Almost Sure Local Limit Theorem for the Dickman distribution
aa r X i v : . [ m a t h . P R ] S e p Almost Sure Local Limit Theorem for the Dickman distribution
Rita Giuliano ∗ Zbigniew Szewczak † Michel Weber ‡ October 10, 2018
Abstract
In this paper we present a new correlation inequality and use it for proving an Almost SureLocal Limit Theorem for the so–called Dickman distribution. Several related results are alsoproved.
Keywords : General almost sure limit theorem, almost sure local limit theorem, Dickman func-tion, Dickman distribution, characteristic function, correlation inequality, cumulants, Poissondistribution, Bernoulli distribution. : Primary 60F15, Secondaries 62H20, 11N37.
The Dickman function plays an important role in analytic number theory; see [5],[11] for furtherinformation; see also the very recent paper [12], where a new application is given.Besides its importance in number theory, the Dickman function also appears in a large number ofproblems in several other fields: probability, informatics, algebra; we refer to the paper [6] (wherethe new example of Hoare’s quickselect algorithm is illustrated) and the references therein.In the same paper [6], a Local Limit Theorem concerning the Dickman function is stated withoutproof (the authors only refer to Corollary 2.8 in [1]; but the use of this result is not at all easy).Whenever a Local Limit Theorem exists, one can wonder whether it can be accompanied by anAlmost Sure version: see for instance the papers [3], [4] and [13] for examples concerning theLocal Limit Theorem and the Almost Sure Local Limit Theorem for partial sums of i.i.d. randomvariables; see also [2] for examples of an Almost Sure Local Limit Theorem for Markov chains.The purpose of the present paper is twofold: first, we give a detailed proof of the Local LimitTheorem announced in [6]; second, we answer affirmatively the natural question whether some sortof Almost Sure Local Limit Theorem can be stated and proved.It is worth noting that both our proof of the Local Theorem and the proof of the Almost SureLocal Theorem rely on a new result (Proposition 3.2) that links the behaviour of the distributionfunction of the involved partial sums with their local behaviour.As it often happens in the theory of Almost Sure Theorems, the crucial point for the proof of ourAlmost Sure Local Theorem is a new correlation inequality that can have some interest on its own. ∗ Address: Dipartimento di Matematica ”L. Tonelli”, Universit`a di Pisa, Largo Bruno Pontecorvo 5, I-56127 Pisa,Italy. e-mail: [email protected] † Address: Nicolaus Copernicus University, Faculty of Mathematics and Computer Science, ul. Chopina 12/1887-100 Toru´n, Poland. e-mail: [email protected] ‡ Address: IRMA, UMR 7501, 7, rue Ren´e–Descartes, 67084 Strasbourg Cedex, France. e-mail: [email protected]
Notation . By the symbol C we denote a positive constant, the value of which may change fromone case to another. We shall not make any distinction between absolute constants and constantsdepending on some parameter of the problem. Let ρ be the Dickman function, i.e. the function defined on [0 , + ∞ ) by the two conditions(i) ρ ( x ) = 1 , x
1; (ii) xρ ′ ( x ) + ρ ( x −
1) = 0 , x > . It is known (see [5], Lemma 2.6) that Z ∞ ρ ( x ) dx = e γ , where γ is the Euler–Mascheroni constant; hence x e − γ ρ ( x ), x >
0, is a probability density,known as the
Dickman density . The distribution function with this density is called the
Dickmandistribution and will be denoted with D . Thus its probability density is D ′ ( x ) = e − γ ρ ( x ). Someproperties of the Dickman function and the Dickman distributions will be discussed in the nextsection.We are interested in the probabilistic model introduced in [6]: precisely, let ( Z k ) k > be independentand such that, for each k , Z k = ( k − k . For every pair of integers ( m, n ) with 0 m < n denote T nm = n X k = m +1 kZ k . For simplicity we also put T n = T n .Here are the main results of this paper. Theorem 2.1 (Local Limit Theorem).
Let ( κ n ) n > be any sequence of integers with lim n →∞ κ n n = x > . Then lim n →∞ nP ( T n = κ n ) = e − γ ρ ( x ) . Theorem 2.2 (Almost Sure Local Limit Theorem).
Let x > be fixed and let κ = ( κ n ) n > be a strictly increasing sequence with lim n →∞ κ n n = x > . Then lim N →∞ N N X n =1 { T n = κ n } = e − γ ρ ( x ) , a.s. The proof of Theorem 2.2 is in Section 7.
