Abstract
It is shown that the limit μ of a commutative infinitesimal triangular system Δ on a totally disconnected locally compact groupG is embeddable in a continuous one-parameter convolution semigroup if either (1)G is a compact extension of a closed solvable normal subgroup or (2)G is discrete and Δ is normal or (3)G is a discrete linear group over a field of characteristic zero. For a special triangular system of convolution powers\(\left( {\mu _\nu ^{\kappa _\nu } \to \mu ,\mu _n \to \delta _3 } \right)\), the above is shown to hold without any of the conditions (1)–(3). For a general locally compact groupG necessary conditions are obtained for the embeddability of a shift of limit μ of Δ; in particular, the conditions are trivially satisfied whenG is abelian. Also, the embedding of a limit of a symmetric system onG is shown to hold under condition (1) as above.
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[B] A. Borel,Linear Algebraic Groups, W. A. Benjamin, New York, 1969.
[C] H. Carnal,Deux theoremes sur les groupes stochastiques compacts, Commentarii Mathematici Helvetici40 (1966), 237–246.
[D] S. G. Dani,On ergodic quasi-invariant measures of group automorphisms, Israel Journal of Mathematics43 (1982), 62–74.
[DM] S. G. Dani and M. McCrudden,Embeddability of infinitely divisible distributions on linear Lie groups, Inventiones Mathematicae110 (1992), 237–261.
[DS] S. G. Dani and Riddhi Shah,Concentration functions of probability measures on Lie groups, TIFR preprint.
[DG] Y. Derriennic and Y. Guivarc'h,Théorème de renouvellement pour les groupes non moyennables, Comptes Rendus de l'Académie des Sciences, Paris277 (1973), 613–616.
[G] R. Gangolli,Isotropic infinitely divisible measures on symmetric spaces, Acta Mathematica111 (1964), 213–246.
[Gr] U. Grenander,Probabilities on Algebraic Structures, Almquist and Wiksell, Stockholm-Göteberg-Uppsala, 1963.
[H] W. Hazod,Probabilities on totally disconnected compact groups, in Symposia Mathematica21, Academic Press, London and New York, 1977.
[HS] W. Hazod and E. Siebert,Continuous automorphism groups on a locally compact group contracting modulo a compact subgroup and applications to stable convolution semigroups, Semigroup Forum33 (1986), 111–143.
[HR] E. Hewitt and K. A. Ross,Abstract Harmonic Analysis, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963.
[He1] H. Heyer,Fourier transforms and probabilities on locally compact groups, Jahresbericht der Deutschen Mathematiken Vereinigung70 (1968), 109–147.
[He2] H. Heyer,Probability Measures on Locally Compact Groups, Springer-Verlag, Berlin-Heidelberg, 1977.
[Ho] G. P. Hochschild,Basic Theory of Algebraic Groups and Lie Algebras, Springer-Verlag, Berlin-Heidelberg, 1981.
[Hu] J. E. Humphreys,Linear Algebraic Groups, Springer-Verlag, Berlin-Heidelberg, 1975.
[JRW] W. Jaworski, J. Rosenblatt and G. Willis,Concentration functions in locally compact groups, Mathematische Annalen305 (1996), 673–691.
[Mc] M. McCrudden,Factors and roots of large measures on connected Lie groups, Mathematische Zeitschrift177 (1981), 315–322.
[N1] D. Neuenschwander,Limits of commutative triangular systems on simply connected step-2 nilpotent groups, Journal of Theoretical Probability5 (1992), 217–222.
[N2] D. Neuenschwander,Triangular Systems on discrete subgroups of simply connected nilpotent Lie groups, Publications Mathematicae Debrecen47 (1995), 329–333.
[P] K. R. Parthasarathy,Probability Measures on Metric Spaces, Academic Press, New York-London, 1967.
[PRV] K. R. Parthasarathy, R. Ranga Rao and S. R. S. Varadhan,Probability distributions on locally compact abelian groups, Illinois Journal of Mathematics7 (1963), 337–369.
[Ru] W. Rudin,Functional Analysis, Tata-McGraw-Hill Publ. Co. Ltd., New Delhi, 1974.
[R1] I. Z. Ruzsa,Infinite divisibility, Advances in Mathematics69 (1988), 115–132.
[R2] I. Z. Ruzsa,Infinite divisibility II, Journal of Theoretical Probability1 (1988), 327–339.
[RS] I. Z. Ruzsa and G. J. Szekely,Theory of decomposition in semigroups, Advances in Mathematics56 (1985), 9–27.
[S1] Riddhi Shah,Infinitely divisible measure on p-adic groups, Journal of Theoretical Probability4 (1991), 391–405.
[S2] Riddhi Shah,Semistable measures and limit theorems on real and p-adic groups, Monatshefte für Mathematik115 (1993), 191–213.
[S3] Riddhi Shah,Limits of commutative triangular systems on real and p-adic groups, Mathematical Proceedings of the Cambridge Philosophical Society120 (1996), 181–192.
[Sh] Y. Shalom,The growth of linear groups, Journal of Algebra199 (1998), 169–174.
[St] J. Štěpán,On the family of translations of a tight probability measure on a topological group, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete15 (1970), 131–138.
[T] J. Tits,Free subgroups in linear groups, Journal of Algebra20 (1972), 250–270.
[V] V. S. Varadarajan,Lie Groups, Lie Algebras, and Their Representations, Springer-Verlag, New York-Berlin-Heidelberg, 1984.
[Z] R. J. Zimmer,Ergodic Theory and Semisimple Groups, Monographs in Mathematics Vol.81, Birkhäuser, Boston-Basel-Stuttgart, 1984.
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Shah, R. The central limit problem on locally compact groups. Isr. J. Math. 110, 189–218 (1999). https://doi.org/10.1007/BF02808181
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DOI: https://doi.org/10.1007/BF02808181