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in (1.4). It turns out to be purposeful to decompose ℓn,t(θ) =Mn,t(θ) +Bn,t(θ) (2.4) into a martingale term and a drift term by adding and subtracting the sum of the F(k−1)/n- conditional expectation of each increment of ℓn,t(θ), i.e. Bn,t(θ) =⌊nt⌋X k=1Eρ0 log pρ0+θ 1/n(X(k−1)/n,Xk/n) pρ0 1/n(X(k−1)/n,Xk/n) X(k−... | https://arxiv.org/abs/2503.07022v1 |
summands that appear in the definition of ℓn(θ)andBn(θ). We already incorporate in the statement that parts of I2(θ)andI8(θ)equal±log(β2/α2)on their respective interval which are the only contributing expressions to the variance (for θ≪1/√n) and have to be dealt with seperately. As will be seen later, these are exactly... | https://arxiv.org/abs/2503.07022v1 |
thorough control of the drift Bn(θ)on each slice is analytically highly challenging: As soon as |θ−ρ0| ≍n−1/2,alltermsI1(θ),...,I9(θ)are of the same stochastic order of magnitude. At the same time, the likelihood function ℓn(θ)cannot be expanded into a Taylor series due to its discontinuities in the parameter θ. Althou... | https://arxiv.org/abs/2503.07022v1 |
1{Lρ0 1(X)>0}=OPρ01√n . (3.5) For the proof of this√n-consistency, recall the normalized log-likelihood ℓn(θ)given in (1.4) and fix an arbitrary ϵ >0. Asℓn(0) = 0 andˆρn−ρ0∈Argsupz∈Rℓn(z/n), we know that |ˆρn−ρ0|> K/√nimplies that sup|θ|>K/√nℓn(θ)≥0. Consequently, we are going to show that limsup K→∞limsup n→∞Pρ0 √n|... | https://arxiv.org/abs/2503.07022v1 |
1(X)>0)≤Pρ0(Ac 6|A3) +Pρ0(Ac 3|Lρ0 1(X)>0) 14 and Pρ0(Ac 3|Lρ0 1(X)>0)≤Pρ0(Ac 3|A2) +Pρ0(Ac 2|Lρ0 1(X)>0). Both of the last estimates follow by the law of total probability applied to P(· |Lρ0 1(X)>0) and noting that A2,A3⊂ {Lρ0 1(X)>0}. After those preliminaries, we now start with the main part of the proof of (3.5), ... | https://arxiv.org/abs/2503.07022v1 |
(recall dα,β<0) −sup θ∈Sn,jNn(θ)≥ − sup θ∈Sn,jnX k=1dα,β 1{ρ0+L/√n≤X(k−1)/n<ρ0+θ} =−dα,βnX k=11{ρ0+L/√n≤X(k−1)/n<ρ0+2j/√n} To continue bounding supθNL n(θ)onA(n)and establish Claim II, we distinguish the two cases 2j≤n1/4and2j> n1/4when counting the observations in the interval [ρ0+ L/√n,ρ0+ 2j/√n). The reason for trea... | https://arxiv.org/abs/2503.07022v1 |
log( n)) C1n(1∧2j/(2√n)−CFLn2/3 ≤C3n−1/8(log(n))2+C3X j≥Mj2−j/2, for a constant C3=C3(α,β,γ,ξ, Γ)>0andnlarge enough. To complete the discussion for the first summand in (3.6), we now make the choice of Mby taking it large enough to satisfiy C4P j≥Mj2−j/2< ϵand then to take nlarge enough to have the first summand also <... | https://arxiv.org/abs/2503.07022v1 |
variation. Due to Scheff ´e’s theorem (Tsybakov (2009), Lemma 2.1) the second inequality in the statement can then be deduced from the inequality Z R pρ0+θ 1/n(X(k−1)/n,y)−pρ0 1/n(X(k−1)/n,y) dy ≥C1 α,βθ√nexp −(X(k−1)/n−ρ0)2 min{α2,β2}/n−C2 α,βK! ,(3.14) where C1 α,β,C2 α,β>0are suitable constants independent of nandθ.... | https://arxiv.org/abs/2503.07022v1 |
is somewhat remarkable as NL nin (F.4) was given as a part of I5(θ)which was the dominant one outside the n−1/2-environment of ρ0, but is no longer part of the dominant term M1 nwithin the n−1/2-environment. Defining n-CONSISTENCY AND STABLE POISSON-TYPE CONVERGENCE OF THE MLE 21 the metric ρn(θ,θ′) =p C1n|θ−θ′|, by th... | https://arxiv.org/abs/2503.07022v1 |
property, which was introduced in R´enyi (1963) and studied in Aldous and Eagleson (1978), is (slightly) stronger than mere convergence in law. It applies in particular for Ebeing the Skorohod space D([0,1]). 4.2. Proof of Proposition 1.3. Before starting the proof, we want to draw attention to a particular feature of ... | https://arxiv.org/abs/2503.07022v1 |
infill asymptotics and does not require a certain nestedness condition on the filtration (that is not valid in our setup). As the described result in Jacod (1997) only covers a continuous (in time) limit, we have to do some modification of Theorem 4.1in Jacod (2003) that covers limit processes with jumps but does not a... | https://arxiv.org/abs/2503.07022v1 |
reveals that the F-conditional law of the processes ℓin Proposition 4.1 and the first coordinate of (4.7) are the same. In order to derive the characteristics of (4.7), we need a preliminary result. Here, we slightly abuse notation and passagewise explicitly highlight the dependence of random vari- ables on their respe... | https://arxiv.org/abs/2503.07022v1 |
