$\mathbb{R}^d$ 中更新定理的收敛速率
概率论
2016-07-12 v2 动力系统
摘要
设 ρ \rho ρ S L d ( R ) \mathrm{SL}_d(\mathbb{R}) SL d ( R ) ρ \rho ρ R d ∖ { 0 } \mathbb{R}^d\setminus\{0\} R d ∖ { 0 } ( X n ) (X_n) ( X n ) x ∈ R d ∖ { 0 } x\in \mathbb{R}^d\setminus\{0\} x ∈ R d ∖ { 0 } X 0 = x X_0 =x X 0 = x X n + 1 = g n + 1 X n X_{n+1} = g_{n+1} X_n X n + 1 = g n + 1 X n ( g n ) (g_n) ( g n ) ρ \rho ρ S L d ( R ) \mathrm{SL}_d(\mathbb{R}) SL d ( R ) ρ \rho ρ f f f R d \mathbb{R}^d R d G f ( x ) : = E x ∑ n = 0 + ∞ f ( X n ) Gf(x) := \mathbb{E}_x \sum_{n=0}^{+\infty} f(X_n) G f ( x ) := E x ∑ n = 0 + ∞ f ( X n ) x ∈ R d ∖ { 0 } x\in \mathbb{R}^d \setminus\{0\} x ∈ R d ∖ { 0 } G f Gf G f 0 0 0 ( P ( i t ) ) t ∈ R (P(it))_{t\in \mathbb{R}} ( P ( i t ) ) t ∈ R S d − 1 \mathbb{S}^{d-1} S d − 1 f f f x ∈ S d − 1 x\in \mathbb{S}^{d-1} x ∈ S d − 1 P ( i t ) f ( x ) = ∫ S L d ( R ) e − i t ln ∥ g x ∥ ∥ x ∥ f ( g x ∥ g x ∥ ) d ρ ( g ) P(it) f(x) = \int_{\mathrm{SL}_d(\mathbb{R})} e^{-it \ln \frac{ \|gx\|}{\|x\|}} f\left(\frac{gx}{\|gx\|}\right) \mathrm{d}\rho(g) P ( i t ) f ( x ) = ∫ SL d ( R ) e − i t l n ∥ x ∥ ∥ g x ∥ f ( ∥ g x ∥ g x ) d ρ ( g ) L ∈ R L\in \mathbb{R} L ∈ R t 0 ∈ R + ∗ t_0 \in \mathbb{R}_+^\ast t 0 ∈ R + ∗ sup t ∈ R ∣ t ∣ ⩾ t 0 1 ∣ t ∣ L ∥ ( I d − P ( i t ) ) − 1 ∥ 有限 \sup_{\substack{t\in \mathbb{R}\ |t| \geqslant t_0}} \frac{ 1 }{|t|^L} \left\| (I_d-P(it))^{-1} \right\| \text{ 有限} t ∈ R ∣ t ∣ ⩾ t 0 sup ∣ t ∣ L 1 ( I d − P ( i t ) ) − 1 有限
引用
@article{arxiv.1603.07214,
title = {The rate of convergence for the renewal theorem in $\mathbb{R}^d$},
author = {Jean-Baptiste Boyer},
journal= {arXiv preprint arXiv:1603.07214},
year = {2016}
}