A local limit theorem for continued fractions

A local limit theorem for continued fractions Zbigniew S. Szewczak∗ 15 December 2009

Abstract It is shown that functionals of digits in continued fraction expansion satisfy either the DeMoivre–Gnedenko or the Shepp–Stone limit theorems if and only if their marginals are in the domain of attraction of the normal law. Key words: local limit theorem, domains of attraction, ψ–mixing, continued fractions. Mathematics Subject Classification (2000): 60F99, 11K50.

1

Introduction and result

Any irrational number x ∈ (0, 1] can be uniquely expressed as a simple non-terminating continued fraction x = [0 ; a1 (x), a2 (x), a3 (x) . . .], where [0 ; x1 , . . . , xn ] =

1 x1 +

,

1

n ∈ N = {1, 2, . . .}

1

x2 + x3 +

..

. + x1n

and an (x) ∈ N (cf. [24], Theorem 14). It follows from Euclid’s algorithm (cf. [25], p.334) that the continued fraction transformation T defined by 1 1 1 T (x) = − = − max{n ∈ N ; nx ≤ 1}, x ∈ (0, 1], x x x generates natural numbers an (x), called partial quotients, or digits via 1 a1 (x) = , an+1 (x) = a1 (T n (x)) x (cf. [22], p.14). Let P denote Gauss’ measure, i.e. Z 1 λ(dx) , P(A) = ln 2 A 1 + x where A ⊆ (0, 1) and λ is Lebesgue: measurable set and measure, respectively. It is well-known that P is an invariant measure for T (cf. [22], Theorem 1.2.1). ∗ Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, ul. Chopina 12/18, 87-100 Toru´ n, Poland, e-mail: [email protected]

1

Besides, if Fk m denote the σ–algebra generated by ak , ak+1 , . . . , am then by Corollary in [21] (see also Corollary 1.3.15 in [22]) P(A ∩ B) ∞ − 1|; P(A) · P(B) > 0, A ∈ F1 k , B ∈ Fn+k } ≤ C%n , P(A) · P(B) k∈N (1.1) √ 7−4 2 2 ln√ 2 where C ≤ 7−4 2 , % = and ψ1 ≤ 2 ln 2 − 1 < 0.39 (i.e. {an } is an 2 exponentially fast ψ–mixing sequence (cf. [6])). It turns out that some classical limit theorems for the i.i.d. random variables have the full analogs for functionals of digits of continued fraction expansion (see e.g. [34], [39]). The principal result of this note states that this is also the case for normal local limit theorems, i.e. we obtain the continued fraction analog of Theorem 4.2.1 in [20] and Theorem 8.4.2 in [5] (see also [36]; [31]). ThereP exist a rich literature devoted to the asymptotic behaviour of Birkhoff’s k sums n−1 k=0 g ◦ T for some measurable transformations T and functions g. In particular local limit theorems were obtained for (among other transformations) Lasota–Yorke maps with g of bounded variation (cf. [33]; [28]) and for mixing Gibbs–Markov maps (which are exponentially ψ–mixing) with Lipschitz or (locally) Hölder continuous g (cf. [15]; [1]; [2]; [3]; [41]; [13]). In these works variants of Nagaev’s method (cf. [29], [30], [17]) including weak perturbation method (cf. [23], [19]) are used. Since the continued fraction transformation is an example of Gibbs–Markov map (see [10]) thus the above papers include special cases of the presented local limit theorem, which gives necessary and sufficient conditions. On the other hand many papers deal with limit theorems for mixing strictly stationary sequences {Xk }, which can be always represented by a measure preserving transformation as Xk = X0 (T k ). In this case the usual way is to impose only assumptions on distribution of X1 (preferably as in the i.i.d. case) ([9], page 144, problem 1). Furthermore, often the exponential rate of ψ–mixing can be weakened. The reason is that some classical inequalities such as Kolmogorov, Khinchine (cf. [39]) Lévy–Ottaviani–Etemadi, HoffmannJørgensen (cf. [40]) hold without restriction on mixing rate. Nevertheless, for some weaker mixing conditions the Keller–Liverani theorem (cf. [19]) applies at the cost of the conditional structure of the limit theorem ([38], Theorems 3 &4). However, it is worth noting that for the particular limit theorem yet another technique may yield more general results, an example is the Bergström approach to the Berry–Esseen theorem (cf. [32], [19]). In order to state the main result fix a Borel function f taking values on the real line R and such that E[f (a1 )] = 0. Define also the sequence bn as follows: P √ if E[f 2 (a1 )] < ∞ then bn = σ n where σ 2 = E[f 2 (a1 )] + 2 ∞ E[f (a1 )f (ak )] k=2 2 (cf. [20], Theorem 18.5.2 and [12], Remark 5) and if E[f (a1 )] = ∞ then bn = sup{x > 0 ; x−2 E[f 2 (a1 )I[|f (a1 )|≤x] ] ≥ n1 }. ψn = sup sup{|