t(X)z+ 1/β2 Fs =κ+ 1N(s,z+ 1)−κ+ 1Lρ0s(X)z+ 1/β2 which yields ˆE Mt|Fs⊗ F′′ s =Msa.s. Let(sn)n∈Nbe a decreasing sequence with sn↘s. Then by right-continuity of Mand the tower property of conditional expectation, together with dominated convergece that is applicable by the Burkholder-Davis-Gundy inequality as Mis squ... | https://arxiv.org/abs/2503.07022v1 |
In order to apply Proposition G.1 we have to establish the following for every t∈[0,1]: sup s≤t NX j=1κjBn,s(zj/n)−NX j=1κj 1{zj≥0}bα,β+ 1{zj<0}b′ α,β |zj|Lρ0s(X) −→Pρ00, (4.9) ⌊nt⌋X k=1Eρ0 (ynk−Eρ0[ynk|F(k−1)/n])2|F(k−1)/n −→Pρ0logα2 β221 α2Lρ0 t(X)mX i=1 κ− i+···+κ− m2 z− i−z− i−1 + logβ2 α221 β2Lρ0 t(X)MX ... | https://arxiv.org/abs/2503.07022v1 |
and by similar estimations, we get Eρ0[ 1Jk]≤Cα,β(z− m,...,z+ M)1√ nk. (4.15) From this, the boundedness and Lipschitz assumption on g(where Lgdenotes the Lipschitz constant of g) and Markov’s inequality, it follows for any ϵ >0that Eρ0 ⌊nt⌋X k=1Eρ0 (g(ynk,wnk)−g(ynk,0)) 1Jk|F(k−1)/n ≤⌊nt⌋X k=1ϵEρ0[ 1Jk] +∥g∥s... | https://arxiv.org/abs/2503.07022v1 |
np 2π/n⌊nt⌋X k=11{X(k−1)/n<ρ0+z− m/n}exp −(X(k−1)/n−ρ0)2 2α2/n! +oPρ0(1) 34 =2 α+ββ α|z− l−z− l−1| np 2π/n⌊nt⌋X k=11{X(k−1)/n<ρ0}exp −(X(k−1)/n−ρ0)2 2α2/n! +oPρ0(1), where the first step follows by a Taylor expansion and is given in detail for the treatment ofS31(k)within the verification of (4.10) in Appendix H and th... | https://arxiv.org/abs/2503.07022v1 |
in the second one, P′ sup z>2Mℓ(z)≥0 ≤X j≥MP′ sup z∈Sjlogβ2 α2 N(cz/β2)−cz β2 ≥ −bα,βc2j! ≤logβ2 α22 (−bα,β)cX j≥M2−jE′" N(c2j+1/β2)−c2j+1 β22#1 2 = logβ2 α22 (−bα,β)β√cX j≥M2−j2(j+1)/2M→∞−→ 0. An analogous result holds true for z <0. Recalling ℓ(0) = 0 , this proves that the random set Argsupz∈Rℓ(zLρ0 1(X)... | https://arxiv.org/abs/2503.07022v1 |
(1999). Convergence of probability measures , second ed. Wiley Series in Probability and Statistics: Probability and Statistics . John Wiley & Sons, Inc., New York A Wiley-Interscience Publication. https://doi.org/10.1002/9780470316962 MR1700749 BLANCHARD , P., R ¨OCKNER , M. and R USSO , F. (2010). Probabilistic repre... | https://arxiv.org/abs/2503.07022v1 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 B Preliminary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 C Local time estimator for OBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 D Expansion of the normalized log-likelihood function . . .... | https://arxiv.org/abs/2503.07022v1 |
i(x,y;t)is continuous (and differentiable) in the parameter ρ. MSC2020 subject classifications: Primary 60F05, 62M05; secondary 62F12, 62E20. Keywords and phrases: Stable Poisson convergence, infill asymptotics, n-consistency, MLE. 1 2 J. BRUTSCHE AND A. ROHDE B. Preliminary results. LEMMA B.1. The transition density (... | https://arxiv.org/abs/2503.07022v1 |
+ 1Jc 2(X(k−1)/n)λ(J2)√nexp −ν(J2,X(k−1)/n) 4max{α2,β2}/n ≤Cα,β(c) Eρ0 1J1(X(k−1)/n) ∧Eρ0h 1J1∩J2(X(k−1)/n) + 1J1∩Jc 2(X(k−1)/n)λ(J2)√nexp −ν(J2,X(k−1)/n) 4max{α2,β2}/n ≤Cα,β(c) Eρ0 1J1(X(k−1)/n) ∧ Eρ0 1J2(X(k−1)/n) +λ(J2)√nEρ0 exp −ν(J2,X(k−1)/n) 4max{α2,β2}/n ≤Cα,β(c) 1∧λ(J1)p k/n∧λ(J2)p k/n! .(... | https://arxiv.org/abs/2503.07022v1 |
k/n≤ρ0}+ 1{ρ0−K/n<X (k−1)/n<ρ0} By Lemma B.3, Eρ0 2√nnX k=1 1{ρ0−K/n<X k/n≤ρ0}+ 1{ρ0−K/n<X (k−1)/n<ρ0}!2 ≤Cα,β(K)1 n and thus, sup ρ0−K/n≤ρ≤ρ0 ˆLρ n−ˆLρ0n −→Pρ00. The same argument works for supρ0≤ρ≤ρ0+K/n|ˆLρ n−ˆLρ0n|and (C.1) follows. D. Expansion of the normalized log-likelihood function. In this section, we... | https://arxiv.org/abs/2503.07022v1 |
B.4, log pρ0+θ 1/n(X(k−1)/n,Xk/n) pρ0+θ′ 1/n(X(k−1)/n,Xk/n) 1{Xk/n≤ρ0+θ′≤X(k−1)/n<ρ0+θ} ≤Cα,βn|θ−θ′| |θ−θ′|+|Xk/n−ρ0−θ| 1{Xk/n≤ρ0+θ′≤X(k−1)/n<ρ0+θ}.(D.9) •Iθ′,θ 5,k. We treat ρ0+θ′≤X(k−1)/n< ρ0+θ,ρ0+θ′< X k/n≤ρ0+θand find log pρ0+θ 1/n(X(k−1)/n,Xk/n) pρ0+θ′ 1/n(X(k−1)/n,Xk/n) 1{ρ0+θ′≤X(k−1)/n<ρ0+θ,ρ0+θ′<Xk/... | https://arxiv.org/abs/2503.07022v1 |
the remaining two cases we first have for x,y > ρ that ∂ ∂y σρ(y)2pρ t(x,y) =∂ ∂y σρ(y)2Pρ 2(x,y;t) =β√ 2πt −y−x tβ2exp −(y−x)2 2tβ2 −α−β α+βy−2ρ+x tβ2exp −(y−2ρ+x)2 2tβ2 and secondly for y < ρ < x that ∂ ∂y σρ(y)2pρ t(x,y) =∂ ∂y σρ(y)2Pρ 4(x,y;t) =−1√ 2πt2β α+βy−ρ tα−x−ρ tβ exp −1 2ty−ρ α−x−ρ β2! . B... | https://arxiv.org/abs/2503.07022v1 |
(D.2) (with θ′= 0), Eρ0 log pρ0+θ 1/n(X(k−1)/n,Xk/n) pρ0 1/n(X(k−1)/n,Xk/n) 1Iθ 1,k X(k−1)/n =− 1{X(k−1)/n<ρ0}θZρ0 −∞α−β α+β4(y−2ρ0+X(k−1)/n) (2α2/n)p 2πα2/nexp −(y−2ρ0+X(k−1)/n)2 2α2/n! dy +Eρ0 R1(k,0,θ)|X(k−1)/n . Evaluation of the integral reveals −θZρ0 −∞α−β α+β4(y−2ρ0+X(k−1)/n) 2α2/n1p 2πα2/nexp −(y−2ρ... | https://arxiv.org/abs/2503.07022v1 |
