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corresponding proof, 𝑄𝜃 ⊗2=0if𝑋is obtained by discrete sampling from an affine process. For the following statement we introduce some more notation. Set ̂𝑋𝜃(𝑡)∶=̂𝑋𝜃(𝑡,𝑡−1), 𝑉𝜃(𝑡)∶=𝑉𝜃(𝑡,𝑡−1),𝑊𝜃(𝑡)∶=𝑊𝜃(𝑡,𝑡−1),𝑋𝜃(𝑡)∶=(𝑋(𝑡),̂𝑋𝜃(𝑡),𝑉𝜃(𝑡))as well as 𝐾𝜃∶=𝐴𝜃̂Σ𝜃 ∶,o(̂Σ𝜃 o)−1, 𝐹𝜃∶=𝐴𝜃−...	https://arxiv.org/abs/2503.05590v1
𝑡𝑡 ∑ 𝑠=1𝑡 ∑ 𝑢=1Cov𝜗[𝑍𝜗(𝑠,𝑠−1),𝑍𝜗(𝑢,𝑢−1)] using a non-parametric kernel estimator, also called heteroskedastic autocorrelation consis- tent (HAC) estimator or Newey–West estimator, see Andrews [8] or Newey and West [87]. Alternatively, extending a result of Francq et al. [58], Boubacar Ma ¨ınassara and Fra...	https://arxiv.org/abs/2503.05590v1
of the form 𝑈𝜗=∫(𝑔𝜗(𝑥)𝑔𝜗(𝑥)⊤−ℎ𝜗(𝑥)ℎ𝜗(𝑥)⊤)𝜇𝜗(𝑑𝑥), 𝑊(𝜗)=∫̃𝑓𝜗(𝑥)𝑑𝜇𝜗(𝑑𝑥). As a consequence, these matrices can be computed as follows if Assumption A’ holds: Algorithm 3.12. Step 1: Since𝑋is a polynomial state space model of order 4, we have that 𝔼𝜃(𝑋(𝑡)𝜆\|\|F𝑡−1)=∑ 𝜇∈ℕ𝑑 4𝑏𝜆,𝜇𝑋(𝑡−1)𝜇,...	https://arxiv.org/abs/2503.05590v1
sequence (̂𝜃𝑐(𝑡))𝑡∈ℕis weakly𝜗-consistent and more- over that (√𝑡(̂𝜃(𝑡)−𝜗),√𝑡(̂𝜃𝑐(𝑡)−𝜗))𝑡∈ℕis jointly asymptotically normal. In this case, since 𝜗∈ int(Θ) , the Lagrange multiplier theorem (see e.g. Fitzpatrick [57, Theorem 17.17]) and consistency imply that, with probability converging to 1, ̂𝜃𝑐(𝑡)s...	https://arxiv.org/abs/2503.05590v1
of a general multivariate L ´evy-driven Ornstein-Uhlenbeck process. 4.1 The Heston stochastic volatility model In the stochastic volatility model of Heston [63], the asset 𝑆=(𝑆(𝑡))𝑡∈ℝ+incorporates a stochas- tic variance component 𝑣=(𝑣(𝑡))𝑡∈ℝ+which is driven by a square-root diffusion. More specif- ically, it i...	https://arxiv.org/abs/2503.05590v1
polynomial state space model in the sense of Definition 2.1. For computational simplicity, we consider fixed drift parameter 𝜇= 0and volatility response parameter 𝛿=1 2so that the log-spot process 𝑌from (4.3) has the form d𝑌(𝑡)=√ 𝑣(𝑡)d𝑊(1)(𝑡), (4.4) 22 which is a martingale. Then Δ𝑌=(𝑌(𝑡)−𝑌(𝑡−1))𝑡∈ℕis th...	https://arxiv.org/abs/2503.05590v1
[88] together with the expressions from Section 3.3 in Al `os and Lorite [6]. 23 estimation framework. We did not test the effect of these model modifications concerning the efficiency of the resulting quasi-maximum likelihood estimator, say by examining the result- ing differences in size of the asymptotic variances o...	https://arxiv.org/abs/2503.05590v1
̂ 𝜎(𝑡)is larger by approximately a factor of 1.5 compared to the isolated estimation. Interestingly however, 05106 10106 15106 20106 161718192021Standard deviation for (t) Isolated estimation of the parameters 05106 10106 15106 20106 16182022Joint estimation of the parameters 05106 10106 15106 20106 0.420.440.46Stand...	https://arxiv.org/abs/2503.05590v1
15106 20106 Number of observations t0.2 0.1 0.00.10.20.30.40.50.6 Corr[,m] Corr[,] Corr*[m,] Figure 4: For each pair of estimator components, the figure displays five independent se- quences of estimator correlations, respectively obtained from the covariance estimates ̂𝑉𝑡[̂𝜃(𝑡)] for𝑡varying between 1 and 20⋅106...	https://arxiv.org/abs/2503.05590v1
of rejection under the null 𝐻0∶𝜎∗= 0.3for the Wald, the LM and the LR test. Asymptotically, these sizes of the tests fall theoretically within the given confidence intervals in the fifth column with a probability of 95%. 𝑝-values of the sizes are given below. scale. To this end, one may define the polynomial state s...	https://arxiv.org/abs/2503.05590v1
Proposition 4.4. Let𝑋be an Ornstein–Uhlenbeck process with background driving L ´evy process 𝐿. Let𝑝∈[1,∞)and suppose that 𝔼(‖𝐿(1)‖𝑝)<∞(or equivalently ∫‖𝑥‖≥1‖𝑥‖𝑝𝜈𝐿(d𝑥)<∞, where 𝜈𝐿is the L ´evy measure of 𝐿). Then 𝔼(‖𝑋(𝑡)‖𝑝)<∞for any𝑡∈ℝ+. If moreover 𝛼(𝑄)>0, i.e.𝑄 has only eigenvalues with positi...	https://arxiv.org/abs/2503.05590v1
on strong mixing properties and all in all less elementary than those introduced in Section 2.1. We will now briefly treat the fulfilment of the Assumptions A’, B, C’ from Section 2.1 for a parametric Ornstein–Uhlenbeck polynomial state space model 𝑋parameterised by some 𝜃∈ Θ, whereΘis a convex and compact subset of ...	https://arxiv.org/abs/2503.05590v1
