Asymptotic version of the parametrix method for Markov chains converging to diffusions

I. Bitter; V. Konakov

REVIEW 2 major objections 3 minor 27 references

Asymptotic version of the parametrix method for Markov chains converging to diffusions

T0 review · 2 major / 3 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proves a uniform rate of convergence of inhomogeneous Markov-chain transition densities to diffusion transition densities when coefficients match only asymptotically and drift grows at most linearly.

desk verdict A serious extension of the parametrix method with a real flaw: the main theorem fails for adjacent grid points because Lemma 3.1 uses the wrong local-limit rate; the fix is a time-lag restriction or Gaussian-close noise. read the letter →

arxiv 2505.24548 v1 pith:XOXGK4IR submitted 2025-05-28 math.PR

I. Bitter , V. Konakov This is my paper

classification math.PR MSC 60F0560J35

keywords inhomogeneousMarkovchainslocallimittheoremparametrixmethodtransitiondensityestimatesdiffusionapproximationunboundeddrifttransportingflowquantitativeconvergencerate

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends the local-limit-theorem approach for inhomogeneous Markov chains to the case where the chain coefficients and the diffusion coefficients coincide only asymptotically, and where the drift may be unbounded with at most linear growth. Its main result is a uniform upper bound on $|p(t_i,t_j,x,y)-p^n(t_i,t_j,x,y)|$: the error is controlled by $\Delta_n = n^{-\min\{\gamma/2,\alpha,\beta\}}+\Delta_b+\Delta_a$, up to a factor $\ln(e(j-i))$, polynomial weights in $x$ and $y$, and two polynomial Gaussian kernels centred at the deterministic flows $\theta_{t_i,t_j}(y)$ and $\theta^n_{t_i,t_j}(y)$. A sympathetic reader would take this as a quantitative local limit theorem, giving the rate at which transition densities of the discrete process approach those of the diffusion, rather than only weak convergence of the processes.

What carries the argument

The machinery is the parametrix expansion in continuous and discrete form. The diffusion density is written as $\tilde p + \sum_{r\ge 1}\tilde p\otimes H^r$, and the chain density as the finite discrete analogue $\sum_{r=0}^{j-i}\tilde p^n \otimes_n H^{n,r}$, where $\tilde p$ and $\tilde p^n$ are densities of 'frozen' processes whose coefficients are evaluated along the backward transporting flow, the deterministic solution of the drift equation with terminal condition $y$. The kernels $H$ and $H^n$ measure the difference between the full and frozen generators. The proof compares the two expansions term by term, estimating differences between frozen densities, between continuous and discrete convolutions, and between the two kernels; the $\ln(e(j-i))$ factor comes from the harmonic sum $\sum_{k=1}^{j-i} k^{-1}$ in the discrete convolution bound.

What would settle it

Choose adjacent grid points $t_i=i/n$, $t_{i+1}=(i+1)/n$ and let the noise density in (A4) be a smooth zero-mean non-Gaussian density with covariance $a$, for instance a Gaussian with a small higher-cumulant perturbation. Then the one-step chain density is $n^{d/2}q^n_{t_i,x}(\sqrt n(y-x)-b_n/\sqrt n)$ while the diffusion density is $n^{d/2}\varphi_a(\sqrt n(y-x)+O(1/\sqrt n))$; at points where the argument is $O(1)$ the absolute difference is of order $n^{d/2}$, whereas the right side of (1.4) is $O(\Delta_n)$, so the claimed uniform bound fails for this pair.

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Extended reading notes

Core claim

The paper claims that Theorem 1.1 holds for every $n$ and every pair of grid points: for a constant $C$ independent of the grid pair, $$|p(t_i,t_j,x,y)-p^n(t_i,t_j,x,y)| \le C\ln(e(j-i))(1+|x|^{S-d-2}+|y|^{S-d-2})\Delta_n \left(Q_{S-d-6}\!\left(\frac{x-\$\theta$^n_{t_i,t_j}(y)}{\sqrt{t_j-t_i}}\right)+Q_{S-d-6}\!\left(\frac{x-\theta_{t_i,t_j}(y)}{\sqrt{t_j-t_i}}\right)\right),$$ with $\Delta_n$ as above. The polynomial kernels $Q_S$ decay like $(1+|z|)^{-S}$, and the transported terminal states $\theta$ encode the effect of the unbounded drift. In words, the transition density of the Markov chain is close to the diffusion transition density at a rate determined by the worse of the regularity exponents $\gamma,\alpha,\beta$, the sup-norm coefficient mismatches $\Delta_b,\Delta_a$, and the step size. The logarithmic factor is the only price paid for summing over intermediate time grid points.