n(t,θ) +r3,2 n(t,θ), where r3,1 n(t,θ) :=− 1{X(k−1)/n<ρ0}θ1 α−1 β2α α+β1p 2π/n ·⌊nt⌋X k=1" exp −(θ−β α(X(k−1)/n−ρ0))2 2β2/n! −exp −(X(k−1)/n−ρ0)2 2α2/n!# , r3,2 n(t,θ) :=−nθ2 21 α−1 β2⌊nt⌋X k=11{X(k−1)/n<ρ0}Eρ0 1{Xk/n>ρ0+θ}|X(k−1)/n . In what follows, we are going to prove the moment condition on both r3,1 n(t,θ)... | https://arxiv.org/abs/2503.07022v1 |
s≤t|r7,1 n(s,θ′)|# ≤Cα,βnθ2⌊nt⌋X k=1Eρ0 1{Xk/n≤ρ0≤X(k−1)/n} ≤Cα,βn3/2θ2√ t. To eliminate the dependence of the indicator 1{X(k−1)/n≥ρ0+θ}in the first-order term on θ, we now define r7,2 n:=−θ1 α−1 β2β α+β1p 2π/n⌊nt⌋X k=1 1{X(k−1)/n≥ρ0+θ}− 1{X(k−1)/n≥ρ0} ·exp −(X(k−1)/n−ρ0)2 2β2/n! . Since 1{X(k−1)/n≥ρ0+θ}− 1{X(k−... | https://arxiv.org/abs/2503.07022v1 |
n(t,θ)with r9,1 n(t,θ) :=θα−β α+β2p 2πβ2/n⌊nt⌋X k=11{X(k−1)/n≥ρ0+θ}" exp −(X(k−1)/n−ρ0+θ)2 2β2/n! −exp −(X(k−1)/n−ρ0)2 2β2/n!# , r9,2 n(t,θ) :=θα−β α+β2p 2πβ2/n⌊nt⌋X k=1 1{X(k−1)/n≥ρ0+θ}− 1{X(k−1)/n≥ρ0} ·exp −(X(k−1)/n−ρ0)2 2β2/n! , 26 J. BRUTSCHE AND A. ROHDE r9,3 n(t,θ) :=⌊nt⌋X k=1Eρ0 R9(k,0,θ)|X(k−1)/n . We disc... | https://arxiv.org/abs/2503.07022v1 |
expectation of this bound under Pρ0, we have to distinguish the cases θ′<0andθ′≥0in order to perform the calculation with the correct regime of the transition density. −θ′<0. Here, {X(k−1)/n< ρ0+θ′,Xk/n≤ρ0+θ′} ⊂ { X(k−1)/n,Xk/n≤ρ0}. Then, using (E.4), we find from (E.8), (E.9) and the Gaussian tail inequality, Eρ0h Z1 ... | https://arxiv.org/abs/2503.07022v1 |
−(y−ρ0−θ′)2 2max{α2,β2}/n dy ≤Cα,β(K)|θ−θ′|dnd/21{ρ0+θ′≤X(k−1)/n<ρ0+θ}Z0 −∞(1 +|y|)dexp −y2 2max{α2,β2} dy 32 J. BRUTSCHE AND A. ROHDE ≤Cα,β(D,K )|θ−θ′|dnd/21{ρ0+θ′≤X(k−1)/n<ρ0+θ}. In particular, we find with Lemma B.1 and√n|θ−θ′| ≤2K, Eρ0h Z4 k(θ′,θ) di ≤Cα,β(D,K )|θ−θ′|dnd/2Zρ0+θ ρ0+θ′1p (k−1)/nexp −(y−x0)2 2max{... | https://arxiv.org/abs/2503.07022v1 |
the n-consistency of Subsection 3.2 may be found. Recall the definition of Iθ′,θ j,kgiven in (2.2) and remember that Cα,βdenotes some real and positive constant that only depends on αandβbut may change from line to line. F.1. Remaining proofs of Subsection 3.1. PROOF OF LEMMA 3.1. The proof makes use of the inequality ... | https://arxiv.org/abs/2503.07022v1 |
Pρ0+θ 1(X(k−1)/n,Xk/n;1/n) Pρ0 2(X(k−1)/n,Xk/n;1/n) =β αexp −(Xk/n−X(k−1)/n)2 2α2/n exp −(Xk/n−X(k−1)/n)2 2β2/n1−α−β α+βexp −2 α2/n(Xk/n−ρ0−θ)(X(k−1)/n−ρ0−θ) 1 +α−β α+βexp −2 β2/n(Xk/n−ρ0)(X(k−1)/n−ρ0) . Taking the logarithm on both sides then yields log Pρ0+θ 1(X(k−1)/n,Xk/n;1/n) Pρ0 2(X(k−1)/n,Xk/n;1/n)! = lo... | https://arxiv.org/abs/2503.07022v1 |
BRUTSCHE AND A. ROHDE is larger than 1−3ϵfornsufficiently large. Here, we used that on A′ 3we have Ly 1(X)> ξl forρ0−γ/2≤y≤ρ0+γ/2and consequently by the occupation times formula √nZ1 0Gn,θ(Xs)ds≥√n max{α2,β2}Z1 0Gn,θ(Xs)σ(Xs)2ds =√n max{α2,β2}Z RGn,θ(y)Ly 1(X)dy ≥√n max{α2,β2}Z0 −∞exp2y α2/√n Ly+ρ0+θ 1 dy ≥√nξl max{α... | https://arxiv.org/abs/2503.07022v1 |
˜Ξ(5,3) n≤4Cα,βξu min{α2,β2}n2/3. −Ξ(5,4) n. By Lemma B.1, Corollary B.2 and boundedness of x7→xexp(−x2/4), Eρ0h |gk| 1{Xk/n<ρ0≤ρ0+L/√n≤X(k−1)/n}i ≤Cα,βEρ0 1{X(k−1)/n≥ρ0+L/√n}Zρ0 −∞ 1 +n(y−X(k−1)/n)2 ·√nexp −(y−X(k−1)/n)2 2max{α2,β2}/n! dy# ≤Cα,βEρ0" 1{X(k−1)/n≥ρ0+L/√n}Z−√n(X(k−1)/n−ρ0) −∞ 1 +y2 ·exp −y2 2max{α2,... | https://arxiv.org/abs/2503.07022v1 |
The first expectation on the right-hand side of (F.2) can be rewritten as Eρ0 n(Xk/n−X(k−1)/n)2−β2 1Bk 1{Xk/n<ρ0} X(k−1)/n = 1Bk Zρ0 −∞ n(y−X(k−1)/n)2−β21p 2π/n2 α+ββ αexp −n 2y−ρ0 α−X(k−1)/n−ρ0 β2! dy = 1Bk2α2β α+βZ−(X(k−1)/n−ρ0)/√ β2/n −∞ y−(β−α)(X(k−1)/n−ρ0) αβ/√n2 −β2 α2! 1√ 2πexp −y2 2 dy. Now, using ... | https://arxiv.org/abs/2503.07022v1 |
of generality, we assume θ∈ Tnand set 0 =θn 0∈ Tn. Furthermore, we define δn 0:= sup θ∈Tnρn(θ,θn 0)≤q cn2j+1/√n=p C22(j+1)/2n1/4 andδn k:= 2−kδn 0fork∈N. Now we inductively define maximal subsets Tn k⊂ Tnsuch that Tn k−1⊂ Tn k and ρn(θ,θ′)≤δn kfor different θ,θ′∈ Tn k. In particular, the cardinality #Tn kis bounded by ... | https://arxiv.org/abs/2503.07022v1 |
2max{α2,β2}/n! dy =Cα,βZ Ry4exp −y2 2max{α2,β2} dy≤Cα,β,(F.11) and(a+b)2≤2a2+ 2b2yields again with Lemma B.1 Eρ0 dk(θ,θ′)2 ≤21 α2−1 β22 Eρ0h n2(Xk/n−X(k−1)/n)4+n2Eρ0[(Xk/n−X(k−1)/n)2|X(k−1)/n]2 · 1{ρ0+θ′≤X(k−1)/n<ρ0+θ}i ≤41 α2−1 β22 Eρ0 Eρ0 n2(Xk/n−X(k−1)/n)4|X(k−1)/n 1{ρ0+θ′≤X(k−1)/n<ρ0+θ} ≤4Cα,β1 α2−1 ... | https://arxiv.org/abs/2503.07022v1 |
this, we find on E 1√nnX k=11[a+θ−L/√n,a+θ](X(k−1)/n)−√nZ1 01[a+θ−L/√n,a+θ](Xs)ds ≤1√nnX k=11[a+θj0−L/√n,a+θj0+1](X(k−1)/n)−√nZ1 01[a+θj0+1−L/√n,a+θj0](Xs)ds ≤1√nnX k=11[a+θj0−L/√n,a+θj0+1](X(k−1)/n)−√nZ1 01[a+θj0−L/√n,a+θj0+1](Xs)ds +√nZ1 01[a+θj0−L/√n,a+θj0+1](Xs)− 1[a+θj0+1−L/√n,a+θj0](Xs)ds ≤1√nnX k=11[a+θj0−L/√n,a... | https://arxiv.org/abs/2503.07022v1 |
