Large deviations for additive functionals of Markov chains

120 Pages·2014·0.67 MB·English

by  Alejandro D. De Acosta

MEMOIRS
of the
American Mathematical Society
Volume 228 • Number 1070 (second of 5 numbers) • March 2014
Large Deviations
for Additive Functionals
of Markov Chains
Alejandro D. de Acosta
Peter Ney
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
MEMOIRS
of the
American Mathematical Society
Volume 228 • Number 1070 (second of 5 numbers) • March 2014
Large Deviations
for Additive Functionals
of Markov Chains
Alejandro D. de Acosta
Peter Ney
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Providence, Rhode Island
Library of Congress Cataloging-in-Publication Data
Acosta,AlejandroD.de,1941-author.
Large deviations for additive functionals of Markov chains / Alejandro D. de Acosta, Peter
Ney.
pagescm. –(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;number1070)
“March2014,volume228,number1070(secondof5numbers).”
Includesbibliographicalreferences.
ISBN978-0-8218-9089-9(alk. paper)
1. Large deviations. 2. Markov processes. 3. Additive functions. I. Ney, Peter, 1930- au-
thor. II.Title.
QA273.67.A262014
519.2(cid:2)33–dc23 2013042546
DOI:http://dx.doi.org/10.1090/memo/1070
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Contents
Chapter 1. Introduction 1
Chapter 2. The transform kernels K and their convergence parameters 9
g
2.1. Irreducibility 9
2.2. Small functions and measures 10
2.3. The convergence parameter 12
2.4. The period of K and aperiodicity 19
g
Chapter 3. Comparison of Λ(g) and φ (g) 25
μ
Chapter 4. Proof of Theorem 1 31
Chapter5. AcharacteristicequationandtheanalyticityofΛ : thecasewhen
f
P has an atom C ∈S+ satisfying λ∗(C)>0 33
Chapter 6. Characteristic equations and the analyticity of Λ : the general
f
case when P is geometrically ergodic 41
Chapter 7. Differentiation formulas for u and Λ in the general case and
g f
their consequences 51
Chapter 8. Proof of Theorem 2 63
Chapter 9. Proof of Theorem 3 67
Chapter 10. Examples 71
Chapter 11. Applications to an autoregressive process and to reflected
random walk 77
11.1. Application of Theorem 1 to an autoregressive process 77
11.2. Application of Theorem 2 to reflected random walk 83
Appendix 93
AI. Renewal sequences 93
AII. Complex kernels and their associated renewal sequences 94
AIII. Renewal characterization of the convergence parameter 95
AIV. Some consequences of ergodicity 96
AV. Geometric ergodicity 98
Background comments 105
References 107
iii

Abstract
For a Markov chain {Xj} w(cid:2)ith general state space S and f :S →Rd, the large
deviation principle for {n−1 n f(X )} is proved under a condition on the chain
j=1 j
which is weaker than uniform recurrence but stronger than geometric recurrence
and an integrability condition on f, for a broad class of initial distributions. This
result is extended to the case when f takes values in a separable Banach space.
Assuming only geometric ergodicity and under a non-degeneracy condition, a local
largedeviationresultisprovedforboundedf. Acentralanalyticaltoolisthetrans-
form kernel, whose required properties, including new results, are established. The
rate function in the large deviation results is expressed in terms of the convergence
parameter of the transform kernel.
ReceivedbytheeditorMay20,2011.
ArticleelectronicallypublishedonJuly15,2013.
DOI:http://dx.doi.org/10.1090/memo/1070
2010 MathematicsSubjectClassification. 60J05,60F10.
Key words and phrases. Large deviations, Markov chains, additive functionals, transform
kernels,convergenceparameter,geometricergodicity.
Affiliations at time of publication: Alejandro D. de Acosta, Department of Mathematics,
CaseWesternReserveUniversity,Cleveland,Ohio44106;PeterNey,DepartmentofMathematics,
UniversityofWisconsin,Madison,Wisconsin53706.
(cid:3)c2013 American Mathematical Society
v

