Probability Distributions on Bicompact Topological Groups

Kloss, B. M.

doi:10.1137/1104026

Cited by 45 publications

(17 citation statements)

References 15 publications

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“…Q.E.D. Our final theorem in this paper answers a question of Kloss [8] and extends a result of his on compact groups. Theorem 6.…”

Section: Let F = Y4-1z) D (F X G X Y) Then By (3) F Is Compact Sisupporting

confidence: 72%

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Limit Theorems for Convolution Iterates of a Probability Measure on Completely Simple or Compact Semigroups

Mukherjea¹

1977

Transactions of the American Mathematical Society

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Abstract. This paper extends the study (initiated by M. Rosenblatt) of the asymptotic behavior of the convolution sequence of a probability measure on compact or completely simple semigroups. Let S be a locally compact second countable Hausdorff topological semigroup. Let ji be a regular probability measure on the Borel subsets of S such that S does not have a proper closed subsemigroup containing the support F of /x. It is shown in this paper that when S is completely simple with its usual product representation X X G X Y, then the convolution sequence ¡i" converges to zero vaguely if and only if the group factor G is noncompact. When the group factor G is compact, ¡i" converges weakly if and only if limn_>00 F" is nonempty. This last result remains true for an arbitrary compact semigroup S generated by F. Furthermore, we show that in this case there exist elements an G S such that ¡i" « 5ĉ onverges weakly, where 5a> is the point mass at an. This result cannot be extended to the locally compact case, even when 5 is a group.1. Throughout this paper, 5 will denote a locally compact second countable semigroup (i.e. an algebraic semigroup with locally compact Hausdorff topology and jointly continuous multiplication).By a measure on S, we will mean a finite regular nonnegative measure on the class of all Borel sets (generated by open sets) of S. Let C(S) be the realvalued continuous functions with compact support. A net of measures (ßk) is said to converge in the weak*-sense (or vaguely) to a measure ß if and only if for each/ E C(S), ffdßk converges to ffdß. By Banach-Alaoglu's theorem, the set B(S) of all measures ß with ß(S) < 1 is compact in the weak*-topology. Also B(S) is an algebraic semigroup under the usual convolution (*) operation of measures.

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“…Q.E.D. Our final theorem in this paper answers a question of Kloss [8] and extends a result of his on compact groups. Theorem 6.…”

Section: Let F = Y4-1z) D (F X G X Y) Then By (3) F Is Compact Sisupporting

confidence: 72%

“…In §3, we will also show that in a compact semigroup there exist elements an such that the sequence p" * 8aii converges vaguely as n -» oo. This result is the extension of a similar result by Kloss [8] for compact groups, who had left the case of compact semigroups as an open question.…”

supporting

confidence: 79%

Limit Theorems for Convolution Iterates of a Probability Measure on Completely Simple or Compact Semigroups

Mukherjea¹

1977

Transactions of the American Mathematical Society

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“…This is a compact group, and, in general, a random walk on a compact group converges to a uniform distribution. More specifically, the assumption that the speed vector has a density implies that b( V n ) also has one with respect to the uniform measure on the torus, and by [15,Section 5.2], this implies that the distribution of M n converges weakly to uniform distribution on the torus.…”

Section: Proof Of Lemmamentioning

confidence: 99%

The Random Trip Model: Stability, Stationary Regime, and Perfect Simulation

Boudec

Vojnović

2006

IEEE/ACM Trans. Networking

119

114

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Abstract-We define "random trip", a generic mobility model for random, independent node motions, which contains as special cases: the random waypoint on convex or non convex domains, random walk on torus, billiards, city section, space graph, intercity and other models. We show that, for this model, a necessary and sufficient condition for a timestationary regime to exist is that the mean trip duration (sampled at trip endpoints) is finite. When this holds, we show that the distribution of node mobility state converges to the time-stationary distribution, starting from origin of an arbitrary trip. For the special case of random waypoint, we provide for the first time a proof and a sufficient and necessary condition of the existence of a stationary regime, thus closing a long standing issue. We show that random walk on torus and billiards belong to the random trip class of models, and establish that the time-limit distribution of node location for these two models is uniform, for any initial distribution, even in cases where the speed vector does not have circular symmetry. Using Palm calculus, we establish properties of time-stationary regime, when the condition for its existence holds. We provide an algorithm to sample the simulation state from a time-stationary distribution at time 0 ("perfect simulation"), without computing geometric constants. For random waypoint on the sphere, random walk on torus and billiards, we show that, in the time-stationary regime, the node location is uniform. Our perfect sampling algorithm is implemented to use with ns-2, and is available to download from http://ica1www.epfl.ch/RandomTrip.

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“…Hence this class is a Borel field and must coincide with 03(5) (see [4]). By Theorem 14 of [3] we already know that an idempotent probability measure must have a completely simple semigroup as its support. From this point on let us take 5 a compact completely simple semigroup with representation TXXX Fand corresponding function Suppose ¡t is an idempotent measure on 5 with support 5.…”

Section: Corollarymentioning

confidence: 99%

“…Kernel semigroups K are rather special semigroups and are often referred to as completely simple semigroups [ó]. Every compact completely simple semigroup can be represented as the product space TXXX F of a compact topological group T and compact Hausdorff spaces X, Y where the multiplication of two elements s = (t, x, y), s' = (t', x', y') is given by (3) ss' = (/, x, y) (t\ x', y') = (td>(x, y')t', x', y)…”

mentioning

confidence: 99%