=nX k=1 1 +n 2(Xk/n−X(k−1)/n)2 1{X(k−1)/n<ρ0+θ≤Xk/n} −Eρ0h 1 +n 2(Xk/n−X(k−1)/n)2 1{X(k−1)/n<ρ0+θ≤Xk/n} X(k−1)/ni . Then for some constant Cα,β>0we have Eρ0" sup 0≤θ≤n−1/4|Mn(θ)|# ≤Cα,βn3/8log(n). PROOF . The proof relies on a chaining argument similar to that in Subsection 3.1. First, for0≤θ′≤θ, we rewrite 1{X(k−1... | https://arxiv.org/abs/2503.07022v1 |
A′ n(κ) :=A2∩A3∩En∩( sup θ′≤κ/√n|rn(1,θ′)| ≤Crnκ) . By Markov’s inequality, Pρ0 sup |θ′|≤κ/√n|rn(1,θ′)|> C rnκ! ≤Cα,βκ Crκ=Cα,β Cr, and we can choose Crindependently of nandκlarge enough such that Pρ0(A′ n(κ))≥1−ϵ/2 forn≥n0. Then we have for |θ| ≤κ/√n, Bn(θ) 1A′ n(κ)≤ −n|θ| 1{θ≥0}Fα,β+ 1{θ<0}˜Fα,βξ(α+β) 4αβ−Crκ . F... | https://arxiv.org/abs/2503.07022v1 |
4(X(k−1)/n,·) 1(−∞,ρ0](·) L1(λ) ≥ Pρ0+θ 1(X(k−1)/n,·) 1(−∞,ρ0](·) L1(λ)− Pρ0 4(X(k−1)/n,·) 1(−∞,ρ0](·) L1(λ) .(F.23) By substitution, we find Pρ0+θ 1(X(k−1)/n,·) 1(−∞,ρ0](·) L1(λ) = Φ −X(k−1)/n−ρ0 α/√n −α−β α+βΦX(k−1)/n−ρ0−2θ α/√n SUPPLEMENT TO ”STABLE LIMIT THEORY FOR THE MLE OF OBM” 63 and Pρ0 4(X(k−1)/n,·) 1(−∞... | https://arxiv.org/abs/2503.07022v1 |
we find the upper bound dk(θ,θ′)2≤26logβ2 α22h 1{ρ0+θ′≤Xk/n≤ρ0+θ}+Eρ0 1{ρ0+θ′≤Xk/n≤ρ0+θ}|X(k−1)/n2 + 1{ρ0+θ′≤Xk/n≤ρ0+θ}+Eρ0 1{ρ0+θ′≤Xk/n≤ρ0+θ}|X(k−1)/n2 + 1{ρ0+θ′≤X(k−1)/n≤ρ0+θ}+Eρ0 1{ρ0+θ′≤X(k−1)/n≤ρ0+θ}|X(k−1)/n2i . 66 J. BRUTSCHE AND A. ROHDE With Lemma B.1 we then find the moment bounds Eρ0 1{ρ0+θ′≤Xk/n≤ρ0... | https://arxiv.org/abs/2503.07022v1 |
For each n∈N, letXnbe aFn-semimartingale with Xn t=⌊nt⌋X k=1χnk, where χnkisFn k/n-measurable and square-integrable. • We now consider the Fn-semimartingale (Xn,Wn). Its first characteristic Bn, its sec- ond modified characteristic Cnand its third characteristic νnare given as (see Theo- remII.3.11(b) and II.3.18in Jac... | https://arxiv.org/abs/2503.07022v1 |
for symmetric and positive semidefinite bilinear forms, ⟨˜Xn,˜Wn⟩t− ⟨Xn,Wn⟩t = ⟨˜Xn−Xn,˜Wn⟩t+⟨Xn,˜Wn−Wn⟩t ≤q ⟨˜Xn−Xn,˜Xn−Xn⟩t⟨˜Wn,˜Wn⟩t +q ⟨˜Wn−Wn,˜Wn−Wn⟩t⟨˜Xn,˜Xn⟩t. The first summand converges to zero in probability by (G.8), for the second one it follows by ⟨˜Wn−Wn,˜Wn−Wn⟩t ≤⌊nt⌋X k=1(Wk/n−W(k−1)/n)21{|Wk/n−W(k−1)/n... | https://arxiv.org/abs/2503.07022v1 |
define N(n)m= (N(n)m t)t∈[0,1]viaN(n)m t=Nm ⌊nt⌋/n. AsNm is bounded, we clearly have supω,t,n|N(n)m t(ω)|<∞and because Nmis continuous by the martingale representation property, we find ˜Wn,N(n)1,...,N (n)m −→P W,N1,...,Nm in the Skorohod space D([0,1],Rm+1). 72 J. BRUTSCHE AND A. ROHDE We can consider N= (Nm)m∈Na... | https://arxiv.org/abs/2503.07022v1 |
the read- ers convenience. From (vii) we know that the sequence (˜Xn,N(n))converges in law to (X,N). In particular, if f:D([0,1],R)−→Ris a bounded continuous function we find (denoting with ˜Ethe expectation with respect to ˜P), Eh f(˜Xn)N(n)m 1i −→E[f(X)Nm 1], since N(n)mis a component of N(n)that is uniformly bounded... | https://arxiv.org/abs/2503.07022v1 |
the first two, we bound there L1(Pρ0)-norm as both will turn out to be negligable. The third one gives the main contributing term and is evaluated more explicitly. •S1(k).Using |ab| ≤(a2+b2)/2, we obtain by Proposition 2.4, Eρ0 Eρ0 ZZ′|F(k−1)/n ≤1 2Eρ0 Z2+ (Z′)2 ≤Cα,βmax{|z− m|,...,|z+ M|}21 n√ k. In particular, ... | https://arxiv.org/abs/2503.07022v1 |
1{X(k−1)/n<ρ0} 78 J. BRUTSCHE AND A. ROHDE ·mX l=1 κ− l+···+κ− m2 z− l−z− l−1 +r31 nk+s31 nk, where r31 nk+s31 nkis a remainder with the L1(Pρ0)-bound Eρ0 |r31 nk+s31 nk| ≤Cα,β(m,z− 1,...,z− m)1 n√ k. −S32(k).First, we note that mX l=1κ− l1 Iz− l/n,0 8,k!2 = mX l=1 κ− l+···+κ− m 1{ρ0+z− l/n<X k/n≤ρ0+z− l−1/n≤ρ0≤X... | https://arxiv.org/abs/2503.07022v1 |
OF OBM” 81 By the moment bound (H.6), we then deduce n−1/2P⌊nt⌋ k=1p |rnk| −→ Pρ00and finally get ⌊nt⌋X k=1Eρ0 ynk Wk/n−W(k−1)/n |F(k−1)/n −→Pρ00. VERIFICATION OF (4.13). First, we observe that |ynk|= mX l=1κ− l −9X j=1Zj k(z− l/n,0) + logα2 β2 1 Iz− l/n,0 2,k+ 1 Iz− l/n,0 8,k +MX l=1κ+ l 9X j=1Zj k(0,z+... | https://arxiv.org/abs/2503.07022v1 |
(H.11) and the moment bound (H.10) for Bn. By (H.9), Eρ0[|ℓn(z1/n)−ℓn(z2/n)||ℓn(z2/n)−ℓn(z3/n)|] ≤Cα,βEρ0" nX k=11{X(k−1)/n≤ρ0+z1/n≤Xk/n≤ρ0+z2/n}+ 1{ρ0+z1/n≤Xk/n≤ρ0+z2/n≤X(k−1)/n}! · nX l=11{X(l−1)/n≤ρ0+z2/n≤Xl/n≤ρ0+z3/n}+ 1{ρ0+z2/n≤Xl/n≤ρ0+z3/n≤X(l−1)/n}!# +Cα,β|z1−z2|Eρ0" Bn(z1,z2) nX l=11{X(l−1)/n≤ρ0+z2/n≤Xl/n≤ρ0+z3... | https://arxiv.org/abs/2503.07022v1 |
