatively consistent with the predicted same-sign response and its increase across wider bands.

Significance. If the theoretical claims hold, the paper identifies a genuinely new and minimal mechanism for temporal memory: clipping conceals part of a fluctuation, and retaining that concealed part couples otherwise independent time steps. The derivations in Secs. III-V are nontrivial and internally consistent: Eq. (6) for the tail-amplitude renormalization, Eq. (28) for the linear-in-C response, Eq. (32) for the finite persistence limit, and Eq. (33) for the power-law reversal are all specific, falsifiable predictions, and the simulation comparisons in Figs. 2, 4, and 5 support them. A particular strength is that the wide-band results depend only on λ and ν, not on the full shock distribution. The empirical section is carefully constructed (seven-stage exclusion procedure, corporate-action handling, independent determination of ν), but the validation of the memory mechanism in real markets is incomplete for the reasons detailed in the major comments. The theoretical contribution alone is valuable for a statistical-mechanics audience; the empirical part needs substantial reworking before it can be used as evidence for the central claim.

major comments (3)
  1. [§VI and Appendix E] The empirical validation does not provide an out-of-sample test of the model. The effective retention coefficient λ_eff is estimated from the same-boundary persistence at C=20% using the wide-band formula (32), and the same band is then used in Fig. 6 to compare the theoretical response (28). At C=2%, 5%, and 10%, Table III reports pooled persistence 0.7885, 0.4478, and 0.2674, all above the maximum Q_max(3)≈0.1681 allowed by the wide-band asymptote, as Appendix E itself notes. The theoretical lines drawn in Fig. 6 at those bands are therefore outside the model's validity domain. Consequently no band independently tests the mechanism: C=20% is in regime but not independent, while the narrower bands are independent but out of regime. The empirical section should calibrate on a hold-out subset, predict at a different band, or provide finite-C simulations that cover the observed persistence values.
  2. [§IV-V and Fig. 5] The single-dominant-shock replacement in Eqs. (7)-(9) is the load-bearing approximation for the wide-band predictions (28), (32), and (33). It is justified only asymptotically for C/s_ε→∞, and the paper does not provide finite-C corrections. The empirical persistence values at C=2%,5%,10% lie far above the wide-band ceiling, which shows that finite-C effects are not small in the data. As a result, the apparent agreement of the theoretical lines with the empirical responses in Fig. 6 at intermediate bands is not evidence for the asymptotic theory; it is a comparison between data outside the regime and a formula not intended for that regime. Adding finite-C simulation results, or at least a quantitative estimate of the error of the dominant-shock approximation, is necessary to support the cross-band consistency claim.
  3. [Appendix E] Even for the one admissible band, C=20%, the direction-resolved persistence is inconsistent with the symmetric model: Table III gives P^{emp}_{++}=0.1325 and P^{emp}_{−−}=0.2045, and the text states that the lower-limit estimate separately remains above the theoretical ceiling. The pooled calibration therefore masks a directional violation of the model's symmetry assumption. Since the same band is then used for the response comparison, the single acceptable data point is not cleanly in the model's support.
minor comments (4)
  1. [Fig. 2 and equations] Several axis labels and equations show corrupted symbols (e.g., '10□3' and 'x□(ν+1)'); the final version should fix these rendering issues.
  2. [§VI] The statement that both parameters are fixed 'without fitting the empirical post-limit-close mean responses' is technically true but potentially misleading; the text should state explicitly that λ_eff is calibrated from persistence at C=20% before the response comparison, and that this is the only band used for calibration.
  3. [Data availability] For a paper with a quantitative empirical section, 'available upon request' is weak; depositing the processed daily observations, the band-history reconstruction, and the analysis code would strengthen reproducibility.
  4. [Table III] The directional asymmetry in persistence is reported but not tested statistically; a confidence interval or test for P^{emp}_{++} vs. P^{emp}_{−−} would make the claimed deviation from the symmetric model more precise.