the short rate, where the short rate can be thought of as some sort of instantaneous forward interest rate with infinitesimal horizon. For details about interest rate models in general and the example considered here see Chapter 14 and Example 14.10 in Eberlein and Kallsen [44], respectively. In our case we assume that...	https://arxiv.org/abs/2503.05590v1
components assumed to be known. Figure 6 shows ten independent sequences (̂𝛿(𝑡))𝑡∈{1,…,𝑇}of the quasi-maximum likelihood estimator, first if the latent mean 𝑚is assumed to be unobservable and secondly if 𝑚is assumed to be observable. Again as in the case of the Heston model, introducing additional observable comp...	https://arxiv.org/abs/2503.05590v1
ten se- quences of asymptotic estimator correlations obtained from ̂𝑉𝑡[̂𝜃(𝑡)]. Black lines show values obtained from the explicit calculations detailed in Section 3.3. of𝛿with the latent mean 𝑚treated as unobservable. Next to a kernel density estimate, we again show the Gaussian density corresponding to a mean of...	https://arxiv.org/abs/2503.05590v1
the state transition matrix 𝐴𝜃of𝑋is given by 𝐴𝜃 𝑖𝑗=−i𝜕𝑢𝑖Ψ𝜃 𝑗(0)for𝑖,𝑗∈{1,…,𝑑}, which follows by differentiation in (5.1) and 𝜑𝜃 𝑡(0)=1 . Moreover, 𝔼𝜃[𝑋𝑖(𝑡+1)𝑋𝑗(𝑡+1)\|F𝑡]=−𝜕𝑢𝑖𝜕𝑢𝑗𝜑𝜃 𝑡(0)=−𝜕𝑢𝑖[(𝜕𝑢𝑗𝜓𝜃(𝑢,𝑡))𝜑𝜃 𝑡(𝑢)]\|\|\|𝑢=0 =−𝜕𝑢𝑖𝜕𝑢𝑗𝜓𝜃(0,𝑡)−(𝜕𝑢𝑖𝜓𝜃(0,𝑡))(𝜕𝑢𝑗𝜓�...	https://arxiv.org/abs/2503.05590v1
definite. 5.2 Proofs for Section 3.1 Proof of Proposition 3.5. For𝑡∈ℕ∗the matrices ̂Σ𝜃(𝑡+1,𝑡)and̂Σ𝜃(𝑡,𝑡)are positive definite because the matrices 𝐶𝜃(𝑡)are positive definite (see Proposition 2.13) and hence all pseudoin- verses occurring in Proposition 2.8 are proper inverses. The result now follows by differ...	https://arxiv.org/abs/2503.05590v1
= −𝑡̃𝑍𝜗(𝑡)−11√𝑡𝑍𝜗(𝑡)and the right-hand side converges in ℙ𝜗-law to−𝑊(𝜗)−1𝑍 by Slutsky’s theorem. Since ℙ𝜗(𝐶𝑡)→1as𝑡→∞, this concludes. The main goal of Sections 5.2.1–5.2.3 is to establish the various conditions from Proposi- tions 5.2 and 5.3 in order to prove Theorems 3.3 and 3.4. The following Section...	https://arxiv.org/abs/2503.05590v1
or Schlemm and Stelzer [100], our approach of using Markovianity of an augmented version of the original process seems to be novel in the context of quasi-maximum likelihood estimation. The usual path to proving asymptotic normality of the quasi-score process consists in using certain strong mixing prop- erties for the...	https://arxiv.org/abs/2503.05590v1
and 𝐷𝜃(𝑡)∶=𝐶𝜃(𝑡)−𝐶𝜃, see Anderson and Moore [7, Problem 4.5]. By iterating this equation one obtains ̂Σ𝜃(𝑡)−̂Σ𝜃=(𝐴𝜃−𝐾𝜃𝐻)𝑡−1(̂Σ𝜃(1)−̂Σ𝜃)Ψ𝜃⊤ 𝑡,1+𝑡−2 ∑ 𝑠=0(𝐴𝜃−𝐾𝜃𝐻)𝑠𝐷𝜃(𝑡−𝑠)Ψ𝜃⊤ 𝑡,𝑡−𝑠. (5.10) Let𝐹𝜃∶=𝐴𝜃−𝐾𝜃𝐻. We now argue that 𝜌(𝐹𝜃)≤𝛼for any𝜃∈ Θ and some𝛼∈ [0,1). Analogously to...	https://arxiv.org/abs/2503.05590v1
tingale difference sequence for 𝑋𝜃from (5.5). Then there exists a matrix 𝐴𝜃∈ℝ𝑑(𝑘+2)×𝑑(𝑘+2)with 𝜌(𝐴𝜃)<1such that𝐴𝜃(𝑡)→𝐴𝜃uniformly in 𝜃at a geometric rate. Moreover, the homogeneous Markov chain 𝑋𝜃,homwith𝑋𝜃,hom(0)∶=𝑋𝜃(0)and𝑋𝜃,hom(𝑡)∶=̂ 𝑎𝜃 1+𝐴𝜃𝑋𝜃,hom(𝑡−1)+𝑁𝜗(𝑡)is a polynomial state spa...	https://arxiv.org/abs/2503.05590v1
let⊕denote the usual Minkowski sum of subsets of ℝ𝑑. Define the set R𝜃∶=∞ ⨁ 𝑠=0(𝐹𝜃)𝑠𝐺𝜃(𝐸)∈B(ℝ𝑑) (5.15) and letR𝜃 𝑎∶=R𝜃+𝑎𝜃. Since𝐸is a connected smooth manifold containing 0, R𝜃∋ 0is a connected smooth manifold too, and the partial sums in (5.15) form an increasing sequence of sets. Suppose that the sta...	https://arxiv.org/abs/2503.05590v1
any𝑝∈[1,∞). Since𝑋has uniformly bounded moments of order 4+𝛿, the claim follows by establishing that ̂𝑋𝜃,aux,𝑎has uniformly bounded moments of order 4+𝛿. This however holds by Lemma A.14. By the same argument, the processes ̃𝑋𝜃and𝑋𝜃,homare also bounded in𝐿4+𝛿. Sincesupp(𝜓)⊇ 𝐻 has non-empty interior, all ...	https://arxiv.org/abs/2503.05590v1
a geometric rate. Proof. As before let 𝑎𝜃denote the state transition vector of 𝑋𝜃and𝑋𝜃,hom, let𝐴𝜃(𝑡)and𝐴𝜃 denote the state transition matrices of 𝑋𝜃and𝑋𝜃,hom, respectively, and let 𝑁𝜗denote the martingale difference sequence in the state space representation of 𝑋𝜃and𝑋𝜃,hom(all under ℙ𝜗). Then𝑋𝜃(...	https://arxiv.org/abs/2503.05590v1
Then the Poisson equation 𝑓(𝑥) =𝑃𝜃𝑔(𝑥)−𝑔(𝑥)+𝑟is fulfilled whenever 𝛼⊤ 𝑔(𝐴𝜃 ⊗2−I) =𝛼⊤ 𝑓. This equation has the unique solution 𝛼⊤ 𝑔=𝛼⊤ 𝑓(𝐴𝜃 ⊗2−I)−1, where the inverse is well-defined because𝜌(𝐴𝜃 ⊗2)<1by Remark 5.7. This finishes the proof. The preceding lemma is the main ingredient of the followi...	https://arxiv.org/abs/2503.05590v1