arXiv:2503.07088v1 [math.ST] 10 Mar 2025Asymptotic normality and strong consistency of kernel regression estimation in q-calculus Emmanuel De Dieu Nkou(a)andFridolin Melong(b) (a)Laboratoire de Probabilités, Statistique et Informatique (LPSI), Unité de Recherche en Mathématiques et Informatique (URMI) , Université des ... | https://arxiv.org/abs/2503.07088v1 |
1] and extend this work. More precisely, under relaxed assumptions, we investigate the strong convergenc e of this estimator and determine its rate of convergence. Furthermore, we apply the same q-calculus operations to investigate the asymptotic normality and almost sure convergence of the reg ression function estimat... | https://arxiv.org/abs/2503.07088v1 |
s/greaterorequalslant1. To derive some of our results, we will use the q-Taylor formula for a function fbeing differentiable stimes, which can be found in [ 9,11]. This formula is given by f(b) =s/summationdisplay k=0(b−a)k q [k]q!/parenleftbig Dk qf/parenrightbig (a) +Rs(f, b, a, q ), (2.2) where Rs(f, b, a, q ) is the... | https://arxiv.org/abs/2503.07088v1 |
2 In the classic case of regression function r(x), due to his quotient form, controlling its denominator for very small values is essential to obtain ing valid results. For overcoming this technical difficulty, some studies have introduced assumpti ons on f(x), such that f(x)> cwith c >0[18] or by replacing f(x)with fbn(... | https://arxiv.org/abs/2503.07088v1 |
known as the uniform, Epa nechnikov (or quadratic), biweight (or quartic) and triweight kernels respectively. 4. Assumptions and results In this section, we first present the assumptions, followed b y the results establishing the asymp- totic normality of the q-kernel estimator /hatwidern(x). We then conclude with the r... | https://arxiv.org/abs/2503.07088v1 |
Theorem. From the result of the previous theorem, we can easily obtain the strong convergence of the estimator/hatwidern. This result is illustrated in Theorem 3below. Theorem 3 Under Assumptions 1,2,3and4, for all t >0andn/greaterorequalslant1, we have sup x∈R|/hatwidern(x)−r(x)|=Oa.s/parenleftBig qwh2 n+/radicalBig v... | https://arxiv.org/abs/2503.07088v1 |
The following result states that the sequence of random vari ables Zi(x) satisfy the q-Lyapunov condition 4.5. The details of its proof are provided in Section 5. Lemma 4.1 Under Assumptions 1,2,3and4,x∈Rand0< q < 1. The sequence Z1(x),· · ·, Zn(x) random variables satisfies the q-Lyapunov condition lim n→+∞1 L2+δnqn/su... | https://arxiv.org/abs/2503.07088v1 |
=/integraldisplay R/integraldisplayx+hnν x−hnν/parenleftbig v−r(x)/parenrightbig Kq/parenleftbiggx−u hn/parenrightbigg fX,Y(u, v)dqu dqv =hn/integraldisplayν −νg(x−hnu)Kq(u)dqu−r(x)hn/integraldisplayν −νKq(u)f(x−hnu)dqu. Applying a second-order q-Taylor series expansion about xresults in g(x−hnu), we obtain /integraldi... | https://arxiv.org/abs/2503.07088v1 |
there exist two con- stants v >0andc >0such thatn/summationdisplay i=1Eq/parenleftbig X2 i/parenrightbig /lessorequalslantvand ∀k/greaterorequalslant3,n/summationdisplay i=1Eq/bracketleftBig (Xi)k +/bracketrightBig /lessorequalslantv[k]q!ck−2 [2]q. Then for all λ∈[0,1/c[, lnqEq/bracketleftbig expq/braceleftbig λ/parenl... | https://arxiv.org/abs/2503.07088v1 |
It follows that |/hatwidern(x)−r(x)|/lessorequalslant2b−2 n/bardblf/bardblL∞ q|/hatwidegn(x)−g(x)|+b−2 n/bardblg/bardblL∞ q/vextendsingle/vextendsingle/vextendsingle/hatwidefn(x)−f(x)/vextendsingle/vextendsingle/vextendsingle. Consequently, sup x∈Rp|/hatwidern(x)−r(x)|/lessorequalslant2e−2 n/parenleftbigg /bardblf/bard... | https://arxiv.org/abs/2503.07088v1 |
Estimation of Local Geometric Structure on Manifolds from Noisy Data Yariv Aizenbud∗1and Barak Sober∗2 1Department of Mathematics, Tel Aviv University 2Department of Statistics and Data Science, Center of Digital Humanities, The Hebrew University of Jerusalem 11th March 2025 Abstract A common observation in data-driven... | https://arxiv.org/abs/2503.07220v1 |
on the creation of Delaunay complexes through Voronoi diagrams in the ambient space. Harvesting the idea of tangential Delaunay complexes, Boissonat and Ghosh [13] have provided a method reconstructing a simplicial complex which is computationally tractable (i.e., its complexity has linear dependency in the ambient dim... | https://arxiv.org/abs/2503.07220v1 |
rates were shown to be optimal. 2 These results were later refined to a class of H¨ older-like smooth manifolds by Aamari and Lev- rard [2]. They come to the conclusion that the optimal rate of convergence for such k-times smooth manifold estimation is O(n−k/d) for the noiseless case and is bounded from below by eO(n−k... | https://arxiv.org/abs/2503.07220v1 |
