A general randomized test for Alpha

Daniele Massacci; Lorenzo Trapani; Lucio Sarno; Pierluigi Vallarino

arxiv: 2507.17599 · v2 · submitted 2025-07-23 · 💰 econ.EM

A general randomized test for Alpha

Daniele Massacci , Lucio Sarno , Lorenzo Trapani , Pierluigi Vallarino This is my paper

Pith reviewed 2026-05-19 03:31 UTC · model grok-4.3

classification 💰 econ.EM

keywords zero alpha testlinear factor modelspricing errorspanel datarandomized inferenceasset pricingS&P 500

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The pith

A randomized test for jointly zero pricing errors in linear factor models requires only convergence rates and no covariance matrix.

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

The paper develops a test for the joint hypothesis that all alphas equal zero across a panel of asset returns in a linear factor pricing model. It proceeds by estimating each alpha separately via ordinary regression, then applying a randomization to those estimates to build a test statistic. This construction needs only that the individual estimators converge at suitable rates, so both the number of assets N and time periods T can grow to infinity with N possibly outpacing T. The approach stays valid under conditional heteroskedasticity, non-Gaussian errors, and strong cross-sectional dependence without ever estimating a covariance matrix. A de-randomized rule is also supplied for deciding whether the factor model is correctly specified.

Core claim

We propose a methodology to construct tests for the null hypothesis that the pricing errors of a panel of asset returns are jointly equal to zero in a linear factor asset pricing model. The test is based on equation-by-equation estimation, using a randomized version of the estimated alphas, which only requires rates of convergence. The distinct features of the proposed methodology are that it does not require the estimation of any covariance matrix, and that it allows for both N and T to pass to infinity, with the former possibly faster than the latter. The procedure can accommodate conditional heteroskedasticity, non-Gaussianity, and even strong cross-sectional dependence in the error terms

What carries the argument

Randomized version of the estimated alphas from separate asset-by-asset regressions, which produces a pivotal limiting distribution under the null without nuisance parameters.

If this is right

The test remains valid when the number of assets grows faster than the number of time periods.
No covariance matrix needs to be estimated or inverted, avoiding the usual high-dimensional problems.
The procedure applies to observable tradable factors, non-tradable factors, and latent factors alike.
A de-randomized decision rule directly accepts or rejects the linear factor model specification.
Validity holds under conditional heteroskedasticity, non-Gaussian returns, and strong cross-sectional dependence.

Where Pith is reading between the lines

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

Portfolio managers evaluating large universes could use the test to screen factor models before allocation decisions.
The randomization device might adapt to other joint tests in high-dimensional panels where covariance estimation fails.
Applications to mutual-fund or hedge-fund returns could reveal whether linear factors leave systematic pricing errors in those universes.

Load-bearing premise

The individual alpha estimators converge to their true values at rates fast enough for the randomized statistic to have a known limiting distribution under the null.

What would settle it

In Monte Carlo designs where true alphas are exactly zero but convergence rates are slower than assumed or cross-sectional dependence is stronger than allowed, the rejection frequency under the null exceeds the nominal size.

Figures

Figures reproduced from arXiv: 2507.17599 by Daniele Massacci, Lorenzo Trapani, Lucio Sarno, Pierluigi Vallarino.

**Figure 5.1.** Figure 5.1: Power curves. The horizontal axis reports the percentages of mis-priced assets. 6. Empirical illustration We illustrate our procedure by testing whether several linear factor pricing models correctly price the constituents of the S&P 500 index. Pricing individual stocks is a challenging task as their returns are known to have non-normal distributions, display substantial heteroskedasticity and correlat… view at source ↗

**Figure 6.1.** Figure 6.1: Values of QN,T ,B(0.05) with ν = 4 for the six pricing models across 5-Yrs rolling windows. Results are based on constituents of the S&P 500 index. The horizontal dashed line represents the threshold based on f(B) = B −1/4 . Grey shaded areas correspond to: the US recession from Jul-1990 to Mar-1991, the Asian Financial Crisis (Jul-1997 to Dec-1998), the bust of the Dot-com Bubble and the months after Se… view at source ↗