CHAPTER 1
Introduction
Let P be a Markov kernel and μ a probability measure on a measurable space
(S,S) with S countably generated. Our basic framework will be the canonical
Markov chain with transition kernel P and initial distribution μ, given by (Ω =
SN0,SN0,Pμ,{Xj}j≥0), where N0 is the set of non-negative integers, {Xj}j≥0
are the coordinate functions on SN0, and Pμ is the unique probability measure
on (SN0,SN0) such that {Xj}j≥0 is a Markov chain with transition kernel P and
initial distribution μ.
Assume that P is positive Harris recurrent (see A.IV) and let π be its unique
invariant(cid:2)probability measure. For a measurable function f :S →Rd, let
S (f)= n f(X ). By the ergodic theorem for functionals of Markov chains, if
n j=1 j
π((cid:5)f(cid:5))<∞ then for any μ
limn−1S (f)=π(f) P a.s. .
n μ
n
The main objective of the present paper is to study the large deviations associated
with this result; that is, to determine under what conditions sharp asymptotic
bounds can be obtained for
n−1logP [n−1S (f)∈B],
μ n
whereBisaBorelsetinRd,andinparticularwhentheprobabilitiesP [n−1S (f)∈/
μ n
U] decay exponentially and at what rate, where U is an open set in Rd which con-
tains π(f).
In order to place our results in the right context we need to refer to some
previous work (Theorems A and B below). In these theorems we obtained lower
and upper large deviation bounds, resp., for P [n−1S (f) ∈ B] in terms of rate
μ n
functions which are structural objects but which are in general different in the two
cases. A central task in our work is to establish conditions under which the lower
and upper rate functions coincide.
We now introduce certain analytical objects that will play a crucial role in the
formulationandproofofourresults(andwhichareneededtostateTheoremsAand
B).Thetransform kernelassociatedwiththeMarkovkernelP andameasurable
function g :S →R is defined to be
(cid:3)
K (x,A)= eg(y)P(x,dy), x∈S, A∈S,
g
A
and, in particular, for g =(cid:7)f,ξ(cid:8) with f as above, ξ ∈Rd,
(cid:3)
Kf,ξ(x,A)=K(cid:5)f,ξ(cid:6)(x,A)= e(cid:5)f(y),ξ(cid:6)P(x,dy).
A
ThisdefinitionextendsintheobviouswaytothecasewhenE isaseparableBanach
space, f :S →E is measurable and ξ ∈E∗, the dual space of E. The kernels K
f,ξ
1
2 A.DEACOSTAANDPETERNEY
are a natural extension to Markov kernels of the Laplace transform of probability
measures on Rd (or on E).
If P is irreducible, then so is K , and its convergence parameter R(K ) exists
g g
(see Chapter 2 for these notions). We define
Λ(g)=−logR(K )
g
and for f :S →Rd (resp., E), ξ ∈Rd (resp., E∗),
Λ (ξ)=Λ((cid:7)f,ξ(cid:8))=−logR(K ).
f f,ξ
Λ∗ :Rd →[0,∞] is defined by
f
Λ∗(u)= sup[(cid:7)u,ξ(cid:8)−Λ (ξ)], u∈Rd;
f f
ξ∈Rd
that is, Λ∗ is the convex conjugate of Λ . Λ∗ :E →[0,∞] is defined similarly in
f f f
the case f :S →E.
The following lower bound was proved in de Acosta-Ney [deA,N].
Theorem A. Let P be irreducible and let f : S → Rd be measurable. Then
for every probability measure μ on (S,S) and every open set G in Rd,
limn−1logP [n−1S (f)∈G]≥− inf Λ∗(u).
n μ n u∈G f
More generally, the result was proved in [deA,N] for f :S →E.
For a probability measure μ on (S,S) and a measurable function g : S → R,
we define (cid:4) (cid:5)
φ (g)=limn−1logE expS (g) ;
μ μ n
n
if x∈S and μ=δ , we write
x (cid:4) (cid:5)
φ (g)=limn−1logE expS (g) .
x x n
n
For f :S →Rd (resp., E), ξ ∈Rd (resp., E∗)
φ (ξ)=φ ((cid:7)f,ξ(cid:8))=limn−1logE (exp(cid:7)S (f),ξ(cid:8)).
f,μ μ μ n
n
φ∗ :Rd →[0,∞] is defined by
f,μ
φ∗ (u)= sup[(cid:7)u,ξ(cid:8)−φ (ξ)], u∈Rd;
f,μ f,μ