have both simplified the algorithm and refined the theoretical analysis. The algorithm presented in this paper is divided into two steps. In Step 1, we find an initial local coordinate system. It is proved in Theorem 3.2 that this coordinate system is a “reasonable” approximation to the tangent of the manifold at some ... | https://arxiv.org/abs/2503.07220v1 |
not obscure the geometrical shape. In accordance with that, we limit our discussion to manifolds with a reach bounded away from zero (notice that in the case of flat manifolds, the reach is infinite) and with a noise model that limits the noise level from above by the reach. Our noise model in the analysis is as follow... | https://arxiv.org/abs/2503.07220v1 |
we try to estimate the manifold locally as a function over this approximated domain, the noise in the sample is biased with respect to this coordinate system (see Figure 2). To account for the bias, in our altered version of the algorithm, we perform the second step iteratively, taking the tangent estimate at each iter... | https://arxiv.org/abs/2503.07220v1 |
2 - The iterated projection Given ( q, H)∈RD×Gr(d, D) we define the following minimization scheme, known as local polynomial regression (e.g., [16, 41]): Find π∈Πd7→D k−1a polynomial of total degree deg(π)≤k−1 fromRdtoRD−dwhich minimizes J2(π|q, H) =1 Nq,HX ri∈Un ROI∥ri−(xi, π(xi))H∥2, (8) where xi∈Rdare the projection... | https://arxiv.org/abs/2503.07220v1 |
value of the samples, aligns with the estimated function. However, in our case this assumption does not hold, since the noise model is tubular with respect to the manifold and unless the coordinate system is aligned with the tangent, the expected value of Ygiven X=xdoes not equal to fℓ(x) (see Figure 2); i.e., the samp... | https://arxiv.org/abs/2503.07220v1 |
Algorithm 3 Step 1: in practice 1:Input: {ri}N i=1, r, d, τ, σ, ϵ 2:Output: q- anndimensional vector U- ann×dmatrix whose columns are {uj}d j=1▷ H=q+Span{uj}d j=1 3:define Rto be an n×Nmatrix whose columns are ri 4:initialize Uwith the first dprincipal components of the spatially weighted PCA 5:q←r 6:repeat 7: qprev=q ... | https://arxiv.org/abs/2503.07220v1 |
probability of at least 1−δ. Furthermore, we have ∥p−q∗∥ ≤3σ The proof of this theorem can be found in Section 4.1. Theorem 3.3. Assume that M > C τ√DlogD. Denote p=Proj M(r)where r is the input point for Algorithm 2. Let (q, H)be an initial coordinate system for which ∥q−p∥ ≤3σand ∠max(H, T pM)≤p cM/M, where p=Proj M(... | https://arxiv.org/abs/2503.07220v1 |
of Theorem 3.2. The proof can be described by the following three arguments which are proven in Lemmas 4.1, 4.3. Arg. 1: Denote pr=Proj M(r). Then, since r−pr⊥TprMandr∈ M σ, we have that q=pr along with H=TprMare in the search space defined by the constraints of (7). 12 Arg. 2: From Lemma 4.1 it follows that for large ... | https://arxiv.org/abs/2503.07220v1 |
From Lemma 4.2, we have that there is a constant CMsuch that for any α <π 2and M=CM/α2, and for any τandσmaintainingτ σ> M the following hold: For any δ > 0 arbitrarily small there is Nδsufficiently large such that for all n > N δ,alllinear spaces Hwith ∠max(H, T prM)> α, yield a score J1(r;pr, H)≥109·σ2, with probabil... | https://arxiv.org/abs/2503.07220v1 |
result of the regression will estimate efℓrather than fℓitself (see Figure 2). Thus, when we wish to estimate fℓor its derivatives using local polynomial regression, we estimate a different function, for which the noise has zero mean (see Figure 4). Figure 4: Illustration of Mas a graph of a func- tionfℓ(marked by the ... | https://arxiv.org/abs/2503.07220v1 |
have that Pr (∥f0(0)∥ ≤10σ)≥1−δ1. This means that P(A0)≥1−δ1(note that the failure probability comes from Theorem 3.2, and once the result of the theorem holds, the conditions of Lemmas C.1, C.3 are met). Step(ii): Denote by Aℓthe event that ∠max(T0fℓ, Hℓ)≤α02−ℓ=αℓand also ∥fℓ(0)∥ ≤ ∥ fℓ−1(0)∥ 1 + 40 α2 ℓ−1 . We now ... | https://arxiv.org/abs/2503.07220v1 |
bound the two parts in the right hand side separately. The first part, ∥pκ−p∥is bounded in (27). Furthermore, assuming event Aκholds, since α=α02−κ<1/D, there is N3such that for n > N 3we have from Lemma C.23 that ∥ˆpn−pκ∥ ≤8σDα2 12−2κ+cln 1 δ nr0 18 holds with probability at least 1 −δ/(κ+ 1), where cis some general ... | https://arxiv.org/abs/2503.07220v1 |