∑ 𝑠=1𝑈𝜗 𝑡(𝑠)𝑈𝜗⊤ 𝑡(𝑠)=1 𝑡𝑡 ∑ 𝑠=1[𝑃𝜗𝑔𝜗(𝑋𝜗,hom(𝑠−1))−𝑔𝜗(𝑋𝜗,hom(𝑠))][𝑃𝜗𝑔𝜗(𝑋𝜗,hom(𝑠−1))−𝑔𝜗(𝑋𝜗,hom(𝑠))]⊤ , where each summand is an ℝ𝑘×𝑘-valued random variable whose entries are quartic polynomi- als in𝑋𝜗,hom(𝑠−1)and𝑋𝜗,hom(𝑠). Since𝑋𝜗,homis weakly𝑓-ergodic for any quartic polyno...	https://arxiv.org/abs/2503.05590v1
the context of hypothesis testing using the Lagrange Multiplier test, see Section 3.4. This is remedied by a different estimator in Boubacar Ma ¨ınassara [24] and Boubacar Ma¨ınassara et al. [25]. Theorem 5.11 establishes the asymptotic normality condition needed in Proposition 5.3, which is taken from Jacod and Sørens...	https://arxiv.org/abs/2503.05590v1
or 3.6 and similar algebraic manipulations as in equation (5.21) show for any 𝑖,𝑗,𝑙∈{1,…,𝑘}we have 𝜕𝜃𝑙𝑅𝜃 𝑖𝑗(𝑡,𝑡−1)=𝐹𝜃(𝑡−1)𝜕𝜃𝑙𝑅𝜃 𝑖𝑗(𝑡,𝑡−1)𝐹𝜃(𝑡−1)⊤+terms uniformly bounded in 𝜃(5.22) because all third derivatives of 𝐶𝜃(𝑡)are uniformly bounded in 𝜃by Assumption A. Lemma A.10.1 then yields ...	https://arxiv.org/abs/2503.05590v1
in 𝑡and 𝜃for any𝑙∈ {1,…,𝑘}. Since all partial derivatives 𝜕𝜃𝑙𝔼𝜗[𝑓(𝑋𝜃(𝑡)) ∣𝑋(0) =𝑥]are uniformly bounded in 𝑡and𝜃, the sequence (𝔼𝜗[𝑓(𝑋𝜃(𝑡)) ∣𝑋(0) =𝑥])𝑡∈ℕ∗is equicontinuous by the multivariate mean value theorem, see for example Theorem A.12, and it converges pointwise for𝜆𝐸-almost any 𝑥∈𝐸,...	https://arxiv.org/abs/2503.05590v1
Simple differen- tiation as in Propositions 3.5 or 3.6 yields that 𝜕𝜃𝑙𝐹𝜃(𝑡)and𝜕𝜃𝑙𝐺𝜃(𝑡)depend on 𝑡only through 51 ̂Σ𝜃(𝑡,𝑡−1) and its inverse, 𝑆𝜃 𝑗(𝑡,𝑡−1),𝑅𝜃 𝑖𝑗(𝑡,𝑡−1) as well as𝜕𝜃𝑙𝑅𝜃 𝑖𝑗(𝑡,𝑡−1) for𝑖,𝑗,𝑙∈{1,…,𝑘}. Similar manipulations as in equation (5.21) show 𝜕𝜃𝑙𝑅𝜃 𝑖𝑗(𝑡,𝑡...	https://arxiv.org/abs/2503.05590v1
Theorem A.20 are fulfilled and we can deduce sup𝜃∈Θ‖1 𝑡∑𝑡 𝑠=1𝑓(𝑋𝜃(𝑠))−∫𝑓d𝜇𝜃‖→0inℙ𝜗-probability. We are now ready to prove the main result of this section, assumed in Proposition 5.2: Proposition 5.16. There exists a continuous matrix-valued function 𝑊∶ Θ→ℝ𝑘×𝑘such that sup𝜃∈Θ‖1 𝑡∇𝜃𝑍𝜃(𝑡)−𝑊(𝜃)‖→0inℙ...	https://arxiv.org/abs/2503.05590v1
The closed-form expressions for the score process ̃𝑍𝜃 𝑌(𝑡)and the observed Fisher information ∇𝜃̃𝑍𝜃 𝑌(𝑡)are again analogous to the ones given in Propositions 3.5 and 3.6 with 𝐶𝜃in place of 𝐶𝜃(𝑡). We know that 𝐿𝜃(𝑡,𝑡−1)=log𝑞𝜃 𝑡∣𝑡−1(𝑋o(𝑡)∣𝑋o(1),…,𝑋o(𝑡−1)) is a quadratic polynomial in𝑋𝜃(𝑡)wit...	https://arxiv.org/abs/2503.05590v1
𝑡𝑍𝜃(𝑡), respectively. In view of the uniform convergence to 𝑊(𝜃)proven in Proposition 5.16, it follows from Lemma A.21 that 𝑊(𝜃) = ∇𝜃𝐺(𝜃)as well as 𝐺(𝜃) = ∇𝜃𝑄(𝜃), and that sup𝜃∈Θ‖1 𝑡𝑍𝜃(𝑡)−𝐺(𝜃)‖ℙ𝜗− −→0as well assup𝜃∈Θ‖1 𝑡𝐿𝜃(𝑡)−𝑄(𝜃)‖ℙ𝜗− −→0. This establishes the uniform convergence propert...	https://arxiv.org/abs/2503.05590v1
culminates in the following Azuma–Hoeffding-type inequality for 𝑌stat o: Proposition 5.23. Fix𝑢∈ℕ∗and𝜃∈Θand let𝑓∶ℝ(𝑑−𝑚)(𝑢+1)→ℝbe bounded and measur- able. Then there is a constant 𝐾 >0depending on 𝑢,𝑓,𝜃such that we have ℙ𝜃(\|\|\|\|1 𝑡𝑡 ∑ 𝑠=1𝑓[𝑌stat o(𝑠),…𝑌stat o(𝑠+𝑢)]−𝔼𝜃[𝑓(𝑌stat o(0),…,𝑌stat o(𝑢)...	https://arxiv.org/abs/2503.05590v1
𝜃. Since𝑌statis a Gaussian process, Kallsen and Richert [73, Proposition 3.2] yields that the conditional den- sity𝑓𝑠∣𝑠−1 𝜃(𝑦𝑠∣𝑦0,…,𝑦𝑠−1)is the density of the normal distribution with mean ̂𝑌𝜃,stat oevaluated at 𝑌stat o(0)=𝑦0,…,𝑌stat o(𝑠−1)=𝑦𝑠−1and covariance matrix ̂Σ𝜃,stat o(𝑠,𝑠−1) given by the ...	https://arxiv.org/abs/2503.05590v1
by ̂𝜃(𝑡),𝑊(̂𝜃(𝑡))can be computed explicitly by the calculations in Section 3.3 (if Assumption A’ holds) in order to verify the invertibility assumption from Theorem 3.4. A large part of the statistical literature on asymptotic normality of (quasi-)maximum like- lihood estimators in hidden Markov models or general ...	https://arxiv.org/abs/2503.05590v1
continuity statement in Corollary 5.13. Since (̂𝜃(𝑡))𝑡∈ℕis𝜗-consistent, the first claim follows from the continuous mapping theorem. The second follows from Slutsky’s theorem. 61 Proof of Proposition 3.14. For the Wald test statistic, note that since√𝑡(̂𝜃(𝑡)−𝜗)ℙ𝜗-𝑑− −−→𝑁(0,𝑉𝜗), one obtains that√𝑡[𝑅(̂𝜃(�...	https://arxiv.org/abs/2503.05590v1
Burkholder–Davis–Gundy inequality from Theorem A.22, we obtain ‖‖‖∫𝑡 0e𝑄𝑠d𝐿(𝑠)‖‖‖𝑝 𝐿𝑝≤𝑐‖‖‖∫e𝑄⋅id1[0,𝑡]d𝐿‖‖‖𝑝 𝐻𝑝≤𝑐‖‖e𝑄⋅id1[0,𝑡]‖‖𝑝 𝑆∞‖‖𝐿⋅1[0,𝑡]‖‖𝑝 𝐻𝑝 ≤𝑐(sup 𝑠≤𝑡‖‖e𝑄𝑠‖‖𝑝 )‖‖‖‖[𝑀𝐿,𝑀𝐿](𝑡)‖1 2+‖𝐴𝐿‖(𝑡)‖‖‖𝑝 𝐿𝑝 ≤𝑐[𝔼(‖[𝑀𝐿,𝑀𝐿](𝑡)‖𝑝 2)+𝔼(‖𝐴𝐿‖(𝑡)𝑝)] ≤𝑐[𝔼(sup 𝑠≤𝑡‖𝑀𝐿(𝑠)‖�...	https://arxiv.org/abs/2503.05590v1
transition semigroup (𝑃𝑡)𝑡∈ℕ, which is connected to 𝑋via 𝔼(𝑓(𝑋(𝑠+𝑡))\|F𝑠) = (𝑃𝑡𝑓)(𝑋(𝑠))for all𝑠,𝑡∈ℕand bounded, measurable 𝑓∶𝐸→ℝ. As usual, we use the slightly ambiguous notation 𝑃𝑡(𝑥,𝐴)∶=(𝑃𝑡1𝐴)(𝑥)=ℙ𝑥(𝑋(𝑡)∈𝐴)for the𝑡- step transition function, where ℙ𝜈denotes the measure under which 𝑋(0...	https://arxiv.org/abs/2503.05590v1