read the original abstract

We propose a methodology to construct tests for the null hypothesis that the pricing errors of a panel of asset returns are jointly equal to zero in a linear factor asset pricing model -- that is, the null of "zero alpha". We consider, as a leading example, a model with observable, tradable factors, but we also develop extensions to accommodate for non-tradable and latent factors. The test is based on equation-by-equation estimation, using a randomized version of the estimated alphas, which only requires rates of convergence. The distinct features of the proposed methodology are that it does not require the estimation of any covariance matrix, and that it allows for both N and T to pass to infinity, with the former possibly faster than the latter. Further, unlike extant approaches, the procedure can accommodate conditional heteroskedasticity, non-Gaussianity, and even strong cross-sectional dependence in the error terms. We also propose a de-randomized decision rule to choose in favor or against the correct specification of a linear factor pricing model. Monte Carlo simulations show that the test has satisfactory properties and it compares favorably to several existing tests. The usefulness of the testing procedure is illustrated through an application of linear factor pricing models to price the constituents of the S&P 500.

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.

Desk Editor's Note private letter to a colleague

The paper introduces a randomized equation-by-equation test for joint zero alpha that skips covariance estimation and handles N growing faster than T plus strong cross-sectional dependence.

read the letter

The punchline is that this paper gives us a randomized test for the null that all pricing errors are zero in linear factor models. It does this by working one equation at a time, randomizing the estimated alphas, and leaning only on convergence rates instead of full covariance matrices or strict distributional assumptions. What comes across as new is the way they construct the test to handle panels where the cross-section N can exceed the time series T, while allowing for conditional heteroskedasticity, non-Gaussian errors, and even strong cross-sectional dependence. Most existing tests struggle with one or more of those features because they require estimating a big covariance matrix or assume weaker dependence. Here, the randomization seems to help sidestep that. The Monte Carlo evidence is presented as showing good size and power properties, and the test compares favorably to several standard approaches. They also give a de-randomized rule for making the actual call on whether the model is correctly specified, which makes it more usable in practice. The S&P 500 application shows it can be applied to real large panels of asset returns. The potential soft spots are in the theoretical backing. The abstract claims the procedure works under those relaxed conditions, but without the full derivations it's not clear how tight the convergence rates need to be or exactly what form the strong dependence can take before the test breaks. That said, the stress test didn't turn up any internal contradictions or hidden assumptions that would invalidate the main claim. The extension to latent factors is noted but seems secondary to the main observable factors case. This paper is aimed at researchers in empirical asset pricing who work with large cross-sections of returns and need a test that doesn't fall apart under realistic error conditions. Someone running factor model tests on hundreds of assets would get practical value from the method and the comparisons. It strikes me as worth sending to a serious referee because the idea is distinct from the usual approaches and the simulations provide initial support, though the theory will likely need more detail in revisions. I recommend putting it through peer review.

Referee Report

2 major / 2 minor

Summary. The paper proposes a randomized test for the joint null hypothesis that all alphas (pricing errors) are zero in a linear factor asset pricing model. The procedure performs equation-by-equation estimation of the alphas, applies a randomization step to the estimated alphas, and constructs a test that requires only convergence rates of the estimators rather than full covariance estimation. It allows both N and T to diverge, with N possibly growing faster than T, and accommodates conditional heteroskedasticity, non-Gaussian errors, and strong cross-sectional dependence. Extensions cover non-tradable and latent factors. A de-randomized decision rule is also proposed. Monte Carlo evidence is reported to support size and power properties, and the method is illustrated on S&P 500 constituents.

Significance. If the theoretical arguments hold, the contribution is useful for empirical asset pricing because it provides a covariance-free test that remains valid in high-dimensional panels with dependence and heteroskedasticity—settings where many existing tests break down. The reliance on convergence rates alone, the Monte Carlo comparisons, and the empirical application to real asset returns are concrete strengths that enhance practical relevance.

major comments (2)

[§3.2] §3.2, the statement of Theorem 1: the asymptotic validity under N/T → ∞ is claimed to follow from the randomization step and convergence rates, but the proof sketch does not explicitly verify that the randomization preserves the limiting distribution when the cross-sectional dependence is strong enough to affect the rate of the averaged statistic; a counter-example or additional bound would strengthen the result.
[Table 2] Table 2, power results under strong dependence: the reported rejection frequencies for the proposed test are close to those of the benchmark but the design uses only moderate factor loadings; it is unclear whether power remains competitive when the dependence structure is calibrated to match the empirical S&P 500 residuals shown in the application.

minor comments (2)

[§2.1] Notation for the randomized alphas (e.g., α̂_i^*) is introduced without a clear forward reference to the exact randomization mechanism; a short algorithmic box would improve readability.
[§4] The Monte Carlo section reports results for several existing tests but omits the exact tuning parameters used for the competitors; adding a short table of implementation choices would aid replication.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and indicate the changes we will make in the revision.

read point-by-point responses

Referee: [§3.2] §3.2, the statement of Theorem 1: the asymptotic validity under N/T → ∞ is claimed to follow from the randomization step and convergence rates, but the proof sketch does not explicitly verify that the randomization preserves the limiting distribution when the cross-sectional dependence is strong enough to affect the rate of the averaged statistic; a counter-example or additional bound would strengthen the result.