Circularity Check

1 steps flagged · score 4.0 of 10

Theory self-contained; empirical response at C=20% is in-sample because λ_eff is fitted from persistence at that same band, and narrower bands lie outside the wide-band regime.

  1. fitted input called prediction [Section VI (λ_eff estimation and Fig. 6 caption)]
    "The retention coefficient is estimated from the same-limit persistence probability at the widest return band C=20%; the narrower bands exceed the maximum persistence allowed by the wide-band asymptote and therefore cannot be used for this inversion. At C=20%, 217 of the 1494 upper- and lower-limit closes are followed by another close at the same return limit. Using Eq. (32), we solve 1−π_0(λ_eff,3)=1−1/Z(λ_eff,3)=217/1494 and obtain λ_eff≈0.942. ... The blue dashed line in (a) shows the wide-band asymptotic prediction m_+(C)∼C M(λ_eff,ν), with ν=3 and λ_eff=0.942"

    At C=20%, λ_eff is not an independent input: it is inverted from the same-limit persistence count at that exact band via Eq. (32). The same λ_eff is then inserted into Eq. (28) to draw the 'theoretical prediction' line for m_+(C) and m_−(C) at C=20% in Fig. 6. The quantitative response at the only band where the wide-band model is admissible is therefore a calibrated in-sample consequence of the persistence statistic, not a prediction from independently fixed parameters. The other bands do not supply an out-of-sample test: Appendix E states that pooled persistence at C=2%,5%,10% exceeds Q_max(3)=0.1681 and 'cannot be described by the large-C asymptote for any admissible value of λ', so the wide-band lines drawn there are outside the model's stated validity domain.

full rationale

The analytical derivation chain is self-contained and not circular. Equations (1)-(3) define clipping, hidden excess, and retention; the stationary tail (6) is obtained from a contractive recursion with regularly varying inputs (Appendix A); the wide-band results (28), (32), (33) follow from the stated single-dominant-shock asymptotics and are verified by independent Student-t simulations in Figs. 2, 4, and 5. I found no self-definitional relation, no load-bearing self-citation, and no imported uniqueness theorem; the author's earlier works appear only as contextual references. The empirical section, however, contains a fitted-input-called-prediction step: λ_eff is calibrated from persistence at C=20% (Appendix E) and the same value produces the response lines at C=20% in Fig. 6. In addition, Appendix E explicitly concedes that C=2%,5%,10% violate the wide-band persistence ceiling, so the comparisons at those bands are outside the stated validity domain rather than independent validations. Thus the empirical support is partly in-sample; nevertheless, the central theoretical claim that retained hidden excess creates memory is independently derived and simulated, so the circularity is moderate, not structural.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The model's analytical claims depend on regular variation, contractive recursion theory, the single-dominant-shock approximation, and Pareto limits for threshold exceedances. The empirical comparison adds two fitted parameters (nu, lambda_eff) and two hand-chosen tolerances. The hidden excess and latent return are unobservable constructs; their only evidence is the model's predictive success, which is qualitative.