weak Feller chain bounded in probability on average admits an invariant probability measure, see Meyn and Tweedie [85, Theorem 12.1.2(ii)]. Since 𝐸 ⊆ℝ𝑑 and any compact set in the relative topology on 𝐸is also compact in ℝ𝑑, it suffices to show thatlimsup𝑡∈ℕ∗ℙ𝑥(‖𝑋(𝑡)‖≥𝑀)converges to 0 as 𝑀→∞. But limsup 𝑡→∞ℙ�...	https://arxiv.org/abs/2503.05590v1
that ‖𝐴−1‖−1=𝜎𝑑. Then ‖𝐴−1‖=𝜎−1 𝑑≤𝜎−1 𝑑(𝑑−1 ∏ 𝑘=1𝜎1 𝜎𝑘)=𝜎𝑑−1 1 ∏𝑑 𝑘=1𝜎𝑘=‖𝐴‖𝑑−1 \|det(𝐴)\|. The main content of the following lemmata consists in the fact that powers of a matrix 𝐴 converge geometrically fast to 0 whenever its spectral radius 𝜌(𝐴)is strictly smaller than 1. This basic result exten...	https://arxiv.org/abs/2503.05590v1
particular, 𝐴𝑚→0at a geometric rate as 𝑚→∞. Alternatively, we can also uniformly bound partial derivatives of chains of differentiable matrix functions by a geometric rate, as the following corollary of Lemma A.7 shows: Corollary A.9. Let𝐸be a compact subset of ℝ𝑘and suppose that functions 𝐴𝑡∶𝐸→ℝ𝑑×𝑑and 𝐴∶𝐸→...	https://arxiv.org/abs/2503.05590v1
A.5 we obtain the bound ‖(I𝑑−𝐴(𝑥))−1‖≤‖I𝑑−𝐴(𝑥)‖𝑑−1 \|det(I𝑑−𝐴(𝑥))\|. Since the eigenvalues of I𝑑−𝐴are uniformly bounded away from 0, also \|det(I𝑑−𝐴)\|is uniformly bounded away from 0. Since I𝑑−𝐴is continuous on the compact set 𝐸, it is bounded and so (I𝑑−𝐴)−1is also bounded on 𝐸. Similar to (A.5) we ha...	https://arxiv.org/abs/2503.05590v1
Since𝑑𝑖denotes the dimension of each Jordan block 𝐽𝑖, we have𝑚𝑘=𝑑for𝑚𝑖=𝑑1+⋯+𝑑𝑖. For any𝑥∈𝐸, fix the representation 𝑥=𝑘1(𝑥)𝑣1+⋯+𝑘𝑑(𝑥)𝑣𝑑. Assume now that 𝐴has some eigenvalue 𝜆𝑖with\|𝜆𝑖\|≥1. Since the map 𝐴𝑛is given by 𝐽𝑛with respect to the basis 𝑉, it follows that the components 𝑚𝑖−1+1,…...	https://arxiv.org/abs/2503.05590v1
for \|\|\|𝑀+𝑁\|\|\|≠0: \|\|\|𝑀+𝑁\|\|\|𝑝=𝔼[sup 𝑥∈𝐸‖𝑀𝑥+𝑁𝑥‖𝑝]=𝔼[sup 𝑥∈𝐸(‖𝑀𝑥+𝑁𝑥‖⋅‖𝑀𝑥+𝑁𝑥‖𝑝−1)] ≤𝔼[sup 𝑥∈𝐸(‖𝑀𝑥‖sup 𝑥∈𝐸‖𝑀𝑥+𝑁𝑥‖𝑝−1)]+𝔼[sup 𝑥∈𝐸(‖𝑁𝑥‖sup 𝑥∈𝐸‖𝑀𝑥+𝑁𝑥‖𝑝−1)] ≤(𝔼[sup 𝑥∈𝐸‖𝑀𝑥‖𝑝]1 𝑝+𝔼[sup 𝑥∈𝐸‖𝑁𝑥‖𝑝]1 𝑝 )𝔼[(sup 𝑥∈𝐸‖𝑀𝑥+𝑁𝑥‖𝑝−1)𝑝 𝑝−1 ]𝑝−1 𝑝 =(\|\|\|𝑀\|\|\|+\|\|\|𝑁\|\|\|)\|\|\|...	https://arxiv.org/abs/2503.05590v1
we can assume that 𝑘= 1. Let the function 𝑔∶ℝ𝑑→ℝbe again of the form 𝑔(𝑥) =∑\|𝜆\|≤𝑞𝛼𝜆𝑥𝜆 As in the proof of Lemma A.16, we have ‖𝑔(𝑋(𝑡))−𝑔(𝑌(𝑡))‖‖𝑝 𝑞≤∑\|𝜆\|≤𝑞\|𝛼𝜆\|‖𝑋(𝑡)𝜆−𝑌(𝑡)𝜆‖𝑝 𝑞; so it suffices the show the claim for monomials 𝑔(𝑥)=𝑥𝜆with\|𝜆\|≥1. Now 𝑋(𝑡)𝜆−𝑌(𝑡)𝜆=𝑑 ∏ 𝑗=1𝑋𝑗(𝑡)𝜆𝑗...	https://arxiv.org/abs/2503.05590v1
𝑘\|≤‖𝑡‖‖𝑈(𝑛) 𝑘‖by the Cauchy–Schwarz inequality, we have 𝑘𝑛 ∑ 𝑘=1𝔼[\|𝑡⊤𝑈(𝑛) 𝑛\|21{\|𝑡⊤𝑈(𝑛) 𝑘\|>𝜀}]≤‖𝑡‖2𝑘𝑛 ∑ 𝑘=1𝔼[‖𝑈(𝑛) 𝑘‖21{‖𝑈(𝑛) 𝑘‖≥𝜀 ‖𝑡‖}]𝑛→∞− −−→0 by 2. Thus, the array 𝑡⊤𝑈(𝑛) 𝑘fulfils the conditions 1. and 2. in the case 𝑑= 1, whence 𝑡⊤𝑀𝑛𝑑−→𝑡⊤𝑍. By the Cram ´er–Wold theorem we ...	https://arxiv.org/abs/2503.05590v1
for each 𝑥∈𝐸a sequence of𝑑-dimensional random variables (𝑌𝑥(𝑡))𝑡∈ℕ∗as well as a deterministic 𝑌(𝑥)∈ℝ𝑑are given such that𝑌𝑥(𝑡)and𝑌(𝑥)are continuous in 𝑥. Suppose moreover that 𝑌𝑥(𝑡)ℙ−→𝑌(𝑥)for each𝑥∈𝐸and that for each 𝑥∈𝐸the uniform stochastic equicontinuity condition lim 𝛼→0limsup 𝑡→∞ℙ(sup 𝑑(...	https://arxiv.org/abs/2503.05590v1
less than𝜀 3for some large enough 𝑡independent of 𝑥andℎ, while the term 𝐵becomes less than𝜀 3for any fixed 𝑡if\|ℎ\|is small enough. It follows that 𝜕𝑥𝑗𝑌(𝑥)𝑖=𝑊(𝑥)𝑖𝑗. In Section 4.2 we need multivariate extensions of the well-known Burkholder–Davis–Gun- dy inequality (see Dellacherie and Meyer [36, Theorem ...	https://arxiv.org/abs/2503.05590v1
ℝ𝑚×𝑛-valued process and 𝑋anℝ𝑛-valued semi- martingale. Moreover, let1 𝑝+1 𝑞=1 𝑟for1≤𝑝≤∞and1≤𝑞≤∞. Then ‖‖‖∫𝐻d𝑋‖‖‖𝐻𝑟≤𝑛√𝑚‖𝐻‖𝑆𝑝‖𝑋−𝑋(0)‖𝐻𝑞. Proof. Fix a semimartingale decomposition 𝑋=𝑋(0)+𝑀+𝐴. Then∫𝐻d𝑋=∫𝐻d𝑀+ ∫𝐻d𝐴is a decomposition for ∫𝐻d𝑀. Recall that [∫𝐻d𝑀,∫𝐻d𝑀] =∫𝐻d[𝑀,𝑀]𝐻⊤. We w...	https://arxiv.org/abs/2503.05590v1
Ornstein–Uhlen- beck process for electricity spot price modelling and derivatives pricing. In: Applied Mathematical Finance 14.2, pp. 153–169. [17] F. E. Benth and S. Lavagnini (2021). Correlators of polynomial processes. In: SIAM Journal on Financial Mathematics 12.4, pp. 1374–1415. [18] D. S. Bernstein (2011). Matrix...	https://arxiv.org/abs/2503.05590v1
and R. Van Handel (2011). Consistency of the Maxi- mum Likelihood estimator for general hidden Markov models. In: The Annals of Statis- tics39.1, pp. 474–513. [39] G. R. Duffee (2002). Term premia and interest rate forecasts in affine models. In: The Journal of Finance 57.1, pp. 405–443. [40] D. Duffie, D. Filipovi ´c,...	https://arxiv.org/abs/2503.05590v1