Theorem. Assume E[f 2 (a1 )I[|f (a1 )|≤x] ] is a slowly varying function and E[f (a1 )] = 0. If L(f (a1 )) is arithmetic on {w + kd : k ∈ Z}, Z = {. . . , −1, 0, 1, . . .}, for

2

some w ∈ R and d > 0 the maximal span then n X √ (nw + kd)2 sup | 2πbn P[ f (aν ) = nw + kd] − d exp{− }| = o(1); 2b2n k∈Z

(1.2)

ν=1

if L(f (a1 )) is non-arithmetic then Z n X √ r2 f (aν ) − r)] − exp{− 2 } G(y)dy| = o(1), sup | 2πbn E[G( 2bn r∈R

(1.3)

ν=1

for any continuous function G with a compact support. Conversely, assume that (1.3) or (1.2) holds for some sequence bn , then E[f 2 (a1 )I[f (a1 )≤x] ] is a slowly varying function in the sense of Karamata, the span d is maximal and E[f (a1 )] = 0. The paper is organized as follows: in Section 2 there are three results required for proof of the Theorem in Section 3. This proof does not make the use of the spectral perturbation method (cf. [30], [35], [33], [15], [28], [2], [17], [38]) and it yields λ instead of P. The general strategy is to use Fourier transform inversion formula (Proposition 2) over distributional limit theorem (Theorem 3 in [39]). This is possible provided that we have the proper estimate on characteristic functions of sums (Lemma 1). Moreover, this method is applicable in non-stationary (non-homogeneous) and multidimensional case, too.

2

Preliminaries

P Let Eλ denote the mean value under λ and Sn = nk=1 f (ak ). It is well-known that λ[a1 = i] = i−1 (i + 1)−1 (cf. [24], Ch. III, §12). The first result follows from Theorem 3 in [39], Theorem 1.3.14 in [22], the proof of Theorem 14.2 in [4] (see also p.206) and equivalence of the measure λ and P (cf. [22], p.63, -4).

Proposition 1 Let f be a Borel function.PIn order that there exist sequences

{vn } and {An } such that limn→∞ L(vn−1 ( nk=1 f (ak ) − An )) = N (0, 1) with respect to the Lebesgue measure it is necessary and sufficient that the function Eλ [f 2 (a1 )I[|f (a1 )|≤x] ] is slowly varying in the sense of Karamata. If the latter condition is satisfied, we can set An = nE[f (a1 )] and vn = bn . The second result is

Lemma 1 Let f be a Borel function. Then |Eλ [eiθSn ]| ≤ (1 −

n−1 1 (1 − |Eλ [eiθf (a1 ) ]|2 )) 2 , 324

θ ∈ R, n ∈ N.

Proof of Lemma 1 Recall the incomplete quotients sn s0 (x) = 0,

sn (x) = [0 ; an (x), . . . , a1 (x)],

3

n∈N

(2.4)

and define S0 = {0}, Sk = {(z + i)−1 ; i ∈ N, z ∈ Sk−1 }, k ∈ N. Let Q denote the set of rational numbers. For z, z 0 ∈ Q0 := Q ∩ [0, 1) put 1 z+1 ISk−1 (z)IN ( 0 − z), 1 z + z)(b z 0 c + z + 1)

(k)

λzz 0 =

(b z10 c

k ∈ N.

By Corollary 1.2.8 in [22] (see also p.36) {sn }n∈N∪{0} is a Markov chain on (k)

(Q0 , λ) starting from 0 and defined via the transitions λzz 0 , k ∈ N. Further, if (k) we set µz = (b z1 c)−1 (b z1 c + 1)−1 ISk (z) then for any z, z 0 ∈ Q0 1 (k) (k) (k) µ 0 ISk−1 (z) ≤ λzz 0 ≤ 2µz 0 ISk−1 (z), 3 z

k∈N

and we can adapt the arguments from the proof of Lemma 1.5 in [30]. More precisely, we have (with the convention 00 ≡ 0 ) (

X

zk−1

iθf (b z

X

(k) (k) −1 2 λ(k−1) zk−2 zk−1 λzk−1 zk (µzk ) ) − |

e

1 c) k−1

(k) (k) −1 2 λ(k−1) zk−2 zk−1 λzk−1 zk (µzk ) |

zk−1

X

=

0 zk−1 ,zk−1

(k) (k) (k) −1 (k−1) −1 λ(k−1) λ0 (µ(k) zk−2 zk−1 λzk−1 zk (µzk ) λzk−2 z 0 zk ) k−1 zk−1 zk