Load-bearing premise

The load-bearing premise is that a block of $m=j-i$ increments already has transition density within $O(1/\sqrt n)$ of the Gaussian with the same mean and covariance; for one-step blocks this is not a consequence of the noise assumptions, so the uniformity over arbitrary grid pairs needs an additional lower bound on $t_j-t_i$ or a Gaussian-closeness condition on the noise.

Editorial extensions

If this is right

The uniform error bound gives a quantitative local limit theorem: $\Delta_n = n^{-\min\{\gamma/2,\alpha,\beta\}}+\Delta_b+\Delta_a$ is the rate at which the chain's transition density approaches the diffusion's.
The result covers coefficient pairs that agree only asymptotically, so a numerical scheme built on $b_n,a_n$ can be compared with the continuous model without requiring exact coefficient matching.
The polynomial factors $1+|x|^{S-d-2}+|y|^{S-d-2}$ and the transported arguments $\theta_{t_i,t_j}(y)$ show how the unbounded drift moves the effective centers of the Gaussian estimates.
The logarithmic factor $\ln(e(j-i))$ is the only singularity arising from summing over intermediate grid points, so longer time intervals are handled with the same bound up to the log.

Reading between the lines

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

A direct consequence the paper leaves implicit is that the uniform-over-all-grid-pairs form of (1.4) requires a lower bound on $t_j-t_i$: for adjacent grid points the local limit theorem step compares a one-step density to a Gaussian at scale $\sqrt n$, and generic noise densities satisfying (A4) differ from Gaussians by order $n^{d/2}$ in absolute value.
A minimal way to repair this would be to restrict to pairs with $j-i \ge n^\varepsilon$, equivalently $t_j-t_i \ge n^{\varepsilon-1}$, or to add a Gaussian-closeness condition on $q^n_{t,x}$; then the stated $\Delta_n$ rate should survive.
The flow-transport technique should extend to Euler discretizations of SDEs with unbounded drift on noncompact state spaces, where the same log factor would appear and short steps would dominate the local error.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague—

The paper is a serious attempt to push the parametrix method for Markov-chain-to-diffusion density convergence into two directions at once: unbounded drifts with at most linear growth, and coefficients that only match asymptotically. On those terms it mostly delivers a coherent technical scaffold. The flow arguments for unbounded drift, the polynomial kernel bookkeeping, and the way Δ_b and Δ_a enter the rate are all handled with care. The result is genuinely new relative to Konakov–Mammen (2000), which needed bounded coefficients and exact matching.

The problem is the main theorem as stated is false on short time lags. Lemma 3.1 applies the classical local limit theorem to a block of m = j−i summands and writes the error as O(1/√n). The correct local limit error is O(1/√m) = O(1/√(n(t_j−t_i))). For adjacent grid points, m=1, so the error is O(1), not O(n^{-1/2}). Under (A4), the noise densities need not be Gaussian-close, and for generic smooth centered q^n the one-step transition density differs from the diffusion density by order n^{d/2}. The right-hand side of (1.4) is only of order n^{(d−1)/2} at best (Δ_n ≈ n^{-1/2}), so the inequality fails by a factor of √n. This is not a cosmetic slip: the uniform claim over all grid pairs is load-bearing. The same issue affects any time lag that tends to zero as n grows.

The good news is the fix is visible. Restricting Theorem 1.1 to macroscopic intervals t_j − t_i ≥ c > 0, or adding an assumption that the noise density is Gaussian-close at scale 1/√n, would make the argument work; the rest of the estimate chain appears internally consistent. I did not find circularity: the rate is derived, and the comparison to Bhattacharya–Rao is legitimate. Self-citations are to published work and are not padding.

Who should read this: researchers working on local limit theorems, Euler schemes, or Robbins–Monro; they will want the corrected version. As it stands, the paper should go back for major revision with a clear request to fix Lemma 3.1 and restate the theorem. It deserves a serious referee, but not acceptance.