I. Gordin and B. A. Lif ˇsic (1978). Central limit theorem for stationary Markov pro- cesses. In: Dokladi Akademii Nauk SSSR 239.4, pp. 766–767. [61] J. J. Green (1996). Uniform Convergence to the Spectral Radius and Some Related Prop- erties in Banach Algebras. Doctoral thesis, University of Sheffield. [62] P. Hall an...	https://arxiv.org/abs/2503.05590v1
variables. In: Computational Statistics and Data Analysis 53.4, pp. 853–856. [81] R. Lord, R. Koekkoek, and D. J. C. van Dijk (2010). A comparison of biased simulation schemes for stochastic volatility models. In: Quantitative Finance 10.2, pp. 177–194. [82] K. W. Lu (2022). Calibration for multivariate L ´evy-driven O...	https://arxiv.org/abs/2503.05590v1
Fields 151.1, pp. 173–190. [103] K. Singleton (2001). Estimation of affine asset pricing models using the empirical char- acteristic function. In: Journal of Econometrics 252.1-3, pp. 61–70. [104] M. Sørensen (2012). Estimating functions for diffusion-type processes. In: M. Kessler, A. Lindner, M. Sørensen (Eds.), Stat...	https://arxiv.org/abs/2503.05590v1
arXiv:2503.05880v1 [math.ST] 14 Feb 2025Asymptotic properties of maximum composite likelihood est imators for max-stable Brown-Resnick random ﬁelds over a ﬁxed-domain Nicolas CHENAVIER∗and Christian Y. ROBERT† March 11, 2025 Abstract Likelihoodinferenceformax-stablerandomﬁeldsisingeneralimposs iblebecausetheirﬁnite-dim...	https://arxiv.org/abs/2503.05880v1
be distinguished: microergodic and non-microergodic parameters. A parameter is said to be microergo dic if, for two diﬀerent values of it, the two corresponding Gaussian measures are orthogonal [ 25,33]). It is non-microergodic if, even for two diﬀerent values of it, the two corresponding Gaussian measure s are equival...	https://arxiv.org/abs/2503.05880v1
instead of Xunder a ﬁxed-domain framework. They established that the asymptotic distribution theory for nonaﬃne gis somewhat richer than in the Gaussian case (i.e. whengis an aﬃne transformation). Although the variogram-based estima tors are not MLE or MCLE, this study shows that their asymptotic properties can diﬀer s...	https://arxiv.org/abs/2503.05880v1
this paper, we consider the classof spatial max-stableBrown-R esnickrandomﬁelds ( d= 2) associated with isotropic fractional Brownian random ﬁelds as deﬁned in [ 27]. Because the sites where the random ﬁelds are observed are very rarely on grids in practice, we consider ed a random sampling scheme. We assume a Poisson ...	https://arxiv.org/abs/2503.05880v1
(see Theorem 2 in [ 9]). In this paper we generalize this result to the max-stable Brown-R esnick random ﬁeld which is built as the pointwise maximum of an inﬁnite number of isotro pic fractional Brownian ﬁelds (see Theorem 3). Using approximations of the pairwise and triplewise CL objective fu nctions, we then derive ...	https://arxiv.org/abs/2503.05880v1
has the following form Y(x) = exp(W(x)−γ(x)), x∈R2, (2.2) whereWis an isotropic fractional Brownian ﬁeld. With this choice, the max-st able random ﬁeld ηis stationary while Wis not but has (linear) stationary increments (see [ 27]). 2.2 Delaunay triangulation In this section, we recall some known results on Poisson-Del...	https://arxiv.org/abs/2503.05880v1
( x1,x2) such that the following conditions hold: x1∼x2in Del(PN), x1∈B,andx1/√recedesequalx2, where/√recedesequaldenotes the lexicographic order. When B=C, we simply write EN,C=EN. For a Borel subsetBinR2, letDTN,Bbe the set of triples ( x1,x2,x3) satisfying the following properties ∆(x1,x2,x3)∈Del(PN), x1∈B,andx1/√re...	https://arxiv.org/abs/2503.05880v1
not a surprise since the probability that x1andx2belong to the same cell of the canonical tessellation ofthe max-stablerandomﬁeld tends to 1. Indeed, in a commoncell, t he valuesof the max-stablerandom ﬁeld are generated by the same isotropic fractional Brownian rand om ﬁeld. 3.1.2 Pairs of increments Let us now consid...	https://arxiv.org/abs/2503.05880v1
fV(η) 2,NandV(η) 3,N. Theorem 3 Letα∈(0,1). Then, as N→ ∞, √ 3 3N−(2−α)/4V(η) 2,NP→cV2/summationdisplay j≥1/summationdisplay k>jLZk\j(0) √ 2 2N−(2−α)/4V(η) 3,NP→cV3/summationdisplay j≥1/summationdisplay k>jLZk\j(0). In the above theorem we have assumed that α∈(0,1) whereas, in general, α∈(0,2). Such an assumption is im...	https://arxiv.org/abs/2503.05880v1