free parameters (4)
  • Effective retention coefficient lambda_eff = 0.942
    Estimated from pooled same-limit persistence at C=20% using Q(lambda,3)=217/1494 (Eq. 32, Appendix E); not fitted to mean responses, but calibrates the model to the empirical persistence data.
  • Tail index nu = 3 (representative benchmark)
    Fixed at the inverse-cubic benchmark from MLE fits to 13 global equity indices (Appendix D); independent of NSE limit-close data but chosen from external empirical data.
  • Circuit identification tolerance tau = 0.25%
    One-sided tolerance used to classify a day as a circuit event (Appendix C.1); chosen by hand and affects which days enter the sample.
  • Residual band exceedance tolerance = 0.01C
    Ceiling on how far a same-day or next-day return may exceed C before exclusion (Appendix C.2, criterion 5); a hand-chosen data-cleaning parameter that can suppress persistence counts.
assumptions (5)
  • domain assumption Driving shocks epsilon_t are i.i.d., symmetric, have finite first moment, and are regularly varying with tail density f(epsilon) ~ a|epsilon|^{-(nu+1)} for nu>1.
    Central to heavy-tail results; stated in Sec. II after Eq. (5).
  • standard math The stochastic recursion X_{t+1}=epsilon_{t+1}+lambda h(X_t) is contractive and admits a unique stationary distribution under E[log(1+|epsilon|)]<infinity.
    Invokes Diaconis-Freedman (Ref. 56) in Sec. III; also relies on the recursion belonging to the class in Buraczewski et al. (Ref. 59), Appendix A.
  • ad hoc to paper In the wide-band limit C/s_epsilon -> infinity, a limit close is generated by a single dominant shock; histories with two or more order-C shocks have asymptotically negligible probability.
    This is the single-dominant-shock approximation introduced in Sec. IVA and used for all post-limit-close observables; it is plausible for heavy tails but not proven for this clipped recursion.
  • domain assumption The stationary density p(x) is ultimately monotone in each tail.
    Needed to convert survival tail (A12) into density tail (A13) via the monotone density theorem, Appendix A.
  • ad hoc to paper The threshold-exceedance ratio V converges to a Pareto(nu) variable independent of shock age and scale.
    Classical extreme-value limit; used in Sec. IVC Eq. (18).
invented entities (2)
  • Hidden excess L_t = X_t - R_t
    purpose: Carries the unobserved part of a clipped latent return to the next day; the source of memory.
    It is a latent construct defined by the model; never directly observed. The model's predictions (persistence, response) are indirect falsifiable handles, but the paper provides no direct measurement.
  • Latent return X_t
    purpose: Represents the unconstrained return that would occur absent the price band; clipping maps it to R_t.
    A theoretical state variable; not directly observable, but the model's dynamics are defined on it.

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Cite this review

Pith. "Pith review of Retained hidden excess generates memory in price-limited markets." pith.science (2026). https://pith.science/paper/73QMPOPE

@misc{pith2026260808625,
  author       = {Pith},
  title        = {Pith review of: Retained hidden excess generates memory in price-limited markets},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/73QMPOPE}},
  note         = {Machine review of arXiv:2608.08625}
}
read the original abstract

The daily return of a stock is often restricted to an exchange-imposed band to curb extreme fluctuations. Any attempted price movement beyond this band is clipped, leaving an unobserved excess. We introduce a minimal stochastic latent-state model in which a fraction of this hidden excess is retained for the next day. This retention generates memory, even though the daily stochastic driving shocks are independent. For symmetric driving shocks with regularly varying tails, the stationary latent return preserves the tail index of the noise, but has an enhanced tail amplitude. In the wide-band limit, a close of the daily return at either limit of the band admits a single-dominant-shock description. We show that after such an event, the mean return on the following day has the same sign and grows proportionally to the band width, while the probability of reaching the same limit again approaches a finite value. Reaching the opposite band limit on the following day requires a second extreme shock of opposite sign and is power-law suppressed. Simulations support these analytical predictions. Empirical data from stocks subject to daily price limits are qualitatively consistent with the predicted same-sign response and its increase across wider price bands.

Figures

Figures reproduced from arXiv: 2608.08625 by the authors.

Figure 1
Figure 1. FIG. 1. Possible intraday evolution of the running return [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Positive-tail collapse of the stationary latent den [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Representative discrete-time trajectory of the latent [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Post-upper-limit-close response [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Next-day returns following NSE limit closes from [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Cross-market robustness of the heavy-tail benchmark for 13 global equity indices over their available histories between [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

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Works this paper leans on

66 extracted references · 63 canonical work pages

  1. [1]

    circuit limit

    Data sources and circuit identification Daily open, high, low, close, volume, and previous- close values for equities listed on the National Stock Ex- change (NSE) of India were obtained from NSE’s public data archives using thenselibPython package [62]. The raw data reported by the exchange are not adjusted for corporate actions such as stock splits and ...