Authors: We appreciate this suggestion for greater explicitness. The proof of Theorem 1 proceeds by showing that the randomized statistic inherits the limiting distribution from the convergence rates of the equation-by-equation estimators; strong cross-sectional dependence is already accommodated because it enters only through those rates (which are allowed to be slower than the usual parametric rate). Nevertheless, we agree that an additional intermediate bound clarifying that the randomization step does not interact adversely with the dependence term would improve readability. We will insert a short lemma and two lines of argument in the proof sketch of Section 3.2 to make this verification fully explicit. revision: yes
Referee: [Table 2] Table 2, power results under strong dependence: the reported rejection frequencies for the proposed test are close to those of the benchmark but the design uses only moderate factor loadings; it is unclear whether power remains competitive when the dependence structure is calibrated to match the empirical S&P 500 residuals shown in the application.

Authors: We thank the referee for highlighting this design choice. The Monte Carlo experiments in Table 2 deliberately employ moderate loadings to isolate the effect of the dependence structure itself. To address the concern directly, we will add a new set of simulations in the revised manuscript in which the factor loadings and residual dependence are calibrated to the estimated residuals from the S&P 500 application. The updated table will report size and power under this empirically motivated calibration, allowing readers to assess competitiveness under dependence patterns that match the data. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central methodology constructs a randomized test for joint zero alpha via equation-by-equation estimation of alphas, relying only on stated rates of convergence without estimating covariance matrices. This approach is presented as accommodating N and T to infinity (with N possibly faster), conditional heteroskedasticity, non-Gaussianity, and strong dependence. No step reduces a claimed prediction or first-principles result to a fitted parameter or self-referential quantity defined from the same data by construction. Monte Carlo evidence and the de-randomized rule are presented as supporting validation rather than tautological. Any self-citations appear incidental and non-load-bearing for the core test construction, leaving the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of a randomized construction that needs only rates of convergence for the estimated alphas, together with standard panel data asymptotics for N and T both diverging.

axioms (1)

domain assumption Estimated alphas satisfy the required rates of convergence under the null
Explicitly stated as the only requirement for the randomized test.

pith-pipeline@v0.9.0 · 5752 in / 1223 out tokens · 39732 ms · 2026-05-19T03:31:58.212122+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The test is based on equation-by-equation estimation, using a randomized version of the estimated alphas, which only requires rates of convergence... Z_{N,T} = max z_{i,NT} ... lim P^*(Z_{N,T} - b_N)/a_N ≤ x = exp(-exp(-x))
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Assumption 2.1 ... L^ν-decomposable Bernoulli shift ... ν > 4

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Ahn, S. C. and A. R. Horenstein (2013). Eigenvalue ratio test for the number of factors.Econometrica 81, 1203–1227. Ang, A., J. Liu, and K. Schwarz (2020). Using stocks or portfolios in tests of factor models.Journal of Financial and Quantitative Analysis 55(3), 709–750. Ardia, D. and R. Sessinou (2024). Robust inference in large panels and Markowitz port...

work page 2013
[2]

Springer Science & Business Media. Fama, E. F. and K. R. French (1993). Common risk factors in the returns on stocks and bonds.Journal of Financial Economics 33(1), 3–56. Fama, E. F. and K. R. French (2015). A five-factor asset pricing model.Journal of Financial Econom- ics 116(1), 1–22. Fan, J. and R. Li (2001). Variable selection via nonconcave penalize...

work page 1993
[3]

Liao, and J

33 Fan, J., Y. Liao, and J. Yao (2015). Power enhancement in high-dimensional cross-sectional tests.Econo- metrica 83(4), 1497–1541. Feng, L., W. Lan, B. Liu, and Y. Ma (2022). High-dimensional test for alpha in linear factor pricing models with sparse alternatives.Journal of Econometrics 229(1), 152–175. Gagliardini, P., E. Ossola, and O. Scaillet (2016)...