×(1 − cos(θ(f (b ≥ 3−4

(k−1)

X

µ(k−1) zk−1 ISk−2 (zk−2 )µz 0

k−1

0 zk−1 ,zk−1

0 ) ISk−2 (zk−2

×(1 − cos(θ(f (b −4

=3

XX

(1 − cos(θ(f (ik−1 ) −

1 1 c) − f (b 0 c)))) zk−1 zk−1

1 zk−1

c) − f (b

f (i0k−1 ))))λ[a1

1

c)))) 0 zk−1

= ik−1 ]λ[a1 = i0k−1 ]

ik−1 i0k−1

= 3−4 (1 − |Eλ [eiθf (a1 ) ]|2 ).

(2.5)

(k)

Here we used the fact that µz = λ[a1 = b z1 c] on Sk . On the other hand for any function |g| ≤ 1 XX iθf (b z 1 c) iθf (b z1 c) (k) k−1 k e g(zk )| (2.6) | λ(k−1) zk−2 zk−1 λzk−1 zk e zk−1 zk

≤

X X iθf (b z 1 c) (k) k−1 | λ(k−1) | zk−2 zk−1 λzk−1 zk e zk

≤1−

zk−1

X

µ(k) zk (

zk

X

(k) (k) −1 λ(k−1) zk−2 zk−1 λzk−1 zk (µzk )

zk−1

−|

X

iθf (b z

e

1 c) k−1

(k) (k) −1 λ(k−1) zk−2 zk−1 λzk−1 zk (µzk ) |).

zk−1

Now, since X X iθf (b 1 c) (k) (k) −1 −1 zk−1 λz(k−1) λ(k) (µ(k) +| e λ(k−1) zk−2 zk−1 λzk−1 zk (µzk ) | zk ) k−2 zk−1 zk−1 zk zk−1

zk−1

4

≤ 2(1 −

X

X

(k) (k) −1 λ(k−1) zk−2 zk−1 (1 − λzk−1 zk (µzk ) )) ≤ 2(1 +

λ(k−1) zk−2 zk−1 ) = 4

zk−1

zk−1

thus by (2.5) and (2.6) we obtain |

XX

iθf (b z

(k) λ(k−1) zk−2 zk−1 λzk−1 zk e

1 c) k−1

iθf (b z1 c)

e

k

g(zk )|

zk−1 zk

X 1 eiθf (i) λ[a1 = i]|2 ) = γ(θ). (1 − | 34 22

≤1−

i≥1

With k = 2n − 2 this yields iθf (b z

X

|

e

1 c) 2n−3

iθf (b z

λ(2n−3) z2n−4 z2n−3 e

1 c) 2n−2

λ(2n−2) z2n−3 z2n−2

z2n−3 ,z2n−2 ,z2n−1 ,z2n iθf (b z

×e

1 c) 2n−1

iθf (b z 1 c) (2n) 2n λz2n−1 z2n g(z2n )|

λz(2n−1) e 2n−2 z2n−1 iθf (b z

X

= γ(θ)|

e

1 c) 2n−3

iθf (b z

λz(2n−3) e 2n−4 z2n−3

1 c) 2n−2

λ(2n−2) z2n−3 z2n−2

z2n−3 ,z2n−2

×(

X

iθf (b z

e

1 c) 2n−1

iθf (b z 1 c) (2n) 2n λz2n−1 z2n g(z2n )γ −1 (θ))|

λz(2n−1) e 2n−2 z2n−1

z2n−1 ,z2n

≤ γ 2 (θ). Repeating the above procedure with k = 2n − 4, . . . , 2 we arrive to iθ

P2n

f (b

1

c)

k=1 sk ]| |Eλ [eiθS2n ]| = |Eλ [e X 1 iθf (b z c) (1) iθf (b z1 c) (2) iθf (b z 1 c) (2n) 1 2 2n =| e λ0z1 e λ z1 z2 · · · e λz2n−1 z2n |

z1 ,z2 ,...,z2n

≤ γ n (θ) = (1 −

X 1 (1 − | eiθf (i) λ[a1 = i]|2 ))n . 324 i≥1

Consequently, for some |g| ≤ 1 |Eλ [eiθS2n+1 ]| ≤ γ n (θ)|

X

iθf (b z1 c) (1) 1 λ0z1 g(z1 )|

e

≤ γ n (θ)

z1

X |g(i−1 )| i

i(i + 1)

≤ γ n (θ).