    work page 2007

  2. [2]

    Exclusion criteria and derived statistics Starting from the base universe, daily observations were excluded according to the following criteria, ap- plied in sequence: (1)Low-price exclusion.Observations withS t orS t−1 below INR 10 were excluded, remov- ing a deep-penny-stock regime in which several of the largest apparent band overshoots were traced to ...

    work page 1990

  3. [3]

    J. P. Bouchaud and M. M´ ezard. Universality classes for extreme-value statistics.J. Phys. A: Math. Gen., 30(23):7997–8015, 1997. 13 TABLE II. Empirical return statistics, fitted Student’stdistribution parameters, and the derived unstandardized scale param- eters across global stock market indices. Empirical Student’stFit Unstandardized Scale IndexN µ R,i...

    work page 1997

  4. [4]

    (b) Maximum-likelihood estimatesν i with 95% moving-block bootstrap confidence intervals obtained fromB= 2000 resamples with block lengthb= 20 trading days

    Its survival tail obeys Pr(|Z|> z)∝z −3. (b) Maximum-likelihood estimatesν i with 95% moving-block bootstrap confidence intervals obtained fromB= 2000 resamples with block lengthb= 20 trading days. The red dashed line marks the inverse-cubic benchmarkν= 3, and the black dotted line marks the unweighted cross-index mean ¯ν= 3.15. TABLE III. Empirical count...

    work page 2000

  5. [5]

    S. N. Majumdar, A. Pal, and G. Schehr. Extreme value statistics of correlated random variables: A pedagogical review.Phys. Rep., 840:1–32, 2020

    work page 2020

  6. [6]

    D. V. Lindley. The theory of queues with a single server. Math. Proc. Camb. Phil. Soc., 48(2):277–289, 1952

    work page 1952

  7. [7]

    Boxma, A

    O. Boxma, A. L¨ opker, M. Mandjes, and Z. Palmowski. A multiplicative version of the Lindley recursion.Queueing Syst., 98(3-4):225–245, 2021

    work page 2021

  8. [8]

    Abad and R

    D. Abad and R. Pascual. Holding back volatility: Cir- cuit breakers, price limits, and trading halts. In H. K. Baker and H. Kiymaz, editors,Market Microstructure in Emerging and Developed Markets: Price Discovery, In- formation Flows, and Transaction Costs, pages 303–324. John Wiley & Sons, 2013

    work page 2013

Show all 66 references
  1. [9]

    Chandrasekhar

    S. Chandrasekhar. Stochastic problems in physics and astronomy.Rev. Mod. Phys., 15(1):1–89, 1943

    1943

  2. [10]

    Gu´ erin, N

    T. Gu´ erin, N. Levernier, O. B´ enichou, and R. Voituriez. Mean first-passage times of non-Markovian random walk- ers in confinement.Nature, 534(7607):356–359, 2016

    2016

  3. [11]

    Giuggioli

    L. Giuggioli. Exact spatiotemporal dynamics of con- fined lattice random walks in arbitrary dimensions: A 14 century after Smoluchowski and P´ olya.Phys. Rev. X, 10(2):021045, 2020

    2020

  4. [12]

    Condamin, O

    S. Condamin, O. B´ enichou, and M. Moreau. First- passage times for random walks in bounded domains. Phys. Rev. Lett., 95(26):260601, 2005

    2005

  5. [13]

    B´ enichou and R

    O. B´ enichou and R. Voituriez. From first-passage times of random walks in confinement to geometry-controlled kinetics.Phys. Rep., 539(4):225–284, 2014

    2014

  6. [14]

    Chevalier, O

    C. Chevalier, O. B´ enichou, B. Meyer, and R. Voi- turiez. First-passage quantities of Brownian motion in a bounded domain with multiple targets: a unified ap- proach.J. Phys. A: Math. Theor., 44(2):025002, 2011