Referee Report

2 major / 3 minor

Summary. The paper studies the uniform distance between transition densities of an inhomogeneous Markov chain (1.1) and a diffusion (1.3), in the setting where the coefficients coincide only asymptotically and the drift may grow at most linearly. The main result, Theorem 1.1, claims the bound (1.4) with rate Δ_n = n^{-min(γ/2,α,β)} + Δ_b + Δ_a uniformly over all pairs of grid points (t_i,t_j), with a polynomial weight and a logarithmic factor in j-i. The proof expands both transition densities into parametrix series (Section 2), splits the difference into three terms in (3.1), and estimates them in Lemmas 3.4, 3.6, and 3.7 using a local limit theorem for blocks of increments and stability estimates for Gaussian densities.

Significance. An extension of Konakov–Mammen [15] to asymptotically matching coefficients and unbounded drift would be a useful quantitative contribution to the local limit theory for inhomogeneous Markov chains. The choice of parametrix expansions, backward flows to handle unbounded drift, and polynomial kernel bookkeeping is appropriate, and the proof is largely internally consistent. However, the advertised uniform rate is false for one-step transitions under the stated assumptions, so the main theorem is not established and the paper cannot be accepted in its current form.

major comments (2)

[§1.2, Theorem 1.1, Eq. (1.4); §3, Lemma 3.1, Eq. (3.2)] The proof of Lemma 3.1 invokes Theorem 19.3 of [6] for a block of m = j-i summands. The local limit theorem for the standardized sum gives an error O(m^{-1/2}) = O((n(t_j-t_i))^{-1/2}), not the O(n^{-1/2}) written after Eq. (3.2). For adjacent grid points m=1 this error is of order one. Under Assumption (A4) the density q^n_{t,x} may be any smooth centered density with polynomial decay, so it need not be close to a Gaussian. In the one-dimensional case with b=b^n=0, a=a^n=1, x=y=0, and t_j-t_i=1/n, the left-hand side of (1.4) is n^{1/2}|q^n_{t_i,0}(0)-φ(0)| + o(n^{1/2}), while the right-hand side is O(1) because Δ_n = n^{-1/2} and Q_{S-d-6}(0) contributes n^{1/2}. Thus (1.4) fails for generic non-Gaussian noise. A positive lower bound on t_j-t_i, a Gaussian-close assumption on the noise family, or a rate involving (t_j-t_i)^{-1/2} is required; none is present.
[§3, Lemmas 3.4 and 3.6] Because Lemma 3.1's estimate (3.2) is used in the proof of Lemma 3.4 and in the first step of Lemma 3.6, the missing (t_j-t_i)^{-1/2} factor propagates through the convolutions. Correcting (3.2) to its actual local-limit form introduces an additional singularity near t_j=t_i, and the displayed rate Δ_n in Theorem 1.1 no longer follows from the proof. Hence the problem is not confined to a single display; the whole chain of estimates for small lags is affected.

minor comments (3)

[§1.2, Assumption (A2)] The condition 'B < n0' is not meaningful as written; presumably B is a constant independent of n, and n0 is introduced only later.
[§3, Lemma 3.5] The statement of Lemma 3.5 uses p⊗H - p⊗nH, but the first line of the proof estimates ~p⊗H - ~p⊗nH; the notational discrepancy should be fixed and the passage from ~p to p justified.
[General] The manuscript is written in Russian with an English abstract; for an international venue an English version of the introduction and statements would be expected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the theorem is derived from the classical local limit theorem and published parametrix machinery, without any fitted parameter or conclusion reducible to its own input.

