Limit theorems for compact two-point homogeneous spaces of large dimensions |
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Authors: | Michael Voit |
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Institution: | (1) Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany |
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Abstract: | Let
be the field , , or of real dimension . For each dimensiond2, we study isotropic random walks(Y
1)10 on the projective space
with natural metricD where the random walk starts at some
with jumps at each step of a size depending ond. Then the random variablesX
1
d
:=cosD(Y
1
d
,x
0
d
) form a Markov chain on –1, 1] whose transition probabilities are related to Jacobi convolutions on –1, 1]. We prove that, ford, the random variables (vd/2)(X
l(d)
d
+1) tend in distribution to a noncentral
2-distribution where the noncentrality parameter depends on relations between the numbers of steps and the jump sizes. We also derive another limit theorem for
as well as thed-spheresS
d
ford. |
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Keywords: | Projective spaces d-spheres isotropic random walks central limit theorem noncentral
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2-distribution orthogonal polynomials hypergroups |
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