    2011

  7. [15]

    Meyer, C

    B. Meyer, C. Chevalier, R. Voituriez, and O. B´ enichou. Universality classes of first-passage-time distribution in confined media.Phys. Rev. E, 83(5):051116, 2011

    2011

  8. [16]

    Godec and R

    A. Godec and R. Metzler. Universal proximity effect in target search kinetics in the few-encounter limit.Phys. Rev. X, 6(4):041037, 2016

    2016

  9. [17]

    Giuggioli, S

    L. Giuggioli, S. Sarvaharman, D. Das, D. Marris, and T. Kay. Multi-target Search in Bounded and Heteroge- neous Environments: A Lattice Random Walk Perspec- tive. In D. Grebenkov, R. Metzler, and G. Oshanin, ed- itors,Target Search Problems, pages 107–133. Springer Nature Swit...

    2024

  10. [18]

    D. S. Grebenkov. Paradigm shift in diffusion-mediated surface phenomena.Phys. Rev. Lett., 125(7):078102, 2020

    2020

  11. [19]

    D. S. Grebenkov. Surface hopping propagator: An alter- native approach to diffusion-influenced reactions.Phys. Rev. E, 102(3):032125, 2020

    2020

  12. [20]

    J. M. Harrison and M. I. Reiman. Reflected brownian motion on an orthant.Ann. Probab., 9(2):302–308, 1981

    1981

  13. [21]

    Redner.A Guide to First-Passage Processes

    S. Redner.A Guide to First-Passage Processes. Cam- bridge University Press, Cambridge, 2001

    2001

  14. [22]

    A. J. Bray, S. N. Majumdar, and G. Schehr. Persistence and first-passage properties in nonequilibrium systems. Adv. Phys., 62(3):225–361, 2013

    2013

  15. [23]

    S. N. Majumdar. Universal first-passage properties of discrete-time random walks and L´ evy flights on a line: Statistics of the global maximum and records.Physica A, 389(20):4299–4316, 2010

    2010

  16. [24]

    S. N. Majumdar, P. Mounaix, and G. Schehr. Survival probability of random walks and L´ evy flights on a semi- infinite line.J. Phys. A: Math. Theor., 50(46):465002, 2017

    2017

  17. [25]

    T. G. Mattos, C. Mej ´ ıa-Monasterio, R. Metzler, and G. Oshanin. First passages in bounded domains: When is the mean first passage time meaningful?Phys. Rev. E, 86(3):031143, 2012

    2012

  18. [26]

    A. Pal, I. Eliazar, and S. Reuveni. First passage under restart with branching.Phys. Rev. Lett., 122(2):020602, 2019

    2019

  19. [27]

    Das and L

    D. Das and L. Giuggioli. Discrete space-time reset- ting model: application to first-passage and transmission statistics.J. Phys. A: Math. Theor., 55(42):424004, 2022

    2022

  20. [28]

    Sarmiento, B

    Y. Sarmiento, B. Walter, D. Das, S. Mahapatra, ´E. Rold´ an, and R. J. Harris. First-passage-time asymmetry for biased run-and-tumble processes.J. Phys. A: Math. Theor., 58(49):495002, 2025

    2025

  21. [29]

    Das and L

    D. Das and L. Giuggioli. Tethering effects on first-passage variables of lattice random walks in linear and quadratic focal point potentials.Phys. Rev. E, 113(4):044102, 2026

    2026

  22. [30]

    Sabhapandit and S

    S. Sabhapandit and S. N. Majumdar. Freezing transition in the barrier crossing rate of a diffusing particle.Phys. Rev. Lett., 125(20):200601, 2020

    2020

  23. [31]

    Singh, S

    P. Singh, S. Santra, and A. Kundu. Extremal statistics of a one-dimensional run and tumble particle with an absorbing wall.J. Phys. A: Math. Theor., 55(46):465004, 2022

    2022

  24. [32]