full rationale

The paper's central estimate, Theorem 1.1, is not obtained by fitting or by assuming its conclusion. The proof decomposes both transition densities into parametrix series and then bounds three differences termwise. The key local-limit comparison in Lemma 3.1 is explicitly based on the external Bhattacharya–Rao Theorem 19.3, not on a result of the present authors; the Gaussian comparison density is defined by matching the first two moments of the chain increments, which is the standard content of a local limit theorem rather than a circular construction. The self-citations to [5], [14], [15], and [16] are methodological: they supply the discrete parametrix expansions and convolution kernel estimates, and they are themselves published or preprint sources with stated assumptions and proofs; they do not assume the target inequality (1.4). No parameter is fitted to a subset of the data and then renamed a prediction, and no uniqueness or structural assumption is imported from the authors' prior work to force the result. The skeptic's concern about Lemma 3.1 — that the O(n^{-1/2}) local-limit error is applied to a block of j-i summands and may fail for adjacent grid points — is a quantitative correctness issue about the proof's rate, not a circular dependency. Accordingly, the derivation is self-contained against external benchmarks and the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No constants are fitted to data; all rate parameters come from the Hölder exponents in (A2) and the discrepancies in (A5). The machinery rests on the local limit theorem from [6], the parametrix series from [15,18], and Aronson-type Gaussian bounds, plus the domain assumptions (A1)-(A5). The fragile premise is the uniform O(1/√n) local limit rate for blocks of arbitrary length (Lemma 3.1).

assumptions (4)

standard math Local limit theorem for sums of independent random vectors (Theorem 19.3 of [6]) with the rate claimed by the paper.
Invoked in Lemma 2.1 and Lemma 3.1 to control the density of the normalized sum of j-i increments. The rate claimed by the paper (1/√n) is not what the theorem gives for m=j-i summands (1/√m); this is the fragile premise.
standard math Absolute convergence of the parametrix series (2.2) and its discrete analogue (2.3) for transition densities of nondegenerate diffusions and chains.
Relies on Lemma 2.2 and Corollary 2.3, following Konakov-Mammen [15] and Menozzi-Pesce-Zhang [18]; the kernel H satisfies |H| ≤ C(s-t)^{-1+γ/2} p̄(...).
standard math Aronson-type two-sided Gaussian bounds for the frozen density p̃ and the existence of the majorant density p̄.
Used through (2.5) and the remark after Lemma 2.1 to replace Gaussian kernels by the majorant density p̄ and to control polynomial moments of the Gaussian.
domain assumption Assumptions (A1)-(A5): uniform ellipticity, weak Hölder regularity with exponents γ, α, β, linear growth of drift, polynomial decay of the noise density family with S > 2d+6 and derivatives up to order 4, uniform convergence of coefficients.
These conditions define the class of chains and diffusions treated by the theorem. (A4) is the strongest of them and is used to control the third-order remainder terms in Lemma 3.7.

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Pith. "Pith review of Asymptotic version of the parametrix method for Markov chains converging to diffusions." pith.science (2026). https://pith.science/paper/XOXGK4IR

@misc{pith2026250524548,
  author       = {Pith},
  title        = {Pith review of: Asymptotic version of the parametrix method for Markov chains converging to diffusions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XOXGK4IR}},
  note         = {Machine review of arXiv:2505.24548}
}

read the original abstract

The paper presents a generalization of the local limit theorem on the convergence of inhomogeneous Markov chains to the diffusion limit for the case where the corresponding process coefficients satisfy weak regularity conditions and coincide only asymptotically. In particular, the drift coefficients considered by us can be unbounded with at most linear growth, and the estimates reflect the transfer of the terminal state by an unbounded trend through the corresponding deterministic flow. Our approach is based on the study of the uniform distance between the transition densities of a given inhomogeneous Markov chain and the limit diffusion process, and the convergence rate estimate is obtained using the classical local limit theorem and parametrix-type stability estimates.

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Reference graph

Works this paper leans on

27 extracted references · 26 canonical work pages

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Davydov, Yu. and Egorov, V. Functional limit theorems fo r induced order statistics. Mathematical Methods of Statistics, 9(3):297-313, (Janua ry, 2000)

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замороженного

≤C ∫ Rd qn ti,x (z)(1 + |y| + |z|)3× × ∫ 1 0 (1 −δ)2 ⏐ ⏐ ⏐ ⏐Dη x~pn(ti,t j,x + δ nbn(ti,x ) + δ√nz,y ) −Dη x~pn(ti,t j,x + δ nbn(ti,θ n ti,t j (y)) + δ√nz,y ) ⏐ ⏐ ⏐ ⏐dδdz ≤ ≤C ∫ Rd (1 + |y| + |z|)3 1 + |z|S ∫ 1 0 δ(1 −δ)2 n ⏐ ⏐ ⏐bn(ti,x ) −bn(ti,θ n ti,t j (y))| ⏐ ⏐ ⏐× × ∫ 1 0 d∑ k=1 ⏐ ⏐ ⏐ ⏐Dη+ek x ~pn(ti,t j,x + δ √nz + (1 −τ)δ n bn(ti,x ) + τδ nbn(ti,θ ...

