The Minimal Subgroup of a Random Walk |
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| Authors: | Gerold Alsmeyer |
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| Institution: | (1) Fachbereich Mathematik, Institut für Mathematische Statistik, Westfälische Wilhelms-Universität Münster, Einsteinsstraße 62, D-48149 Münster, Germany |
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| Abstract: | It is proved that for each random walk (S
n
)
n 0 on
d
there exists a smallest measurable subgroup
of
d
, called minimal subgroup of (S
n
)
n0, such that P(S
n
)=1 for all n1.
can be defined as the set of all x
d
for which the difference of the time averages n
–1 
n
k=1
P(S
k
 ) and n
–1
n
k=1
P(S
k
+x) converges to 0 in total variation norm as n  . The related subgroup
* consisting of all x
d
for which lim
n  P(S
n
)–P(S
n
+x)=0 is also considered and shown to be the minimal subgroup of the symmetrization of (S
n
)
n0. In the final section we consider quasi-invariance and admissible shifts of probability measures on
d
. The main result shows that, up to regular linear transformations, the only subgroups of
d
admitting a quasi-invariant measure are those of the form
 1×...×
k
×
l–k
×{0}
d–l
, 0 kld, with
1,...,
k
being countable subgroups of
. The proof is based on a result recently proved by Kharazishvili(3) which states no uncountable proper subgroup of
admits a quasi-invariant measure. |
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| Keywords: | random walk symmetrization minimal subgroup coupling zero-one law admissible shift quasi-invariance |
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