and explain how we deﬁne the MCLEs for the other paramete r. 12 Whenα0is assumed to be known, the pairwise and triplewise maximum (tapered ) CL estimators of σ0, denoted by ˆ σj,N, are respectively deﬁned as a solution of the maximization problems max σ∈Sσℓj,N(σ,α0), j= 2,3. Whenσ0is assumed to be known, the pairwise a...	https://arxiv.org/abs/2503.05880v1
o fσ2 0andα0(when the other parameter is known) are consistent in our inﬁll asymptotic setup. T hey have rates of convergence pro- portional to Nα0/4for ˆσ2 2,Nand log(N)Nα0/4for ˆα2,Nthat diﬀer from the expected ratesof convergence N1/2and log(N)N1/2as in [35] for the isotropic fractional Brownian ﬁeld. Second the typ...	https://arxiv.org/abs/2503.05880v1
is given in ( 2.1). SinceW(x2−x1)∼ N(0,2γ(x2−x1)) and since 2 γ(x2−x1) = σ2\|\|x2−x1\|\|α=σ2dα, we deduce that E/bracketleftbigg I/bracketleftbiggY(x2−x1) z2≤Y(0) z1/bracketrightbigg Y(0)/bracketrightbigg = Φ/parenleftbigg σ−1d−α/2log/parenleftbiggz2 z1/parenrightbigg +1 2σdα/2/parenrightbigg = Φ/parenleftbigg u+1 2σdα/2/p...	https://arxiv.org/abs/2503.05880v1
V(η) 2,N LetH2(u) :=u2−1 be the Hermite polynomial of degree 2 so that V(η) 2,N=1/radicalbig \|EN\|/summationdisplay (x1,x2)∈ENH2/parenleftBig U(η) x1,x2/parenrightBig . To deal with the right-hand side, we write H2(U(η) x1,x2) =H2/parenleftBigg U(η) x1,x2+γ(x2)−γ(x1) σ/bardblx2−x1/bardblα/2/parenrightBigg −2U(η) x1,x2γ(...	https://arxiv.org/abs/2503.05880v1
0,σ2 V2/parenrightbig . To deal with the second term, we consider the tessellation ( Ck,j)k/\egatio\slash=j≥1, whereCk,j⊂Cis deﬁned in Eq. (3.5). Notice also that the set JC={(k,j) :Ck,j/ne}ationslash=∅}is a.s. ﬁnite. Adapting the proof of Proposition 1 in [9], we can prove that, for any ( k,j)∈ JC, √ 3 3N−(2−α)/41/rad...	https://arxiv.org/abs/2503.05880v1
σ/bardblx3−x1/bardblα/2 and N(3) x1,x2,x3=/summationdisplay j≥1/summationdisplay k/\egatio\slash=jI /logicalordisplay i≥1Zi(x1) =Zj(x1),/logicalordisplay i≥1Zi(x2) =Zk(x2),x1orx2/∈Ck,j  ×U(Wj) x1,x2 σ/bardblx3−x1/bardblα/2Zk\j(x1) +/summationdisplay j≥1/summationdisplay k/\egatio\slash=jI /logicalordisplay i≥1Zi(...	https://arxiv.org/abs/2503.05880v1
1+a 2u+o(a)/parenrightBig . 26 We deal below with each term of Eq. ( 5.4). First, we notice that ϕ(v(u))v(−u) =ϕ(u)/parenleftBig 1−a 2u+o(a)/parenrightBig/parenleftBiga 2−u/parenrightBig +o(a) =−uϕ(u)/parenleftBig 1−a 2/parenleftbig u+u−1/parenrightbig +o(a)/parenrightBig +o(a) and that e−auϕ(v(−u))v(u) = (1 −au+o(a))ϕ...	https://arxiv.org/abs/2503.05880v1
1R1 R11/parenrightBigg−1/parenleftBigg v1,2(u1,2) v1,3(u1,3)/parenrightBigg . Since∂ ∂σvi,j(ui,j) =1 σvi,j(−ui,j) and∂ ∂σai,j=1 σai,j, 29 we deduce that σ∂ ∂σlog/parenleftbigg −∂3 ∂z1∂z2∂z3Vx1,x2,x3(z1,z2,z3)/parenrightbigg =−a1,2u1,2−a1,3u1,3−2−1 1−R2 1(v1,2(u1,2)−R1v1,3(u1,3))v1,2(−u1,2) −1 1−R2 1(v1,3(u1,3)−R1v1,2(u...	https://arxiv.org/abs/2503.05880v1
to Proposition 8) 1 \|EN\|/summationdisplay (x1,x2)∈ENk0/summationdisplay k=0k/summationdisplay j=0\|U(η) x1,x2\|j/bardblx2−x1/bardbl(k+1−j)α/2P→0, and1 \|EN\|/summationdisplay (x1,x2)∈EN(U(η) x1,x2)2P→1, we deduce that \|ˆα2,N−α\|logNP→0. From Eq. ( 5.5), we have /parenleftbigg 1+1 2(ˆα2,N−α)logN(1+oP(1)/parenrightbigg1 \|EN\|/...	https://arxiv.org/abs/2503.05880v1
for large kth-nearest neighbor balls. J. Appl. Probab., 59(3):880–894, 2022. [8] N. Chenavier and C. Y. Robert. Central limit theorems for squar ed increment sums of fractional Brownian ﬁelds based on a Delaunay triangulation in 2 D.WP, 2025. [9] N. Chenavier and C. Y. Robert. Limit theorems for squared incre ment sums...	https://arxiv.org/abs/2503.05880v1
Detecting correlation efficiently in stochastic block models: breaking Otter’s threshold by counting decorated trees Guanyi Chen∗Jian Ding†Shuyang Gong‡Zhangsong Li§ March 11, 2025 Abstract Consider a pair of sparse correlated stochastic block models S(n,λ n, ϵ;s) subsampled from a common parent stochastic block model ...	https://arxiv.org/abs/2503.06464v1