onto the estimation of the blue circle. Then at each step, a new point is generated along the circle (the black arrows connect the points. The plot illustrates 30 steps. Figure 7: Sample images from a 3d model of an airplane. 6 Conclusions This paper presents a novel algorithm for point estimation and denoising of mani... | https://arxiv.org/abs/2503.07220v1 |
3/2] 1−1/2x2≥√ 1−x2≥1−x2(34) Remark A.2.Forx∈[0,√ 3/2] 1−1/2x2−1/8x4≥√ 1−x2≥1−1 2x2−x4(35) A.1 Principal angles between linear sub-Spaces The concept of Principal Angles between flats were first introduced by Jordan in 1875 [33]. Below, we use the definition of Principal Angles between subspaces as described in [12]. D... | https://arxiv.org/abs/2503.07220v1 |
any p∈ M , letTpMbe the tangent of Matpand let H∈Gr(d, D) such that ∠max(TpM, H) =α≤π/4. For any x=p+xH, where xH∈H,∥xH∥ ≤cπ/4τfor some constant cπ/4, and y∈H⊥such that ∥x−p∥ ≤τcosα,∥y∥ ≤τ/2and(x+y)∈ M , we have that −τcosα+p τ2−(∥x∥ −τsinα)2≤ ∥y∥ ≤τcosα−p τ2−(∥x∥+τsinα)2 The proof of Lemma A.8 follows directly from ap... | https://arxiv.org/abs/2503.07220v1 |
tan2α≤α, we have that α−2∥xH∥ τ(1 +α) +O(∥xH∥2/τ2)≤β≤α+ 2∥xH∥ τ(1 +α) +O(∥xH∥2/τ2). Lemma A.12 (Mis locally a function graph over a tilted plane) .LetMbe ad-dimensional sub-manifold of RDwith reach τ. For any p∈ M , letTpMbe the tangent of Matp. Let H∈ Gr(d, D), such that ∠max(H, T pM) =α≤π/4. Then M ∩ CylH(p, ρ, τ/ 2)... | https://arxiv.org/abs/2503.07220v1 |
c∥x∥2 τ2+∥x∥ τ(1 +π 4)−π 8, Thus, for x < c 1τwe have that ∠max(Txf, H)< π/ 2 holds. We now turn to show that there is a constant cπ/4for which fis uniquely defined in BH(p, cπ/4τ). From Lemma A.8 we know that for any x∈Hwith∥x∥ ≤τ/2 all the y∈H⊥ such that ( x, y)∈ M and∥y∥ ≤τ/2 must satisfy: ∥y∥ ≤τcosα−p τ2−(∥x∥+τsinα... | https://arxiv.org/abs/2503.07220v1 |
τ/2), a cylinder with base BTprM(pr,√στ+σ)⊂TprMand heights τ/2 inTprM⊥can be written as Γ ϕpr,BTprM(pr,√στ+σ), the graph of ϕpr:TprM → TprM⊥. Since ri are in a tubular neighborhood of M, the proof is concluded. Lemma A.15 (Function version of Lemma A.6) .LetMbe a d-dimensional sub-manifold of RDwith reach τ. For any p∈... | https://arxiv.org/abs/2503.07220v1 |
j=1are orthonormal sets. Then, complete the sets {˜yj}d j=1 and{yj}d j=1to an orthonormal basis of H⊥andTprM⊥through adding the orthonormal sets {˜yj}D−d j=d+1and{yj}D−d j=d+1correspondingly. Note that, since H⊥is aD−ddimensional space, and{eyj}d j=1∈H⊥, we need only to add D−2dvectors to have an orthonormal basis. Exp... | https://arxiv.org/abs/2503.07220v1 |
we get that for α=p CM/M, where M=τ σ, and CMis a constant the following holds: For any δ >0 there is Nsuch that for all n > N R1(r;p, H)≥109σ2, (54) with probability of at least 1 −δ. 8. From Claim 1 above, since J1(r;pr, TprM)≥0 we achieve that with probability of at least 1 −δthere is Nδlarge enough such that for al... | https://arxiv.org/abs/2503.07220v1 |
proof below is independent of the current one, but utilizes the notion of Ugooddefined above). Explicitly, for ri∈Ugood ⟨ri−p,˜yj⟩2≥1 64στsinα0−1 2σ3/2τ1/2. Combining this with Lemma B.4 we get that for any δthere is Nlarge enough such that for alln > N with probability of at least 1 −δ R′ 1(r;p, H)≥n 3·µmax·Vd·(a2√στ)... | https://arxiv.org/abs/2503.07220v1 |
+σ/τ−(1−(a1+a2)2σ/τ))2=τ2(σ/τ+ (a1+a2)2σ/τ)2 =σ2(1 + ( a1+a2)2)2≤2σ2, where the last inequality comes from (58). Since ∠(y˜j,˜y˜j) =β˜jwe can use the Euclidean geometry on L˜j(Figure 10) to get ⟨q,˜y˜j⟩=⟨q,Rot(β˜j)y˜j⟩=⟨Rot(−β˜j)q, y˜j⟩, where Rot( θ) denotes the rotation matrix in RDwith respect to the angle θinL˜j. T... | https://arxiv.org/abs/2503.07220v1 |
nnX j=1Z Bd(p,(1−˜ε)ρ)dν, and µmin·Vol(Bd(p,(1−˜ε)ρ))≤E[Zp]≤µmax·Vol(Bd(p,(1−˜ε)ρ)). (64) Plugging this into (62) we get Pr Zp−µmaxVol (Bd(p,(1−˜ε)ρ))≤ −ε ≤e−2nε2, (65) or, alternatively, since # |X∩Bd(p,(1−˜ε)ρ)|=n·Zpwe get Pr [#|X∩Bd(p,(1−˜ε)ρ)| ≤n(µmaxVol(Bd(p,(1−˜ε)ρ))−ε)]≤e−2nε2. Denoting by Apthe event # |X∩Bd(... | https://arxiv.org/abs/2503.07220v1 |
of radius τ/2we get thatp det(G)is bounded and µminµTpM≤(Proj TpMProj M)∗µ≤µmaxµTpM where µmin, µmaxare constants that depend on τ, and µTpMis the Lebesgue measure on TpM. The constants µmin, µmaxcan be described explicitly to show their exact relationship to τ. Combining Lemma B.1 and Corollary B.3, we have the follow... | https://arxiv.org/abs/2503.07220v1 |
= 2 σ(tanαM+c1 M) (70) and thus, dist(fr∗(0), fq(0))≤r 22σ2(tanαM+c1 M)2+ (2σ)2= 2σr (tanαM+c1 M)2+ 1 From Corollary 3 in [14] we have that ∠max(T0fq, T0f0)≤1 2τ2σr (tanαM+c1 M)2+ 1≤1 Mr (tanαM+c1 M)2+ 1 (71) Combining (71) with (69) and (68), we have ∠max(H0, T0f0)≤3αM 2+1 Mr (tanαM+c1 M)2+ 1≤α forMlarge enough. 42 Fi... | https://arxiv.org/abs/2503.07220v1 |
