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Journal Article
Limit theorems for von Mises statistics of a measure preserving transformation
2014
Overview
For a measure preserving transformation
T
of a probability space
(
X
,
F
,
μ
)
and some
d
≥
1
we investigate almost sure and distributional convergence of random variables of the form
x
→
1
C
n
∑
0
≤
i
1
,
…
,
i
d
<
n
f
(
T
i
1
x
,
…
,
T
i
d
x
)
,
n
=
1
,
2
,
…
,
where
C
1
,
C
2
,
…
are normalizing constants and the kernel
f
belongs to an appropriate subspace in some
L
p
(
X
d
,
F
⊗
d
,
μ
d
)
. We establish a form of the individual ergodic theorem for such sequences. Using a filtration compatible with
T
and the martingale approximation, we prove a central limit theorem in the non-degenerate case; for a class of canonical (totally degenerate) kernels and
d
=
2
, we also show that the convergence holds in distribution towards a quadratic form
∑
m
=
1
∞
λ
m
η
m
2
in independent standard Gaussian variables
η
1
,
η
2
,
…
.
Publisher
Springer Berlin Heidelberg,Springer Nature B.V