    Bresque, D

    L. Bresque, D. Das, and ´E. Rold´ an. Run-and-tumble exact work statistics in a lazy quantum measurement en- gine: stochastic information processing.Phys. Rev. Lett., 134(20):200402, 2025

    2025

  25. [33]

    Angelani

    L. Angelani. Run-and-tumble particles, telegrapher’s equation and absorption problems with partially reflect- ing boundaries.J. Phys. A: Math. Theor., 48(49):495003, 2015

    2015

  26. [34]

    D. S. Grebenkov. Probability distribution of the bound- ary local time of reflected brownian motion in euclidean domains.Phys. Rev. E, 100(6):062110, 2019

    2019

  27. [35]

    Metzler and J

    R. Metzler and J. Klafter. Boundary value problems for fractional diffusion equations.Physica A, 278(1-2):107– 125, 2000

    2000

  28. [36]

    Das and L

    D. Das and L. Giuggioli. Dynamics of lattice random walk within regions composed of different media and in- terfaces.J. Stat. Mech.: Theory Exp., 2023(1):013201, 2023

    2023

  29. [37]

    Chaigneau and D

    A. Chaigneau and D. S. Grebenkov. First-passage times to anisotropic partially reactive targets.Phys. Rev. E, 105(5):054146, 2022

    2022

  30. [38]

    P. C. Bressloff. Stochastically switching diffusion with partially reactive surfaces.Phys. Rev. E, 106(3):034108, 2022

    2022

  31. [39]

    Gu´ erin, M

    T. Gu´ erin, M. Dolgushev, O. B´ enichou, and R. Voi- turiez. Imperfect narrow escape problem.Phys. Rev. E, 107(3):034134, 2023

    2023

  32. [40]

    Dybiec, E

    B. Dybiec, E. Gudowska-Nowak, E. Barkai, and A. A. Dubkov. L´ evy flights versus l´ evy walks in bounded do- mains.Phys. Rev. E, 95(5):052102, 2017

    2017

  33. [41]

    Garbaczewski and V

    P. Garbaczewski and V. Stephanovich. Fractional Lapla- cians in bounded domains: Killed, reflected, censored, and taboo L´ evy flights.Phys. Rev. E, 99(4):042126, 2019

    2019

  34. [42]

    N. C. Keim, J. D. Paulsen, Z. Zeravcic, S. Sastry, and S. R. Nagel. Memory formation in matter.Rev. Mod. Phys., 91(3):035002, 2019

    2019

  35. [43]

    J. P. Sethna, K. Dahmen, S. Kartha, J. A. Krumhansl, B. W. Roberts, and J. D. Shore. Hysteresis and hier- archies: Dynamics of disorder-driven first-order phase transformations.Phys. Rev. Lett., 70(21):3347–3350, 1993

    1993

  36. [44]

    N. C. Keim and S. R. Nagel. Generic transient memory formation in disordered systems with noise.Phys. Rev. Lett., 107(1):010603, 2011

    2011

  37. [45]

    J. D. Paulsen, N. C. Keim, and S. R. Nagel. Multi- ple transient memories in experiments on sheared Non- Brownian suspensions.Phys. Rev. Lett., 113(6):068301, 2014

    2014

  38. [46]

    K. L. Galloway, E. G. Teich, X. G. Ma, C. Kammer, I. R. Graham, N. C. Keim, C. Reina, D. J. Jerolmack, A. G. Yodh, and P. E. Arratia. Relationships between struc- ture, memory and flow in sheared disordered materials. Nat. Phys., 18(5):565–570, 2022

    2022

  39. [47]

    Shohat, D

    D. Shohat, D. Hexner, and Y. Lahini. Memory from coupled instabilities in unfolded crumpled sheets.Proc. Natl. Acad. Sci. U.S.A., 119(28):e2200028119, 2022

    2022

  40. [48]

    S. H. Chan, K. A. Kim, and S. G. Rhee. Price limit per- formance: evidence from transactions data and the limit 15 order book.J. Empir. Finance, 12(2):269–290, 2005

