CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups
CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups
复制标题
紧阿贝尔群上通勤自同态随机游走的 CLT
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
J. Conze
中科院分区：
文献类型：
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作者：
G. Cohen;J. Conze
Let $$mathcal S$$S be an abelian group of automorphisms of a probability space $$(X, {mathcal A}, mu )$$(X,A,μ) with a finite system of generators $$(A_1, ldots , A_d).$$(A1,…,Ad). Let $$A^{{underline{ell }}}$$Aℓ̲ denote $$A_1^{ell _1} ldots A_d^{ell _d}$$A1ℓ1…Adℓd, for $${{underline{ell }}}= (ell _1, ldots , ell _d).$$ℓ̲=(ℓ1,…,ℓd). If $$(Z_k)$$(Zk) is a random walk on $${mathbb {Z}}^d$$Zd, one can study the asymptotic distribution of the sums $$sum _{k=0}^{n-1} , f circ A^{,{Z_k(omega )}}$$∑k=0n-1f∘AZk(ω) and $$sum _{{underline{ell }}in {mathbb {Z}}^d} {mathbb {P}}(Z_n= {underline{ell }}) , A^{underline{ell }}f$$∑ℓ̲∈ZdP(Zn=ℓ̲)Aℓ̲f, for a function f on X. In particular, given a random walk on commuting matrices in $$SL(
ho , {mathbb {Z}})$$SL(ρ,Z) or in $${mathcal M}^*(
ho , {mathbb {Z}})$$M∗(ρ,Z) acting on the torus $${mathbb {T}}^
ho $$Tρ, $$
ho ge 1$$ρ≥1, what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on $${mathbb {T}}^
ho $$Tρ after normalization? In this paper, we prove a central limit theorem when X is a compact abelian connected group G endowed with its Haar measure (e.g., a torus or a connected extension of a torus), $$mathcal S$$S a totally ergodic d-dimensional group of commuting algebraic automorphisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on random walks.
Let $$mathcal S$$S be an abelian group of automorphisms of a probability space $$(X, {mathcal A}, mu )$$(X,A,μ) with a finite system of generators $$(A_1, ldots , A_d).$$(A1,…,Ad). Let $$A^{{underline{ell }}}$$Aℓ̲ denote $$A_1^{ell _1} ldots A_d^{ell _d}$$A1ℓ1…Adℓd, for $${{underline{ell }}}= (ell _1, ldots , ell _d).$$ℓ̲=(ℓ1,…,ℓd). If $$(Z_k)$$(Zk) is a random walk on $${mathbb {Z}}^d$$Zd, one can study the asymptotic distribution of the sums $$sum _{k=0}^{n-1} , f circ A^{,{Z_k(omega )}}$$∑k=0n-1f∘AZk(ω) and $$sum _{{underline{ell }}in {mathbb {Z}}^d} {mathbb {P}}(Z_n= {underline{ell }}) , A^{underline{ell }}f$$∑ℓ̲∈ZdP(Zn=ℓ̲)Aℓ̲f, for a function f on X. In particular, given a random walk on commuting matrices in $$SL(
ho , {mathbb {Z}})$$SL(ρ,Z) or in $${mathcal M}^*(
ho , {mathbb {Z}})$$M∗(ρ,Z) acting on the torus $${mathbb {T}}^
ho $$Tρ, $$
ho ge 1$$ρ≥1, what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on $${mathbb {T}}^
ho $$Tρ after normalization? In this paper, we prove a central limit theorem when X is a compact abelian connected group G endowed with its Haar measure (e.g., a torus or a connected extension of a torus), $$mathcal S$$S a totally ergodic d-dimensional group of commuting algebraic automorphisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on random walks.