J. Ding is partially supported by the National Key R&D program of China (No. 2023YFA1010103), the NSFC Key Program (No. 12231002) and the Now Cornerstone Science Foundation through the XPLORER PRIZE. ‡Peking University. Part of the work was carried out when S. Gong was visiting Duke University. S. Gong would like to th...	https://arxiv.org/abs/2503.06464v1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 C.7 Proof of Lemma 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 C.8 Proof of Lemma C.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 C.9 Proof of Lemma C.4 . . . . . . . . . . . . . . . . . . ...	https://arxiv.org/abs/2503.06464v1
algorithms were proposed when the correlation exceeds an arbitrarily small constant [DL22+, DL23+]. The separation between the sparse and dense regimes above, roughly speaking, depends on whether the average degree grows polynomially or sub-polynomially. In addition, another important direction is to establish lower bo...	https://arxiv.org/abs/2503.06464v1
is to study the following hypothesis testing problem: determine whether ( A, B) is sampled from PnorQn, where Qnis the distribution of two independent stochastic block models S(n,λs n, ϵ). In our previous work [CDGL24+] (where we considered symmetric SBMs with kcommunities for k≥2), we considered the problem of testing...	https://arxiv.org/abs/2503.06464v1
1.5. In [CR24], the authors seem to tend to feel that in the logarithmic degree regime the correlated SBMs belong to the same algorithmic universality class of sparse correlated Erd˝ os-R´ enyi model, and thus all inference tasks are computationally impossible when s <√α. However, our result suggests a different possib...	https://arxiv.org/abs/2503.06464v1
detection threshold is still√α. However, as shown in Theorem 1.3, we show that in the very supercritical regime (i.e., when the signal strength of community is sufficiently large), the correlation detection threshold is indeed influenced by the community structure and thus is strictly below Otter’s threshold. Intuitive...	https://arxiv.org/abs/2503.06464v1
marginally both AandBare Erd˝ os-R´ enyi graphs G(n,λs n) and their edge correlation is given by Cov( Aπ∗(i),π∗(j), Bi,j) = [1+ o(1)]s. A natural attempt is to count the (centered) graphs Hin both AandBfor each unlabeled graph H, i.e., we consider the statistics gH=X S∼=HY (i,j)∈E(S)(Ai,j−λs n)X K∼=HY (i,j)∈E(K)(Bi,...	https://arxiv.org/abs/2503.06464v1
combine the subgraph counts relevant to both tasks. To be more precise, our approach involves counting a carefully chosen family of unlabeled multigraphs, which (informally speaking) is formed by attaching non-backtracking paths to an unlabeled tree; the flexibility to choose the place of attachment enrich the enumerat...	https://arxiv.org/abs/2503.06464v1
edge set of H. We say His a subgraph of G, denoted by H⊂G, ifV(H)⊂V(G) and E(H)⊂E(G). We sayφ:V(H)→V(S) is an injection, if for all ( i, j)∈E(H) we have ( φ(i), φ(j))∈E(S). ForH, S⊂ K n, denote by H∩Sthe graph with vertex set given by V(H)∩V(S) and edge set given by E(H)∩E(S), and denote by S∪Hthe graph with vertex set...	https://arxiv.org/abs/2503.06464v1
and has no cycles. We say a pair ( T,R(T)) is a rooted tree with root R(T), ifTis a tree and R(T)∈V(T). For a rooted tree Tandu∈V(T), we define DepT(u) =Dist T(R(T), u) to be 8 the depth of uinT. For u, v∈V(T), denote by u→v(or equivalently, v←u) ifvis the children of u, and denote by u ,→v(or equivalently, v←- u) ifvi...	https://arxiv.org/abs/2503.06464v1
to choose δ,∆ in Theorem 1.3 and make several assumptions on various auxiliary parameters that will be used throughout the paper. To this end, we first need the following results on several enumeration problems regarding unlabeled trees which were established in [Ott48]. 9 Lemma 2.1. Denote VNas the set of unlabeled tr...	https://arxiv.org/abs/2503.06464v1
and the red vertices constitute the pairing P(H); right: example of a multigraph S⊩AHwhere the yellow parts are the self-avoiding paths attached to T(S). Given S1⊩AH, S2⊩BHwhere His a decorated tree, we may write ϕS1,S2(A, B) =ϕS1(A)ϕS2(B), where for S⊩H ϕS(X) =Y (i,j)∈E(T(S))Xi,j−λs nq λs n(1−λs n)·ιℵY k=1Y (i,j)∈E(Lk...	https://arxiv.org/abs/2503.06464v1
as incorporated in the following lemma. Lemma 2.9. For each T∈ Tℵ, define S(T) ={W1, . . . , W M}where W1, . . . , W Mare given as in Theorem 2.8. Define H=H(ι,ℵ, ℓ, M ;Tℵ,S(Tℵ)). We then have \|H\| ≥ α−1exp ι(log log( ι−1))4ℵ . Proof. The result of Lemma 2.9 follows directly from Theorems 2.7 and 2.8. We now describ...	https://arxiv.org/abs/2503.06464v1