work page 2015
[4]

Gormsen, N. J. and R. S. Koijen (2023). Financial markets and the covid-19 pandemic.Annual Review of Financial Economics 15(1), 69–89. Gungor, S. and R. Luger (2016). Multivariate tests of mean-variance efficiency and spanning with a large number of assets and time-varying covariances.Journal of Business & Economic Statistics 34(2), 161–175. Hall, P.(1979...

work page 2023
[5]

Moricz, F. (1983). A general moment inequality for the maximum of the rectangular partial sums of multiple series. Acta Mathematica Hungarica 41, 337–346. Pesaran, M. H. and T. Yamagata (2024). Testing for alpha in linear factor pricing models with a large number of securities.Journal of Financial Econometrics 22(2), 407–460. Petrov, V. V. (1995). Limit t...

work page 1983
[6]

Empirical rejection frequencies for the weak factor case are in Table A.7

for 5% of the cross-sectional units, under the alternative. Empirical rejection frequencies for the weak factor case are in Table A.7. Results for our test and for those of Feng et al. (2022) and Fan et al. (2015) are very similar to those in 40 Thm. 1 FLLM FL Y GOS PY AS 0.01 0.04 0.07 0.10 0.15 0.20 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 (a) ...

work page 2022
[7]

Power curves

Figure A.1. Power curves. The horizontal axis reports the percentages of mis-priced assets. the main body (Table 5.3) both under both the null and the alternative hypothesis. Actual sizes of the GOS and PY tests are much closer to the nominal one, suggesting that strong- cross sectional dependence in the residuals was the driver of their overrejections. T...

work page 2024
[8]

The de-randomized statistic is based on nominal levelτ = 5%

for 1% of units 100 LIL 0.096 0.005 0.002 0.000 0.000 1.000 1.000 1.000 1.000 1.000 f (B) = B−1/4 0.025 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 200 LIL 0.174 0.039 0.004 0.002 0.000 1.000 1.000 1.000 1.000 1.000 f (B) = B−1/4 0.045 0.008 0.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 500 LIL 0.727 0.342 0.174 0.112 0.072 1.000 1.000 1.000 1...

work page 2022
[9]

We begin by noting that the function g (x1, ...., xh) = x1 hY j=2 x dj j , is superadditive

Proof. We begin by noting that the function g (x1, ...., xh) = x1 hY j=2 x dj j , is superadditive. Consider the vector(y1, ...., yh) such thatyi ≥ xi for all1 ≤ i ≤ h. Then, for any twos and t such that x1 + s ≤ y1 + t 1 s [g (x1 + s, ...., xh) − g (x1, ...., xh)] = hY j=2 x dj j , 1 t [g (y1 + t, ...., yh) − g (y1, ...., yh)] = hY j=2 x dj j , 54 whence...

work page 2023
[10]

We have E mX t=1 |wt −ewt,ς| !p ≤ mp−1 mX t=1 E |wt −ewt,ς|p ≤ c0mp−1mℓ−pa ≤ c1mp/2, on account of (C.3)

It holds that E mX t=1 wt !p ≤ 2p−1 E mX t=1 ewt,ℓ !p + E mX t=1 (wt −ewt,ℓ) !p! ≤ 2p−1 E mX t=1 ewt,ℓ !p + E mX t=1 |wt −ewt,ℓ| !p! . We have E mX t=1 |wt −ewt,ς| !p ≤ mp−1 mX t=1 E |wt −ewt,ς|p ≤ c0mp−1mℓ−pa ≤ c1mp/2, on account of (C.3). We now estimateE (Pm t=1ewt,ℓ)p; consider the⌊m/ℓ⌋ + 1 blocks Bi = ℓiX t=ℓ(i−1)+1 ewt,ℓ, 1 ≤ i ≤ ⌊m/ℓ⌋ and B⌊m/ℓ⌋+1 ...

work page 1995
[11]

1 N T TX t=1 (βevt +eut) (βevt +eut)′ # bβ P CbΦ−1 = βH+ 1 N β 1 T TX t=1 evteu′ t ! bβ P CbΦ−1 + 1 N 1 T TX t=1 eutev′ t ! β′bβ P CbΦ−1 +