ξ∈Rd
that is, φ∗ is the convex conjugate of φ . φ∗ :E →[0,∞] is defined similarly
f,μ f,μ f,μ
in the case f :S →E.
The following upper bound can be obtained from de Acosta [deA1], Theorem
4.2(a), supplemented by [deA3], Lemma 1.
Theorem B. Let f : S → Rd be a measurable function such that for all
ξ ∈Rd,
supK (x,S)<∞.
f,ξ
x∈S
Then for every probability measure μ on (S,S) and every closed set F in Rd,
limn−1logP [n−1S (f)∈F]≤− inf φ∗ (u).
n μ n u∈F f,μ
Moreover, φ∗ is inf-compact; that is, L ={u:φ∗ (u)≤a} is compact for all
f,μ a f,μ
a≥0.
LARGEDEVIATIONSFORMARKOVCHAINS 3
More generally, it is proved in [deA1] that under certain additional conditions
the result holds for f :S →E (see Theorem 3 below).
In view of Theorems A and B, in order to obtain a large deviation principle
for {P [n−1S (f) ∈ ·]} it suffices to show that under suitable assumptions on P,
μ n
f and μ,
∗ ∗
Λ =φ .
f f,μ
If P is irreducible, then one can show that φ∗ ≤ Λ∗, but in general there is no
f,μ f
equality. Forsomeinsightintotheequality,weobservethat,atleastiff isbounded,
ifthelowerlargedeviationboundinTheoremAiscomplementedbyanupperlarge
deviationboundwiththesameratefunction-thatis,if{P [n−1S (f)∈·]}satisfies
μ n
thelargedeviationprinciplewithratefunctionΛ∗ -,thenΛ∗ =φ∗ . For,itfollows
f f f,μ
from Dinwoodie [Din], Theorem 3.1 (see also Theorem 4.1) that if f is bounded
and {P [n−1S (f) ∈ ·]} satisfies the large deviation principle with a convex rate
μ n
function, then that rate function must be φ∗ . Examples in the literature show
f,μ
that equality may fail to hold even under strong conditions on P and bounded f,
depending on μ, with or without the large deviation principle being satisfied:
(i) Dinwoodie [Din], p.226, presents an example in which |S| = 3, P is ir-
reducible (but not in the matrix sense) and uniformly ergodic, f :S →R3 is the
empirical measure functional and for a certain y ∈S, {P [n−1S (f)∈·]} satisfies
y n
the large deviation principle with rate function φ∗ but Λ∗ (cid:11)=φ∗ .
f,y f f,y
(ii) In another example in [Din], p.220, |S| = 4, P is irreducible (again, not
in the matrix sense) and uniformly ergodic, f :S →R4 is the empirical measure
functional and for a certain z ∈S
{P [n−1S (f)∈·]}
z n
satisfiesthelargedeviationprinciplewithanon-convexratefunctionI;afortiori,in
viewof Theorems A and B and theuniqueness of the large deviationrate function,
Λ∗ (cid:11)=φ∗ (and I (cid:11)=Λ∗, I (cid:11)=φ∗ ).
f f,z f f,z
(iii) In Proposition 5 of Bryc-Dembo [Br,D], S is countable, P is irreducible
(in the matrix sense, that is, counting measure on S is an irreducibility measure)
and uniformly ergodic, f is an indicator function, μ=π, the invariant probability
measure of P, and the large deviation principle for {P [n−1S (f) ∈ ·]} does not
π n
hold for any rate function; here again, by Theorems A and B, it follows that Λ∗ (cid:11)=
f
φ∗ .
f,π
Nevertheless,asweshallseebelow,Λ∗ =φ∗ andthelargedeviationprinciple
f f,μ
for {P [n−1S (f)∈·]} with rate function Λ∗ - which does not depend on μ - does
μ n f
hold for a broad class of initial distributions μ under an assumption on P that is
weaker than uniform recurrence and an integrability condition on f.
In order to state our results, we define for C ∈S
τ =τ =inf{n≥1: X ∈C},
C n
(1.1) λ∗(C)=sup{λ≥0: supExeλτ <∞},
x∈C
λ∗ =sup{λ∗(C): C is a P-small set}
(see Chapter 2 for the notion of P-small set). Following Nummelin [Nu], P is
geometrically recurrentifitisHarrisrecurrentandλ∗ >0; itisgeometrically
ergodic if it is ergodic and λ∗ >0.