Invariance principle for additive functionals of Markov chains

TL;DR

This paper establishes conditions under which additive functionals of Markov chains converge in distribution, extending Dynkin's theorem and providing practical criteria based on transition probabilities.

Contribution

It introduces a general sufficient condition for the convergence of additive functionals of Markov chains, linking it to Dynkin's theorem and transition probabilities.

Findings

Provides a new convergence criterion for additive functionals

Extends Dynkin's theorem to a broader setting

Offers practical conditions based on transition probabilities

Abstract

We consider a sequence of additive functionals {\phi_n}, set on a sequence of Markov chains {X_n} that weakly converges to a Markov process X. We give sufficient condition for such a sequence to converge in distribution, formulated in terms of the characteristics of the additive functionals, and related to the Dynkin's theorem on the convergence of W-functionals. As an application of the main theorem, the general sufficient condition for convergence of additive functionals in terms of transition probabilities of the chains X_n is proved.

Equations

$$E[1{\mathrm{I}}_A((X(t_1),\dots ,X(t_k)))|{\mathcal{G}}_s]=P_{s{t}_1\dots t_k}(X(s),A)a.s.,A\in \mathcal{B}({\mathfrak{X}}^k).$$

$$E[1{\mathrm{I}}_{\cdot }(X{|}_s^{\infty })|{\mathcal{G}}_s]=E[1{\mathrm{I}}_{\cdot }(X{|}_s^{\infty })|X(s)],$$

$$X_n(t)=\frac{S_{k-1}}{\sqrt{n}}+(nt-k+1)\left[\frac{S_k}{\sqrt{n}}-\frac{S_{k-1}}{\sqrt{n}}\right],t\in \left[\frac{k-1}{n},\frac{k}{n}\right),k\in \mathbb{N},$$

$$n^{-\frac{1}{\alpha }}[S_n-a_n]\Rightarrow \mathcal{L},a_n=\{\begin{array}{ll}0,& \alpha \in (0,1)\\ nE{\xi }_1,& \alpha \in (1,2)\\ n^{2}E\sin \frac{{\xi }_1}{n},& \alpha =1\end{array}$$

$$X_n(t)=n^{-\frac{1}{\alpha }}{S}_{k-1}+(nt-k+1)\left[n^{-\frac{1}{\alpha }}{S}_k-n^{-\frac{1}{\alpha }}{S}_{k-1}\right],t\in \left[\frac{k-1}{n},\frac{k}{n}\right),k\in \mathbb{N}.$$

$$Z_n\left(\frac{k+1}{n}\right)=Z_n\left(\frac{k}{n}\right)+a\left(Z_n\left(\frac{k}{n}\right)\right)\frac{1}{n}+b\left(Z_n\left(\frac{k}{n}\right)\right)\Delta X_n\left(\frac{k}{n}\right),Z_n(0)=z,$$

$$dZ(t)=a(Z(t))dt+b(Z(t))dX(t),Z(0)=z,$$

$$\Delta (\gamma )=\underset{n\to +\infty }{\mathrm{limsup}}P\left(\underset{i\le \frac{Tn}{K(\gamma ,T)}}{\sup }\rho \left({\hat{Z}}_n\left(\frac{iK(\gamma ,T)}{n}\right),{\hat{Z}}^n\left(\frac{iK(\gamma ,T)}{n}\right)\right)>\gamma \right),$$

$$\Delta (\gamma )\to 0+,\gamma \to 0+.$$

$$\frac{K({\gamma }_k,T)}{n_k}\to 0,P\left(\underset{i\le \frac{T{n}_k}{K({\gamma }_k,T)}}{\sup }\rho \left({\hat{Z}}_n\left(\frac{iK({\gamma }_k,T)}{n_k}\right),{\hat{Z}}^n\left(\frac{iK({\gamma }_k,T)}{n_k}\right)\right)>{\gamma }_k\right)>c.$$

$$P(Z_{\ast }\ne Z^{\ast })>0.$$

$${\varphi }_n^{s,t}(Y)\stackrel{\mathrm{def}}{=}\sum _{k:s\le k/n<t}{F}_n\left(Y\left(\frac{k}{n}\right),Y\left(\frac{k+1}{n}\right),\dots ,Y\left(\frac{k+L-1}{n}\right)\right),0\le s<t,$$

$${\psi }_n^{s,t}={\varphi }_n^{\frac{j-1}{n},\frac{k-1}{n}}+(ns-j+1){\varphi }_n^{\frac{j-1}{n},\frac{j}{n}}+(nt-k+1){\varphi }_n^{\frac{k-1}{n},\frac{k}{n}},s\in \left[\frac{j-1}{n},\frac{j}{n}\right),t\in \left[\frac{k-1}{n},\frac{k}{n}\right).$$