Corollary 2.3
We have lim N →∞ N N X n =1 { T n = n } = e − γ , a.s. As a consequence, for every x > , lim N →∞ P Nn =1 { T n =[ xn ] } P Nn =1 { T n = n } = ρ ( x ) , a.s. Remark 2.4
This corollary suggests two simulation procedures for estimating (i) Euler’s constant γ and (ii) the values of Dickman’s function ρ .Anyway, we have not investigated the goodness of these methods, nor compared them with theexisting ones. For the values of the Dickman function see for instance [5], Corollary 2.3. Let ρ and D be the Dickman function and the Dickman distribution respectively. It is easy to seethat D ( x ) − D ( x −
1) = e − γ xρ ( x ) = xD ′ ( x ) , x > . (1)In fact, denoting provisorily f ( x ) = D ( x ) − D ( x −
1) and e − γ xρ ( x ) = g ( x ), x >
1, we have, by (i)of section 2, f (1) = Z D ′ ( t ) dt = Z e − γ ρ ( t ) dt = e − γ = e − γ ρ (1) = g (1);and by (ii) f ′ ( x ) = ddx (cid:16) Z xx − D ′ ( t ) dt (cid:17) = ddx (cid:16) Z xx − e − γ ρ ( t ) dt (cid:17) = e − γ ρ ( x ) − e − γ ρ ( x − e − γ (cid:16) ρ ( x ) + xρ ′ ( x ) (cid:17) = ddx (cid:16) e − γ xρ ( x ) (cid:17) = g ′ ( x ) , x > . We also recall that the characteristic function of the Dickman distribution is φ ( t ) = exp n Z e itu − u o du, (2)see [5] again, or [6]. 3et ( Z k ) k > and T nm = P nk = m +1 kZ k (0 m < n ) be as in Section 2. The characteristic functionof Z k is φ Z k ( t ) = 1 + e it − k . The characteristic function of T nm is φ T nm ( t ) = n Y k = m +1 φ Z k ( tk ) = n Y k = m +1 (cid:16) e itk − k (cid:17) . The proof of the following result is identical to the one given in [6] (Proposition 1) for the case m n ≡ Proposition 3.1
Let ( m n ) n > be a sequence of integers such that lim n →∞ ( n − m n ) = + ∞ . Then,as n → ∞ , the sequence T nmn n − m n converges in distribution to the Dickman law. Now we present a result that will be crucial for the proof of the correlation inequality of section 5.Its aim is to connect the local behaviour of T nm to the behaviour of its distribution function; it canbe considered as a quantitative version of Theorem 2.6 in [1]. Proposition 3.2
Let ( κ n ) n be any increasing sequence of integers. Then, for n > m ≥ , (cid:12)(cid:12) ( κ n − κ m ) P ( T nm = κ n − κ m ) − P (cid:0) ( κ n − κ m ) − n < T nm ( κ n − κ m ) − ( m + 1) (cid:1)(cid:12)(cid:12) C nm √ n − m . Proof.
We need a preliminary easy result.
Lemma 3.3
Let T be a random variable taking integer values and with characteristic function φ T .For every integers κ , a and b with a < b we have the formula P ( κ − b T κ − a ) = 12 π Z π − π e − itκ (cid:16) b X j = a e itj (cid:17) φ T ( t ) dt. Proof.