This proves Lemma 1. The third result easily follows from the proof of Theorem 2 in [37]. Its proof is provided here for the reader’s convenience.

Proposition 2

Suppose φn (θ) = E[eiθYn ] and φn (vn−1 θ)e−iθAn →n φ(θ),

∀ θ ∈ R,

where vn > 0 and φ(θ) is an absolutely integrable characteristic function. If sup vn |φn (θ)| = o(1), |θ|∈[a,b]

5

∀ a, b ≥ 0,

(2.7)

and

Z lim sup

T →∞ n

ξvn ≥|θ|>T

|φn (vn−1 θ)|dθ = 0,

∀ ξ > 0,

(2.8)

then for any continuous function G with a compact support Z Z r sup |2πvn E[G(Yn − An vn − r)] − e−iθ vn φ(θ)G(y)dθ dy| = o(1). r∈R

Proof of Proposition 2 By [7] (cf. p.30) G admits monotone ε-approximation, i.e. for any ε > 0 there exist functions G+ , G− such that G− ≤ G ≤ G+ , G+ , G− are absolutely integrable, Z (G+ (y) − G− (y))dy ≤ ε,

and ˆ + (θ) = G

Z

ˆ − (θ) = G

eiθy G+ (y)dy,

Z

eiθy G− (y)dy,

are also compactly supported. ˆ + (−θ) so by the inversion formula Since G+ (−y) has Fourier transform G + for G (−y) we get En = E[G+ (Yn − An vn − r)] Z = G+ (y − An vn − r)P [Yn ∈ dy] Z Z 1 ˆ + (−θ)P [Yn ∈ dy]dθ = eiθ(y−An vn −r) G 2π Z Z 1 ˆ + (−θ) eiθ(y−An vn ) P [Yn ∈ dy]dθ = e−iθr G 2π Z 1 ˆ + (−θ)φn (θ)e−iθAn vn dθ = e−iθr G 2π Z 1 ˆ + (−θ)φn (θ)e−iθAn vn dθ = e−iθr G 2π |θ|≤ξ Z 1 ˆ + (−θ)φn (θ)e−iθAn vn dθ + e−iθr G 2π ξT

By the assumption that L(f (a1 )) is non-lattice we obtain Z n−1 θ 2 b lim exp − (1 − |λ( )| ) dθ = 0 n 648 bn ξbn ≥|θ|>η −1 bn and (3.11) follows. By Proposition 2 we get the desired conclusion for λ. It follows from the above that for a sequence of integers such that pn → ∞, pn ≥ 2, pn = o(n), (1.3) holds for Sn+pn − Spn , too. Since by Theorem 1.3.13 in [22], sup{|

λ(A) ln 2 pn −2 − 1| ; A ∈ Fp∞n } ≤ ·% P(A) 2

therefore n X bn ln 2 pn −2 bn sup |Eλ [G(Sn+pn − Spn − r)] − E[G( f (aν ) − r)]| ≤ ·% . 2 r∈R ν=1

Hence, if {pn } is choosen such that the r.h.s of the above tends to 0 then the relation (1.3) holds for P, as well. Now, assume L(f (a1 )) is lattice on {nw+kd}. 8

Since ψ1 < 0.39 thus random variables Sn are almost surely concentrated on {nw + kd} with the same w, d and d λ[Sn = nw + kd ] = 2π

Z

π d

−π d

e−iθ(nw+kd) φn (θ)dθ.

So that one can use the proof on p.262 in [11] (cf. [20], pp.121-124) together with the previous step. Conversely, assume that either (1.2) or (1.3) holds. Whence the convergence of characteristic functions is uniform on finite intervals (cf. [27], p.191). Thus by Theorem 3 in [39] the function E[f 2 (a1 )I[f (a1 )≤x] ] varies slowly. Moreover, by the proof of Theorem 4.2.1 in [20] d is maximal. Theorem is proved under the measure P. By equivalence of the measure λ and P it holds for the Lebesgue measure, too.

Remark 1 In view of the contents of the present note and the results in [8] and [14] non-normal local limit theorems can be obtained (see also [26], [16], [18], [2]). In particular if we denote by g the density function with the characteristic function Z 1

π

eiθy g(y)dy = e− ln 2 (iθγ+ 2 |θ|+iθ ln |θ|) ,

where γ is Euler’s constant, then by Theorem 2 in [16], Lemma 1 and the proof of Theorem 4.2.1 in [20] it follows that sup |nP[a1 + a2 + · · · + an = k] − g( k∈Z

k − log2 n)| = o(1). n

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