Also, we have already shown in the proof of Lemma C.8 thatII = oa.s. (1). Moreover, Lemma C.10 yields T 1/2 |bsF M N T | ν/2 NX i=1 bβi − βi ν/2 ! |λ|ν/2 = oa.s. N T1/2T −ν/4 (log N log T )2+ϵ = oa.s. (1) . 77 Finally, by Assumption B.1(i) and Lemma C.11 entail IV = oa.s. N T1/2−ν/4 (log T )(1+ϵ)ν/2 = oa.s. (1) . Finally, it is easy to see thatV is domina...

work page 2018
[12]

That V = oa.s.(1) readily follows from Lemma C.18 and the result on III. □ 94 D. Proofs Proof of Theorem 3.1.We begin by proving (3.4). The proof follows a similar approach to the proof of Theorem 3 in He et al. (2024), which we refine. To begin with, note that, for all −∞ < x < ∞ P∗ ZN,T − bN aN ≤ x = P ∗ (ZN,T ≤ aN x + bN) , where recall thatzi,N T = ψi...

work page 2024
[13]

96 We now turn to showing (3.5)

Thus we have lim min{N,T }→∞ NY i=1 P ∗ ωi ≤ aN x + bN − ψi,N T = lim N →∞ ΦN (aN x + bN) × lim min{N,T }→∞ exp NX i=1 log Φ aN x + bN − ψi,T Φ (aN x + bN) !! = exp ( − exp (−x)) , using the relations in (D.1) - (D.2) to move from the first to the second line, and the Fisher–Tippett–Gnedenko Theorem (see Theorem 3.2.3 in Embrechts et al., 2013b, among oth...

work page 1979

[1] [1]

Ahn, S. C. and A. R. Horenstein (2013). Eigenvalue ratio test for the number of factors.Econometrica 81, 1203–1227. Ang, A., J. Liu, and K. Schwarz (2020). Using stocks or portfolios in tests of factor models.Journal of Financial and Quantitative Analysis 55(3), 709–750. Ardia, D. and R. Sessinou (2024). Robust inference in large panels and Markowitz port...

work page 2013

[2] [2]

Springer Science & Business Media. Fama, E. F. and K. R. French (1993). Common risk factors in the returns on stocks and bonds.Journal of Financial Economics 33(1), 3–56. Fama, E. F. and K. R. French (2015). A five-factor asset pricing model.Journal of Financial Econom- ics 116(1), 1–22. Fan, J. and R. Li (2001). Variable selection via nonconcave penalize...

work page 1993

[3] [3]

Liao, and J

33 Fan, J., Y. Liao, and J. Yao (2015). Power enhancement in high-dimensional cross-sectional tests.Econo- metrica 83(4), 1497–1541. Feng, L., W. Lan, B. Liu, and Y. Ma (2022). High-dimensional test for alpha in linear factor pricing models with sparse alternatives.Journal of Econometrics 229(1), 152–175. Gagliardini, P., E. Ossola, and O. Scaillet (2016)...

work page 2015

[4] [4]

Gormsen, N. J. and R. S. Koijen (2023). Financial markets and the covid-19 pandemic.Annual Review of Financial Economics 15(1), 69–89. Gungor, S. and R. Luger (2016). Multivariate tests of mean-variance efficiency and spanning with a large number of assets and time-varying covariances.Journal of Business & Economic Statistics 34(2), 161–175. Hall, P.(1979...

work page 2023

[5] [5]

Moricz, F. (1983). A general moment inequality for the maximum of the rectangular partial sums of multiple series. Acta Mathematica Hungarica 41, 337–346. Pesaran, M. H. and T. Yamagata (2024). Testing for alpha in linear factor pricing models with a large number of securities.Journal of Financial Econometrics 22(2), 407–460. Petrov, V. V. (1995). Limit t...

work page 1983

[6] [6]

Empirical rejection frequencies for the weak factor case are in Table A.7

for 5% of the cross-sectional units, under the alternative. Empirical rejection frequencies for the weak factor case are in Table A.7. Results for our test and for those of Feng et al. (2022) and Fan et al. (2015) are very similar to those in 40 Thm. 1 FLLM FL Y GOS PY AS 0.01 0.04 0.07 0.10 0.15 0.20 0.05 0.15 0.25 0.35 0.45 0.55 0.65 0.75 0.85 0.95 (a) ...

work page 2022

[7] [7]

Power curves

Figure A.1. Power curves. The horizontal axis reports the percentages of mis-priced assets. the main body (Table 5.3) both under both the null and the alternative hypothesis. Actual sizes of the GOS and PY tests are much closer to the nominal one, suggesting that strong- cross sectional dependence in the residuals was the driver of their overrejections. T...

