A limit theorem for Birkhoff sums of non-intetrable functions over rotations

YG Sinai; C Ulcigrai

A limit theorem for Birkhoff sums of non-intetrable functions over rotations

YG Sinai, C Ulcigrai

Research output: Chapter in Book/Report/Conference proceeding › Chapter in a book

Abstract

We consider Birkhoﬀ sums of functions with a singularity of type 1/x over rotations and prove the following limit theorem. Let S_N = SN (α, x)be the Nth non-renormalized Birkhoﬀ sum, where α ∈ [0, 1) is the rotation number, x ∈ [0, 1) is the initial point and (α, x) are uniformly distributed. We prove that S_N /N has a joint limiting distribution in (α, x) as N tends to inﬁnity. As a corollary, we get the existence of a limiting distribution for certain trigonometric sums.

Translated title of the contribution	A limit theorem for Birkhoff sums of non-intetrable functions over rotations
Original language	English
Title of host publication	Geometric and Probabilistic Structures in Dynamics
Editors	Keith Burns, Dimitry Dolgopyat, Yakov Pesin
Publisher	American Mathematical Society
Pages	317 - 340
Number of pages	24
Volume	Contemporary Mathematics 469
ISBN (Print)	9780821842867
Publication status	Published - 2008

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title = "A limit theorem for Birkhoff sums of non-intetrable functions over rotations",

abstract = "We consider Birkhoﬀ sums of functions with a singularity of type 1/x over rotations and prove the following limit theorem. Let S_N = SN (α, x)be the Nth non-renormalized Birkhoﬀ sum, where α ∈ [0, 1) is the rotation number, x ∈ [0, 1) is the initial point and (α, x) are uniformly distributed. We prove that S_N /N has a joint limiting distribution in (α, x) as N tends to inﬁnity. As a corollary, we get the existence of a limiting distribution for certain trigonometric sums.",

author = "YG Sinai and C Ulcigrai",

year = "2008",

language = "English",

isbn = "9780821842867",

volume = "Contemporary Mathematics 469",

pages = "317 -- 340",

editor = "Keith Burns and Dimitry Dolgopyat and Yakov Pesin",

booktitle = "Geometric and Probabilistic Structures in Dynamics",

publisher = "American Mathematical Society",

address = "United States",

}

A limit theorem for Birkhoff sums of non-intetrable functions over rotations. / Sinai, YG; Ulcigrai, C.

Geometric and Probabilistic Structures in Dynamics. ed. / Keith Burns; Dimitry Dolgopyat; Yakov Pesin. Vol. Contemporary Mathematics 469 American Mathematical Society, 2008. p. 317 - 340.

Research output: Chapter in Book/Report/Conference proceeding › Chapter in a book

TY - CHAP

T1 - A limit theorem for Birkhoff sums of non-intetrable functions over rotations

AU - Sinai, YG

AU - Ulcigrai, C

PY - 2008

Y1 - 2008

N2 - We consider Birkhoﬀ sums of functions with a singularity of type 1/x over rotations and prove the following limit theorem. Let S_N = SN (α, x)be the Nth non-renormalized Birkhoﬀ sum, where α ∈ [0, 1) is the rotation number, x ∈ [0, 1) is the initial point and (α, x) are uniformly distributed. We prove that S_N /N has a joint limiting distribution in (α, x) as N tends to inﬁnity. As a corollary, we get the existence of a limiting distribution for certain trigonometric sums.

AB - We consider Birkhoﬀ sums of functions with a singularity of type 1/x over rotations and prove the following limit theorem. Let S_N = SN (α, x)be the Nth non-renormalized Birkhoﬀ sum, where α ∈ [0, 1) is the rotation number, x ∈ [0, 1) is the initial point and (α, x) are uniformly distributed. We prove that S_N /N has a joint limiting distribution in (α, x) as N tends to inﬁnity. As a corollary, we get the existence of a limiting distribution for certain trigonometric sums.

M3 - Chapter in a book

SN - 9780821842867

VL - Contemporary Mathematics 469

SP - 317

EP - 340

BT - Geometric and Probabilistic Structures in Dynamics

A2 - Burns, Keith

A2 - Dolgopyat, Dimitry

A2 - Pesin, Yakov

PB - American Mathematical Society

ER -