work page
[1]

⏐ ⏐θti,t j (y1) −θti,t j (y2) ⏐ ⏐+ ⏐ ⏐ ⏐θn ti,t j (y1) −θn ti,t j (y2) ⏐ ⏐ ⏐ ≤C |y1 −y2| ; 8 2) θti,t k(θtk,t j (y)) = θti,t j (y), θn ti,t k(θn tk,t j (y)) = θn ti,t j (y); 3) C − 1⏐ ⏐θtj ,t i(y1) −y2 ⏐ ⏐ ≤ ⏐ ⏐θti,t j (y2) −y1 ⏐ ⏐ ≤C ⏐ ⏐θtj ,t i(y1) −y2 ⏐ ⏐, C − 1 ⏐ ⏐ ⏐θn tj ,t i(y1) −y2 ⏐ ⏐ ⏐ ≤ ⏐ ⏐ ⏐θn ti,t j (y2) −y1 ⏐ ⏐ ⏐ ≤C ⏐ ⏐ ⏐θn tj ,t i(y1) −y2 ⏐ ...

work page
[2]

Для оценки 1) применим разложение в ряд Тейлора первого порядка, а также п редполо- жение (A2) и оценку ( 2.4): 18

≤ ≤ C|x −θn ti,t j (y)| √n ∫ Rd QS(z)(1 + |y| + |z|)2 ∫ 1 0 (1 −δ)2 ⏐ ⏐ ⏐ ⏐Dη x~pn(ti,t j,x + δ nbn(ti,θ n ti,t j (y)) + δ√nz,y ) ⏐ ⏐ ⏐ ⏐dδdz ≤ ≤ C tj −ti (1 + |y|S− d− 2)∆ nQS− d− 6 ( x −θn ti,t j (y) √tj −ti ) . Для оценки 1) применим разложение в ряд Тейлора первого порядка, а также п редполо- жение (A2) и оценку ( 2.4): 18

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Далее, из разложения a1a2a3 −b1b2b3 = =(a1 −b1)a2a3 +b1(a2 −b2)a3 +b1b2(a3 −b3) аналогичным образом следует

≤ C|x −θn ti,t j (y)| √n ∫ Rd (1 + |y| + |z|)3QS(z)× × ∫ 1 0 (1 −δ)2 ⏐ ⏐ ⏐ ⏐Dη x~pn(ti,t j,x + δ nbn(ti,θ n ti,t j (y)) + δ√nz,y ) ⏐ ⏐ ⏐ ⏐dδdz ≤ C|x −θn ti,t j (y)| (tj −ti)3/ 2 QS− d− 4 ( x −θn ti,t j (y) √tj −ti ) × × ∫ Rd (1 + |y| + |z|)3 (1 + |z|S) ∫ 1 0 (1 −δ)2 ( 1 + δ|bn(ti,θ n ti,t j (y))| n + δ|z|√n ) S− d− 4 dδdz ≤ ≤ C tj −ti (1 + |y|S− d− 1)∆ nQ...

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [15]

and Egorov, V

Davydov, Yu. and Egorov, V. Functional limit theorems fo r induced order statistics. Mathematical Methods of Statistics, 9(3):297-313, (Janua ry, 2000)

work page 2000

[2] [6]

замороженного

≤C ∫ Rd qn ti,x (z)(1 + |y| + |z|)3× × ∫ 1 0 (1 −δ)2 ⏐ ⏐ ⏐ ⏐Dη x~pn(ti,t j,x + δ nbn(ti,x ) + δ√nz,y ) −Dη x~pn(ti,t j,x + δ nbn(ti,θ n ti,t j (y)) + δ√nz,y ) ⏐ ⏐ ⏐ ⏐dδdz ≤ ≤C ∫ Rd (1 + |y| + |z|)3 1 + |z|S ∫ 1 0 δ(1 −δ)2 n ⏐ ⏐ ⏐bn(ti,x ) −bn(ti,θ n ti,t j (y))| ⏐ ⏐ ⏐× × ∫ 1 0 d∑ k=1 ⏐ ⏐ ⏐ ⏐Dη+ek x ~pn(ti,t j,x + δ √nz + (1 −τ)δ n bn(ti,x ) + τδ nbn(ti,θ ...