By the inversion formula we can write P ( κ − b T κ − a ) = κ − a X j = κ − b P ( T = j ) = 12 π Z π − π (cid:16) κ − a X j = κ − b e − itj (cid:17) φ T ( t ) dt, and now, for every t ∈ R , κ − a X j = κ − b e − itj = e − it ( k − b ) + e − it ( k − b +1) + · · · + e − it ( k − a ) = e − itk b X j = a e itj (cid:3) We pass to the proof of Proposition 3.2. First, by Lemma 3.3 P (cid:0) ( κ n − κ m ) − n < T nm ( κ n − κ m ) − ( m + 1) (cid:1) = 12 π Z π − π e − iu ( κ n − κ m ) (cid:16) n X k = m +1 e iuk (cid:17) φ T nm ( u ) du. (3)4oreover, integrating by parts,12 πi Z π − π e − iu ( κ n − κ m ) · φ ′ T nm ( u ) du = 12 πi n φ T nm ( u ) e − iu ( κ n − κ m ) (cid:12)(cid:12)(cid:12) π − π + i ( κ n − κ m ) Z π − π φ T nm ( u ) e − iu ( κ n − κ m ) du o = 12 πi n(cid:16) φ T nm ( π ) e − iπ ( κ n − κ m ) − φ T nm ( − π ) e iπ ( κ n − κ m ) (cid:17) + i ( κ n − κ m ) Z π − π φ T nm ( u ) e − i ( κ n − κ m ) u du o = 12 πi n(cid:16) φ T nm ( π ) e − iπ ( κ n − κ m ) − φ T nm ( π ) e − iπ ( κ n − κ m ) (cid:17) + 2 πi ( κ n − κ m ) P ( T nm = κ n − κ m ) o = Im (cid:0) φ T nm ( π ) e − iπ ( κ n − κ m ) (cid:1) π + ( κ n − κ m ) P ( T nm = κ n − κ m ) = ( κ n − κ m ) P ( T nm = κ n − κ m ) , (4)noticing that φ T nm ( π ) and e − iπ ( κ n − κ m ) are real (recall that κ n is an integer). Since φ ′ T nm ( u ) = φ T nm ( u ) n X k = m +1 kφ ′ Z k ( ku ) φ Z k ( ku ) , subtracting (3) from (4) we obtain( κ n − κ m ) P ( T nm = κ n − κ m ) − P (cid:0) ( κ n − κ m ) − n < T nm ( κ n − κ m ) − ( m + 1) (cid:1) = 12 π Z π − π e − iu ( κ n − κ m ) · φ T nm ( u ) (cid:16) n X k = m +1 ki φ ′ Z k ( ku ) φ Z k ( ku ) (cid:17) du − π Z π − π e − iu ( κ n − κ m ) (cid:16) n X k = m +1 e iuk (cid:17) φ T nm ( u ) du = 12 π Z π − π e − iu ( κ n − κ m ) · φ T nm ( u ) (cid:16) n X k = m +1 ki φ ′ Z k ( ku ) φ Z k ( ku ) − n X k = m +1 e iuk (cid:17) du = n − m π Z π − π e − iu ( κ n − κ m ) · φ T nm ( u ) P nk = m +1 ki φ ′ Zk ( ku ) φ Zk ( ku ) − P nk = m +1 e iuk n − m | {z } = γ m,n ( u ) du = 12 π Z π ( n − m ) − π ( n − m ) e − iu κn − κmn − m · φ Tnmn − m ( u ) γ m,n (cid:16) un − m (cid:17) du. Hence (cid:12)(cid:12)(cid:12) ( κ n − κ m ) P ( T nm = κ n − κ m ) − P (cid:0) ( κ n − κ m ) − n < T nm ( κ n − κ m ) − ( m + 1) (cid:1)(cid:12)(cid:12)(cid:12) π Z π ( n − m ) − π ( n − m ) (cid:12)(cid:12)(cid:12) e − iu κn − κmn − m · φ Tnmn − m ( u ) γ m,n (cid:16) un − m (cid:17)(cid:12)(cid:12)(cid:12) du π n Z π ( n − m ) − π ( n − m ) (cid:12)(cid:12)(cid:12) e − iu κn − κmn − m · φ Tnmn − m ( u ) (cid:12)(cid:12)(cid:12) du o n Z π ( n − m ) − π ( n − m ) (cid:12)(cid:12)(cid:12) γ m,n (cid:16) un − m (cid:17) | du o π n Z π ( n − m ) − π ( n − m ) (cid:12)(cid:12)(cid:12) φ Tnmn − m ( u ) (cid:12)(cid:12)(cid:12) du o · n π ( n − m ) sup − π u π | γ m,n ( u ) | o (5)At the end of this proof we shall show thatsup − π u π | γ m,n ( u ) | C nm n − m . (6)5sing (6) in (5), we get n Z π ( n − m ) − π ( n − m ) (cid:12)(cid:12)(cid:12) φ Tnmn − m ( u ) (cid:12)(cid:12)(cid:12) du o · n π ( n − m ) sup − π u π | γ m,n ( u ) | o C n Z π ( n − m ) − π ( n − m ) (cid:12)(cid:12)(cid:12) φ Tnmn − m ( u ) (cid:12)(cid:12)(cid:12) du o · nm √ n − m . Since T nm n − m = T n n · nn − m − P mk =1 kZ k n − m , putting W = − P mk =1 