this with (72) we have ∥fq(0)∥ ≤4σ+y≤5σ (74) forMlarge enough, as αM=p CM/M. Now we combine (70) with (74) and with the fact that dist(q, r∗)≤2σwe conclude the proof. Lemma C.4. For any δand for any n, α 1, r1, Let C0be the constant from Theorem 3.2 of [5]. We have that κ=r1log2(n) +¯Cα1,d−log ln2r1log2(n) + 2 ¯Cα1,d... | https://arxiv.org/abs/2503.07220v1 |
L C.5 ∠max(T0fℓ, T0efℓ)≤α 6 L C.6 ∥Def(0)− Df(0)∥< εαL C.7 Df[0]− D ˜f[0] op≤ε ⇓ sin(∠max(T0f, T0ef))≤εL C.9 σ≤g(0, θ)≤σ+ 4σα2L C.10 ∥∇xg(0, θ)∥=O(σ τsin(α)) L C.11 ∥I−Jexθ∥=3σ τ+O(σsinα τ) L C.12 L C.14 ∥Dexθw+Dxw∥ ≤ O (σ τsinα) ∥Dexθw∥ ≤ O (sinα) ∥Dxw∥ ≤ O (sinα) 1/2σ2≤∆(0,exθ(0))≤2σ2L C.13 Hf(exθ)w=√ 2σ τ+O(σsinα τ)... | https://arxiv.org/abs/2503.07220v1 |
we can breakdown the integrals over Ω( x) to a radial component rand directions on the (D−d−1)-dimensional sphere. Explicitly, ef(x)−f(x) =R SD−d−1g(x,θ)R 0θrrD−d−1drdθ R SD−d−1g(x,θ)R 0rD−d−1drdθ=(D−d)R SD−d−1θg(x, θ)D−d+1dθ (D−d+ 1)R SD−d−1g(x, θ)D−ddθ, where dris the measure over the radial component, rD−d−1is the J... | https://arxiv.org/abs/2503.07220v1 |
1/D, and we also have thatM > C τ√DlogDwe have that (1 + 4 α2)eDis bounded for any D. Thus, we have that for Cτlarge enough, ∥(I)∥ ≤ε 2α (93) for any D. Next we bound ∥(II)∥from (91): eDR SeD−1zg(x, θ)eD+1dθR SeD−1g(x, θ)eD−1∇xgTdθ R SeD−1g(x, θ)eDdθ!2 First, we note that R SeD−1g(x, θ)eD−1∇xgTdθ R SeD−1g(x, θ)eDdθ ≤c1... | https://arxiv.org/abs/2503.07220v1 |
where vi=Proj L2(ui). And, we wish to show that ∠(uj+1, vj+1) =βj+1, where the fact that βj+1≤∠(uj+1, vj+1) results directly from the definition of βj+1. We first note that since uj+1⊥ U j, we have Proj L2(uj+1)/∈Span{Proj L2(ui)}j i=1=Wj, thus, vj+1∈ W⊥ j. From here on we can repeat the same argument as in the basis o... | https://arxiv.org/abs/2503.07220v1 |
write ∇xg(x, θ) = (JexθDf[exθ]T− D f[x]T)¯θ+1√ ∆ (f(exθ)T¯θ−f(x)T¯θ) JexθDf[exθ]T− D f[x]T¯θ −(Id−Jexθ)(x−exθ)−(Df[x]T−JexθDf[exθ]T)(f(x)−f(exθ)) (102) 57 Next, using Lemma C.11, Lemma C.15 and Lemma C.17, we bound ∥Df[0]T−JexθDf[exθ(0)]T∥=∥Df[0]T− D f[exθ(0)]T+Df[exθ(0)]T−Jexθ(0)Df[exθ(0)]T∥ ≤ ∥D f[0]T− D f[exθ(0)... | https://arxiv.org/abs/2503.07220v1 |
∥A−1B∥ ≤ O (σ τ) Proof. We begin by noting that ∥A−1B∥ ≤ ∥ A−1∥∥B∥, where A=Id+Df[exθ(0)]TDx w[0,exθ(0)], B=Df[exθ(0)]T(Dx w[0,exθ(0)] + Dexθ w[0,exθ(0)]) + Hf(exθ(0))w. Moreover, A−1= (Id+Df[exθ(0)]TDx w[0,exθ(0)])−1=Id+∞X t=1(Df[exθ(0)]TDx w[0,exθ(0)])t. From Lemma C.14 we have that ∥Dx w[0,exθ(0)]∥=O(sinα), where we... | https://arxiv.org/abs/2503.07220v1 |
Eq. (100) at x= 0,exθ=xθ(0) using Lemma C.17, and assuming αis smaller than some constant. ∆(0,exθ(0)) = σ2+ (f(0)Tθ−f(exθ(0))Tθ)2− ∥0−exθ(0)∥2− ∥f(0)−f(exθ(0))∥2 ≥σ2− ∥exθ(0)∥2−2∥f(0)−f(exθ(0))∥2 ≥σ2−2σ2sin2α−4σ2sin2α ≥σ2(1−6 sin2α) ≥1 2σ2. (117) Similarly, ∆(0,exθ(0)) = σ2+ (f(0)Tθ−f(exθ(0))Tθ)2− ∥0−exθ(0)∥2− ∥f(0)−f... | https://arxiv.org/abs/2503.07220v1 |
which are meant to be used conveniently in the proof of Lemma C.6. Accordingly, (123) is already achieved in Lemma C.18. Then, from Lemma A.9 we have ∥f(exθ(0))∥ ≤ ∥exθ(0)∥tanα+O(∥exθ(0)∥2/τ). Thus, for αandσ τsmaller than some constants, we achieve Eq. (124). Next, since ∠max(H, T 0f)≤ α, by Lemma C.8 we have (125). F... | https://arxiv.org/abs/2503.07220v1 |
Since for small enough xwe have that 1 /(1−x)≤1 + 2 x, we have for large enough M, β−≤α 1 + 6σ τ(1 +α)−4σ2 τ2 (1 +α)2−cα ≤α 1 + 10σ τ(1 +α) ≤α 1 +c1σ τ (129) On the other hand, taking the left hand side of (128) we obtain similarly α−2∥xθ(0)∥ τ(1 +α) +c∥xθ(0)∥2/τ2≤β(xθ(0)) α−2∥xθ(0)∥ τ(1 +α)≤β α−2σ τβ(1 +α)≤β α... | https://arxiv.org/abs/2503.07220v1 |
Pr(∥∂xjπ∗ qℓ,Hℓ(0)−∂xjefℓ(0)∥>C0ln(1/δ) nr1for any 1 ≤j≤d)< dδ, and thus Pr(∥Dπ∗ qℓ,Hℓ[0]− Defℓ[0]∥op>√ dC0ln(1/δ) nr)< dδ, as required. In order to use convergence rate results of local polynomial regression for vector valued functions as described in Theorems 3.1 and 3.2 of [5] in our case, we need to show that the n... | https://arxiv.org/abs/2503.07220v1 |
probability of at least 1 −δ. Now we focus on bounding ∥efℓ(0)−fℓ(0)∥. From (89) we have that efℓ(x)−fℓ(x) =eDR SeD−1θg(x, θ)eD+1dθ (eD+ 1)R SeD−1g(x, θ)eDdrdθ=eDR SeD−1θ(g(x, θ)eD+1−σeD+1)dθ (eD+ 1)R SeD−1g(x, θ)eDdrdθ or, looking at some direction ⃗ zwe have zT·(efℓ(x)−fℓ(x)) =eDR SeD−1z(g(x, θ)eD+1−σeD+1)dθ (eD+ 1)R... | https://arxiv.org/abs/2503.07220v1 |
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