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A New Proof of the Central Limit Theorem on Stratified Lie Groups

In: Probability Measures on Groups X

Author

Listed:
  • Gyula Pap

    (Lajos Kossuth University, Mathematical Institute)

Abstract

Let G be a stratified Lie group of step s, that is a simply connected nilpotent Lie group whose Lie algebra g has a vector space decomposition % MathType!MTEF!2!1!+- % feaaguart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4zaiabg2 % da9iabgwPifpaaDaaaleaacaWGQbGaeyypa0JaaGymaaqaaiaadMga % aaGccaWGwbWaaSbaaSqaaiaadQgaaeqaaaaa!3FB7! $$ g = \oplus _{j = 1}^i{V_j}\ $$ such that [V i , V j] ⊂ V i+j when i + j > s and [V i, V j] = 0 when i + j > s, and V 1 generates g as an algebra. Let exp: g → G be the exponential mapping (which is now a diffeomorphism). We equip g as well as G with the natural dilations by extending % MathType!MTEF!2!1!+- % feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbiaeaacq % aH0oazdaWgaaWcbaGaamiDaaqabaaabeqaaiaad+gaaaGccaGGOaGa % amiwaiaacMcacqGH9aqpcaWG0bWaaWbaaSqabeaacaWGPbaaaOGaam % iwaaaa!4030! $$ \mathop {{\delta _t}}\limits^o (X) = {t^i}X $$ , t > 0, X ∊ V j by linearity to g and putting % MathType!MTEF!2!1!+- % feaaguart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS % baaSqaaiaacIcacaWG0bGaaiykaaqabaGccaGGOaGaciyzaiaacIha % caGGWbGaamiwaiaacMcacqGH9aqpciGGLbGaaiiEaiaacchacaGGOa % WaaCbiaeaacqaH0oazdaWgaaWcbaGaamiDaaqabaGccaWGybaaleqa % baGaam4BaaaakiaacMcaaaa!4963! $$ {\delta _{(t)}}(\exp X) = \exp (\mathop {{\delta _t}X}\limits^o ) $$ . The family (δt)t>0 is a continuous one-parameter semigroup of automorphisms of G.

Suggested Citation

  • Gyula Pap, 1991. "A New Proof of the Central Limit Theorem on Stratified Lie Groups," Springer Books, in: Herbert Heyer (ed.), Probability Measures on Groups X, pages 329-336, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4899-2364-6_25
    DOI: 10.1007/978-1-4899-2364-6_25
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