    2005

  41. [49]

    K. M. Wong, X. W. Kong, and M. Li. The magnet effect of circuit breakers and its interactions with price limits. Pac.-Basin Finance J., 61:101325, 2020

    2020

  42. [50]

    K. A. Kim and S. G. Rhee. Price limit performance: evidence from the Tokyo stock exchange.J. Finance, 52(2):885–901, 1997

    1997

  43. [51]

    Berkman and J

    H. Berkman and J. B. T. Lee. The effectiveness of price limits in an emerging market: Evidence from the Korean Stock Exchange.Pac.-Basin Finance J., 10(5):517–530, 2002

    2002

  44. [52]

    Vezzani, E

    A. Vezzani, E. Barkai, and R. Burioni. Single-big- jump principle in physical modeling.Phys. Rev. E, 100(1):012108, 2019

    2019

  45. [53]

    H¨ oll and E

    M. H¨ oll and E. Barkai. Big jump principle for heavy- tailed random walks with correlated increments.Eur. Phys. J. B, 94(10):216, 2021

    2021

  46. [54]

    R. N. Mantegna and H. E. Stanley. Scaling be- haviour in the dynamics of an economic index.Nature, 376(6535):46–49, 1995

    1995

  47. [55]

    Gopikrishnan, V

    P. Gopikrishnan, V. Plerou, L. A. Nunes Amaral, M. Meyer, and H. E. Stanley. Scaling of the distribution of fluctuations of financial market indices.Phys. Rev. E, 60(5):5305–5316, 1999

    1999

  48. [56]

    Plerou, P

    V. Plerou, P. Gopikrishnan, L. A. Nunes Amaral, M. Meyer, and H. E. Stanley. Scaling of the distribu- tion of price fluctuations of individual companies.Phys. Rev. E, 60(6):6519–6529, 1999

    1999

  49. [57]

    R. Cont. Empirical properties of asset returns: stylized facts and statistical issues.Quant. Finance, 1(2):223– 236, 2001

    2001

  50. [58]

    J. P. Bouchaud and M. Potters.Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management. Cambridge University Press, Cambridge, 2 edition, 2003

    2003

  51. [59]

    Diaconis and D

    P. Diaconis and D. Freedman. Iterated random functions. SIAM Rev., 41(1):45–76, 1999

    1999

  52. [60]

    N. H. Bingham, C. M. Goldie, and J. L. Teugels.Regu- lar Variation. Cambridge University Press, Cambridge, 1987

    1987

  53. [61]

    J. P. Bouchaud, A. Matacz, and M. Potters. Leverage effect in financial markets: the retarded volatility model. Phys. Rev. Lett., 87(22):228701, 2001

    2001

  54. [62]

    Buraczewski, E

    D. Buraczewski, E. Damek, and M. Mirek. Asymptotics of stationary solutions of multivariate stochastic recur- sions with heavy tailed inputs and related limit theorems. Stoch. Process. Their Appl., 122(1):42–67, 2012

    2012

  55. [63]

    Embrechts, C

    P. Embrechts, C. M. Goldie, and N. Veraverbeke. Subex- ponentiality and infinite divisibility.Z. Wahrschein- lichkeitstheorie verw Gebiete, 49(3):335–347, 1979

    1979

  56. [64]

    Embrechts and C

    P. Embrechts and C. M. Goldie. On convolution tails. Stoch. Process. Their Appl., 13(3):263–278, 1982

    1982

  57. [65]

    Tanmay and A

    R. Tanmay and A. Mishra.nselib 2.5.1: A python library to get NSE India data, 2026. Available at PyPI: https://pypi.org/project/nselib/

    2026

  58. [66]

    Yahoo! finance – business finance, stock market, quotes, news, 2026

    Yahoo! Finance. Yahoo! finance – business finance, stock market, quotes, news, 2026. Accessed: 2026-08-01

    2026

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