we have u̸∈V(T(S))\ L(T(S)). This suggests that L(eS)⊂ L(T(S)). We now show that L(eS) =L(S). Clearly we have L(S)⊂ L(eS). In addition, for all u∈ L(eS)⊂ L(T(S)), denote ( u, v) to be the edge in T(S). For all 1 ≤i≤ιℵ, we must have u̸∈V(Li(S))\ EndP( Li(S)) since otherwise the degree of uineSis at least 2. This suggest...	https://arxiv.org/abs/2503.06464v1
bit more about the construction of our statistic f(A, B) and the seemingly daunting choices of Tℵ,S(Tℵ). Recall that we are in the supercritical region ϵ2λs > 1 where weak community recovery in Aand Bis possible. Also, assuming that all the community labelings in AandB(we denote them as σA andσB) are known to us, it is...	https://arxiv.org/abs/2503.06464v1
the baseline as (non-rigorously speaking) the enumeration of such graphs depends on the number of vertices of G∪and the expectation in each case depends on the number of edges of G∪. •If the two trees T(K1),T(K2) (or T(S1),T(S2)) do not completely overlap (see Figure 2(c)), there will also be additional cycles and thus...	https://arxiv.org/abs/2503.06464v1
(i,j)∈E(S1)∪E(S2)max uχ1(i,j),χ2(i,j), vχ1(i,j),χ2(i,j) . (3.8) Lemma 3.6. Given a tree Tand a subset U⊂V(T)such that Dist T(u,v)≥dfor all u,v∈U (u̸=v), we have Eh ϕT(A)2·Y u∈Uσui ≤2\|U\|·ϵd\|U\|/2. (3.9) 18 The proofs of Lemmas 3.4, 3.5 and 3.6 are incorporated in Sections C.3, C.4 and C.5 of the appendix, respectively. ...	https://arxiv.org/abs/2503.06464v1
Plugging (3.17) into (3.15) yields the desired lower bound for EPid[fA]. For the upper bound, recall that for all S1, S2∈R∗ Hsuch that id∈ A(S1, S2), we have S1∩S2is a tree containing T(S1)∪T(S2). Thus, denoting P(S1) ={(u1, u2), . . . , (u2ιℵ−1, u2ιℵ)}, there must exist paths L1, . . . ,L2ιℵwith ui∈EndP( Li) such that...	https://arxiv.org/abs/2503.06464v1
Using Lemmas 3.2 and 3.5 (with V=∅), we see that (3.23) is bounded by (below we denote E(S)i,j= 0 if ( i, j)̸∈E(S)) 25τ(G∪)+10ℵY (i,j)∈E(G∪)√n√ ϵ2λsE(S1)i,j+E(S2)i,j−2 . Using the fact that X (i,j)∈E(G∪)(E(S1)i,j+E(S2)i,j−2) = 2( ℵ −1 +ιℓℵ)−2\|E(G∪)\|, we obtain that (3.23) is further bounded by 25τ(G∪)+10ℵnℵ−1+ιℓℵ−\|E(...	https://arxiv.org/abs/2503.06464v1
≤ \|E(eS1)\|+\|E(eS2)\| ≤2ℓιℵ+ 2ℵ, we obtain τ(G∪)−\|E(G∪)\| ℓ/2= τ(G∪)−\|E(G∪)\|−ℵ ℓ −\|E(G∪)\|+ℵ ℓ ≥ −2ℵ ℓ−2ℓιℵ+2ℵ ℓ=−2ιℵ −4ℵ ℓ, as desired. We can now complete the proof of Lemma 3.9. Proof of Lemma 3.9. Based on Lemma 3.10, we have (in what follows we say Gsatisfies (3.27) if (3.27) holds after replacing G∪byGand we denote...	https://arxiv.org/abs/2503.06464v1
T(H))2 n2(ℵ+ℓιℵ)X (S,K)∈PHEQ[ϕS(A)ϕK(A)]2 (4.10) +X H,I∈Hs2(ℵ−1)(ϵ2λs)2ℓιℵAut( T(H)) Aut( T(I)) n2(ℵ+ℓιℵ)X (S,K)∈QH,IEQ[ϕS(A)ϕK(A)]2 . (4.11) We first control (4.10). Note that for ( S, K)∈PH, there exist self-avoiding paths {Lu:u∈ Vert(P(S))}satisfying u∈EndP( Lu) for all u∈Vert(P(S)) such that S∩K=T(S)⊕(⊕u∈Vert(P...	https://arxiv.org/abs/2503.06464v1
all (S, K)̸∈P∗ H,Isuch that eS∪eK=G∪andL(S)∪L(K)⊂V(S∩K), we have 2ιℵ −τ(G∪)≤2ℓιℵ+2ℵ−\|E(G∪)\| ℓ/2. (4.16) Proof. The proof is highly similar to the proof of Lemma 3.10, and we omit further details here for simplicity. Using Lemma 4.3, we have X S,K̸∈P∗ H,IEQ[ϕS(A)ϕK(A)]≤X \|E(G∪)\|≤2ℓιℵ+2ℵ G∪satisfies (4.16)X S∪K=G∪EQ[ϕS(A...	https://arxiv.org/abs/2503.06464v1
G≥2 contains T(K1) and the other contains T(K2). In conclusion, when ( S1, S2, K1, K2)∈R∗ H,Iand id∈ A ⋆(S1, S2;K1, K2) one of the two following conditions must hold: (i) S1∩S2, K1∩K2are two 32 disjoint trees containing T(S1)∪T(S2),T(K1)∪T(K2) respectively; (ii) S1∩K2, K1∩S2are two disjoint trees containing T(S1)∪T(K2)...	https://arxiv.org/abs/2503.06464v1
paths. The proof is similar and we omit further details. Proof of Lemma 4.5. Define GT=G≥2∪T(S1)∪T(S2)∪T(K1)∪T(K2). Note that we have assumed L(G∪)⊂V(G≥2). Define Γ to be the number of elements in {Li(S1),Li(S2),Li(K1),Li(K2) : 1≤i≤ιℵ} that are included in GT. It is straightforward to check that τ(G∪)−τ(GT)≥4ιℵ−Γ and \|...	https://arxiv.org/abs/2503.06464v1
. , Mιℵ(S)such that the following conditions hold: (1)V(T(S))⊂[n]\ JA; (2)EndP( Mi(S)) ={ui, vi}where ui, vi∈V(T(S))and the neighbors of ui, viinMi(S)are in JA; (3) Denoting P(S) ={(u1, v1), . . . , (uιℵ, vιℵ)}, there exists a graph isomorphism φ:S→Hsuch thatφmaps T(S)toT(H),maps P(S)toP(H)and maps Mk(S)toMk(H)for1≤k≤ι...	https://arxiv.org/abs/2503.06464v1

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