work page

[3] [1]

⏐ ⏐θti,t j (y1) −θti,t j (y2) ⏐ ⏐+ ⏐ ⏐ ⏐θn ti,t j (y1) −θn ti,t j (y2) ⏐ ⏐ ⏐ ≤C |y1 −y2| ; 8 2) θti,t k(θtk,t j (y)) = θti,t j (y), θn ti,t k(θn tk,t j (y)) = θn ti,t j (y); 3) C − 1⏐ ⏐θtj ,t i(y1) −y2 ⏐ ⏐ ≤ ⏐ ⏐θti,t j (y2) −y1 ⏐ ⏐ ≤C ⏐ ⏐θtj ,t i(y1) −y2 ⏐ ⏐, C − 1 ⏐ ⏐ ⏐θn tj ,t i(y1) −y2 ⏐ ⏐ ⏐ ≤ ⏐ ⏐ ⏐θn ti,t j (y2) −y1 ⏐ ⏐ ⏐ ≤C ⏐ ⏐ ⏐θn tj ,t i(y1) −y2 ⏐ ...

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[4] [2]

Для оценки 1) применим разложение в ряд Тейлора первого порядка, а также п редполо- жение (A2) и оценку ( 2.4): 18

≤ ≤ C|x −θn ti,t j (y)| √n ∫ Rd QS(z)(1 + |y| + |z|)2 ∫ 1 0 (1 −δ)2 ⏐ ⏐ ⏐ ⏐Dη x~pn(ti,t j,x + δ nbn(ti,θ n ti,t j (y)) + δ√nz,y ) ⏐ ⏐ ⏐ ⏐dδdz ≤ ≤ C tj −ti (1 + |y|S− d− 2)∆ nQS− d− 6 ( x −θn ti,t j (y) √tj −ti ) . Для оценки 1) применим разложение в ряд Тейлора первого порядка, а также п редполо- жение (A2) и оценку ( 2.4): 18

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Далее, из разложения a1a2a3 −b1b2b3 = =(a1 −b1)a2a3 +b1(a2 −b2)a3 +b1b2(a3 −b3) аналогичным образом следует

≤ C|x −θn ti,t j (y)| √n ∫ Rd (1 + |y| + |z|)3QS(z)× × ∫ 1 0 (1 −δ)2 ⏐ ⏐ ⏐ ⏐Dη x~pn(ti,t j,x + δ nbn(ti,θ n ti,t j (y)) + δ√nz,y ) ⏐ ⏐ ⏐ ⏐dδdz ≤ C|x −θn ti,t j (y)| (tj −ti)3/ 2 QS− d− 4 ( x −θn ti,t j (y) √tj −ti ) × × ∫ Rd (1 + |y| + |z|)3 (1 + |z|S) ∫ 1 0 (1 −δ)2 ( 1 + δ|bn(ti,θ n ti,t j (y))| n + δ|z|√n ) S− d− 4 dδdz ≤ ≤ C tj −ti (1 + |y|S− d− 1)∆ nQ...

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R. F. Bass, Diﬀusions and Elliptic Operators. Springer, (1997)

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Доказательство

⏐ ⏐ ⏐θti,t j (y) −θn ti,t j (y) ⏐ ⏐ ⏐ ≤C(tj −ti) ( ∆ b + 1 nβ ) (1 + |y|). Доказательство. Доказательство свойства 1) следует из цепочки неравенств |θt,s (x) −θt,s (y)| = ⏐ ⏐ ⏐ ⏐x −y + ∫ s t (b(u,θ u,s (x)) −b(u,θ u,s (y)))du ⏐ ⏐ ⏐ ⏐ ≤ |x −y| +C · s∫ t |θu,s (y) −θu,s (x)|du с последующим применением неравенства Гронуолла. Доказат ельство для дискретного ...

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Локальная предельная теорема для возмущенн ых выборочных траекто- рий индуцированных порядковых статистик // Управление бол ьшими системами

Биттер И.И. Локальная предельная теорема для возмущенн ых выборочных траекто- рий индуцированных порядковых статистик // Управление бол ьшими системами. -

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