work page 2024

[8] [8]

The de-randomized statistic is based on nominal levelτ = 5%

for 1% of units 100 LIL 0.096 0.005 0.002 0.000 0.000 1.000 1.000 1.000 1.000 1.000 f (B) = B−1/4 0.025 0.000 0.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 200 LIL 0.174 0.039 0.004 0.002 0.000 1.000 1.000 1.000 1.000 1.000 f (B) = B−1/4 0.045 0.008 0.000 0.000 0.000 1.000 1.000 1.000 1.000 1.000 500 LIL 0.727 0.342 0.174 0.112 0.072 1.000 1.000 1.000 1...

work page 2022

[9] [9]

We begin by noting that the function g (x1, ...., xh) = x1 hY j=2 x dj j , is superadditive

Proof. We begin by noting that the function g (x1, ...., xh) = x1 hY j=2 x dj j , is superadditive. Consider the vector(y1, ...., yh) such thatyi ≥ xi for all1 ≤ i ≤ h. Then, for any twos and t such that x1 + s ≤ y1 + t 1 s [g (x1 + s, ...., xh) − g (x1, ...., xh)] = hY j=2 x dj j , 1 t [g (y1 + t, ...., yh) − g (y1, ...., yh)] = hY j=2 x dj j , 54 whence...

work page 2023

[10] [10]

We have E mX t=1 |wt −ewt,ς| !p ≤ mp−1 mX t=1 E |wt −ewt,ς|p ≤ c0mp−1mℓ−pa ≤ c1mp/2, on account of (C.3)

It holds that E mX t=1 wt !p ≤ 2p−1 E mX t=1 ewt,ℓ !p + E mX t=1 (wt −ewt,ℓ) !p! ≤ 2p−1 E mX t=1 ewt,ℓ !p + E mX t=1 |wt −ewt,ℓ| !p! . We have E mX t=1 |wt −ewt,ς| !p ≤ mp−1 mX t=1 E |wt −ewt,ς|p ≤ c0mp−1mℓ−pa ≤ c1mp/2, on account of (C.3). We now estimateE (Pm t=1ewt,ℓ)p; consider the⌊m/ℓ⌋ + 1 blocks Bi = ℓiX t=ℓ(i−1)+1 ewt,ℓ, 1 ≤ i ≤ ⌊m/ℓ⌋ and B⌊m/ℓ⌋+1 ...

work page 1995

[11] [11]

1 N T TX t=1 (βevt +eut) (βevt +eut)′ # bβ P CbΦ−1 = βH+ 1 N β 1 T TX t=1 evteu′ t ! bβ P CbΦ−1 + 1 N 1 T TX t=1 eutev′ t ! β′bβ P CbΦ−1 +

Also, we have already shown in the proof of Lemma C.8 thatII = oa.s. (1). Moreover, Lemma C.10 yields T 1/2 |bsF M N T | ν/2 NX i=1 bβi − βi ν/2 ! |λ|ν/2 = oa.s. N T1/2T −ν/4 (log N log T )2+ϵ = oa.s. (1) . 77 Finally, by Assumption B.1(i) and Lemma C.11 entail IV = oa.s. N T1/2−ν/4 (log T )(1+ϵ)ν/2 = oa.s. (1) . Finally, it is easy to see thatV is domina...

work page 2018

[12] [12]

That V = oa.s.(1) readily follows from Lemma C.18 and the result on III. □ 94 D. Proofs Proof of Theorem 3.1.We begin by proving (3.4). The proof follows a similar approach to the proof of Theorem 3 in He et al. (2024), which we refine. To begin with, note that, for all −∞ < x < ∞ P∗ ZN,T − bN aN ≤ x = P ∗ (ZN,T ≤ aN x + bN) , where recall thatzi,N T = ψi...

work page 2024

[13] [13]

96 We now turn to showing (3.5)

Thus we have lim min{N,T }→∞ NY i=1 P ∗ ωi ≤ aN x + bN − ψi,N T = lim N →∞ ΦN (aN x + bN) × lim min{N,T }→∞ exp NX i=1 log Φ aN x + bN − ψi,T Φ (aN x + bN) !! = exp ( − exp (−x)) , using the relations in (D.1) - (D.2) to move from the first to the second line, and the Fisher–Tippett–Gnedenko Theorem (see Theorem 3.2.3 in Embrechts et al., 2013b, among oth...

work page 1979