$$f_n^{s,t}(x)\stackrel{\mathrm{def}}{=}E[{\varphi }_n^{s,t}(Y)|Y(s)=x],s=\frac{i}{n},i\in {\mathbb{Z}}_{+},t>s,x\in \mathfrak{X}.$$

$$\delta (F_n)\stackrel{\mathrm{def}}{=}\sup \{F_n(x_1,\dots ,x_L)|x_1,\dots ,x_L\in \mathfrak{X}\}\to 0,n\to \infty .$$

$$\underset{s=\frac{i}{n},t\in (s,T)}{\sup }\Vert f_n^{s,t}(\cdot )-f^{s,t}(\cdot )\Vert \to 0,n\to \infty ,$$

$$\underset{0\le s\le t<T}{\sup }\left|f^{s,t}(x^{\prime })-f^{s,t}(x^{\prime \prime })\right|\to 0,\left|x^{\prime }-x^{\prime \prime }\right|\to 0.$$

$${\psi }_n(X_n)\Rightarrow \varphi (X)\equiv \{{\varphi }^{s,t}(X),(s,t)\in \mathbb{T}\},$$

$$E\left[{\varphi }^{\frac{Ki}{n},t}|{\mathcal{F}}_{\frac{Ki}{n}}\right]=f^{\frac{Ki}{n},t}\left({\hat{X}}^n\left(\frac{Ki}{n}\right)\right),E\left[{\varphi }_n^{\frac{Ki}{n},t}|{\mathcal{F}}_{\frac{Ki}{n}}\right]=f_n^{\frac{Ki}{n},t}\left({\hat{X}}_n\left(\frac{Ki}{n}\right)\right)$$

$$\underset{n\to \infty }{\mathrm{limsup}}E{\left({\varphi }_n^{s,t}({\hat{X}}_n)-{\varphi }^{s,t}(\hat{X})\right)}^2\le 4\Vert f^{0,T}\Vert G(f,\gamma ,T)+4\sqrt{2\gamma }{\Vert f^{0,T}\Vert }^2,$$

$${\Delta }_i^n\stackrel{\mathrm{def}}{=}{\varphi }_n^{\frac{(i-1)K}{n},(\frac{iK}{n})\bigwedge T},{\tilde{\Delta }}_i^n\stackrel{\mathrm{def}}{=}{\varphi }^{\frac{(i-1)K}{n},\frac{iK}{n}\bigwedge T},i=\overline{1,M_n}.$$

$${\left({\varphi }_n^{0,T}-{\varphi }^{0,T}\right)}^2={\left(\sum _{i=1}^{M_n}{\Delta }_i^n-{\tilde{\Delta }}_i^n\right)}^2={\left(\sum _{i=1}^{M_n}{\Delta }_i^n\right)}^2+{\left(\sum _{i=1}^{M_n}{\tilde{\Delta }}_i^n\right)}^2-2\sum _{i=1}^{M_n}\sum _{j=1}^{M_n}{\Delta }_i^n{\tilde{\Delta }}_j^n={\Sigma }_1^n+2{\Sigma }_2^n,$$

$${\Sigma }_1^n\stackrel{\mathrm{def}}{=}\sum _{i=1}^{M_n}({\Delta }_i^n{)}^2+\sum _{i=1}^M({\tilde{\Delta }}_i^n{)}^2-2\sum _{i=1}^M{\Delta }_i^n{\tilde{\Delta }}_i^n,$$

$${\Sigma }_2^n\stackrel{\mathrm{def}}{=}\left[\sum _{1\le i<l\le M_n}{\Delta }_i^n{\Delta }_l^n-\sum _{1\le i<j\le M_n}{\Delta }_i^n{\tilde{\Delta }}_j^n\right]+\left[\sum _{1\le j<k\le M_n}{\tilde{\Delta }}_j^n{\tilde{\Delta }}_k^n-\sum _{1\le j<i\le M_n}{\Delta }_i^n{\tilde{\Delta }}_j^n\right].$$

$${\Sigma }_1^n\le \sum _{i=1}^{M_n}({\Delta }_i^n{)}^2+\sum _{i=1}^{M_n}({\tilde{\Delta }}_i^n{)}^2.$$

$$E\sum _{i=1}^{M_n}({\Delta }_i^n{)}^2\le E\left(\underset{i=1,M_n}{\sup }{\Delta }_i\right)\sum _{i=1}^{M_n}{\Delta }_i^n\le K{\delta }_n{f}_n^{0,T}\left({\hat{X}}_n(0)\right)\le K{\delta }_n\Vert f_n^{0,T}\Vert \to 0,n\to +\infty ,$$

$$E\sum _{i=1}^{M_n}({\Delta }_i^n{)}^2\le E\left(\underset{i=1,M_n}{\sup }{\Delta }_i\right)\sum _{i=1}^{M_n}{\Delta }_i^n\le K{\delta }_n$$