kZ k n − m we can write, by independence, φ Tnmn − m ( u ) = φ Tnn (cid:16) u nn − m (cid:17) · φ W ( u )which implies n Z π ( n − m ) − π ( n − m ) (cid:12)(cid:12)(cid:12) φ Tnmn − m ( u ) (cid:12)(cid:12)(cid:12) du o · nm √ n − m n Z π ( n − m ) − π ( n − m ) (cid:12)(cid:12)(cid:12) φ Tnn (cid:16) u nn − m (cid:17)(cid:12)(cid:12)(cid:12) du o · nm √ n − m = n − mn n Z πn − πn (cid:12)(cid:12)(cid:12) φ Tnn ( u ) (cid:12)(cid:12)(cid:12) du o · nm √ n − m n Z πn − πn (cid:12)(cid:12)(cid:12) φ Tnn ( u ) (cid:12)(cid:12)(cid:12) du o · nm √ n − m . In Proposition 4.1 of the next section we shall prove that Z πn − πn (cid:12)(cid:12)(cid:12) φ Tnn ( u ) (cid:12)(cid:12)(cid:12) du → Z ∞−∞ | φ ( u ) | du < ∞ , where φ is as in (2); this concludes the proof.It remains to prove (6). Write( n − m ) γ m,n ( u ) = n X k = m +1 (cid:16) ki φ ′ Z k ( ku ) φ Z k ( ku ) − e iuk (cid:17) = n X k = m +1 (cid:16) ki ie iuk k e iuk − k − e iuk (cid:17) = n X k = m +1 e iuk (cid:16) kk − e iuk − (cid:17) = n X k = m +1 e iuk (1 − e iuk ) k − e iuk = n X k = m +1 e iuk (1 − e iuk )( k − e − iuk ) | k − e iuk | . Hence ( n − m ) | γ n ( u ) | n X k = m +1 k | k − e iuk | n X k = m +1 k ( k − C (cid:16) nm (cid:17) , since (cid:12)(cid:12) k − e iuk (cid:12)(cid:12) > (cid:12)(cid:12)(cid:12) ( k − − (cid:12)(cid:12) e iuk (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = k − , k > m + 1 > (cid:12)(cid:12) k − e − iuk (cid:12)(cid:12) ( k −
1) + (cid:12)(cid:12) e − iuk (cid:12)(cid:12) = k, k > n X k = m +1 k ( k − = n − X k = m − k + n − X k = m − k n − X k = m − k + C m − Z n − m − x dx + C C + log n − m − C + log nm − m − m C + log 2 nm = C + log nm C (cid:16) nm (cid:17) . (cid:3) Now we can give the
Proof of Theorem 2.1 . Assume that we are able to prove thatlim n →∞ nP ( T n = e κ n ) = e − γ ρ ( x )for every sequence of integers ( e κ n ) n > with lim n →∞ e κ n n = x . Denote U = Z + 2 Z and notice that U is independent on T n and takes the values 0,1,2,3. Now for each h = 0 , , ,
3, take the sequence( e κ ( h ) n ) n > defined by e κ ( h ) n = κ n − h. Since e κ ( h ) n n = κ n − hn → x as n → ∞ , we have nP ( T n = κ n ) = X h =0 P ( U = h ) (cid:8) nP ( T n = κ n − h ) (cid:9) = X h =0 P ( U = h ) (cid:8) nP ( T n = e κ ( h ) n ) (cid:9) → (cid:16) X h =0 P ( U = h ) (cid:17) e − γ ρ ( x ) = e − γ ρ ( x ) . and the claim is proved. So, let ( e κ n ) n be a sequence with lim n →∞ e κ n n = x . By Proposition 3.1 andthe continuity of the Dickman distribution we have n e κ n · (cid:8) P ( e κ n − n < T n e κ n − (cid:9) = n e κ n · (cid:8) P ( e κ n − nn − < T n n − e κ n − n − (cid:9) → n x (cid:8) D ( x ) − D ( x − (cid:9) = e − γ ρ ( x ) , by (1). Consider the sequence ( κ ′ n ) n defined as κ ′ n = (e κ n n > n = 1 , . The estimation of Proposition 3.2 gives, for n > (cid:12)(cid:12)e κ n P ( T n = e κ n ) − P ( e κ n − n < T n e κ n − (cid:12)(cid:12) = (cid:12)(cid:12) ( κ ′ n − κ ′ ) P ( T n = κ ′ n − κ ′ ) − P ( κ ′ n − κ ′ − n < T n κ ′ n − κ ′ − (cid:12)(cid:12) C n √ n − , and the result follows. (cid:3) Remark 3.4
Concerning the proof of Proposition 3.2, notice that φ ′ Z k ( t ) φ Z k ( t ) = ψ ′ Z k ( t ) , (7)where ψ Z k ( t ) = log φ Z k ( t ), i.e. the second characteristic function of Z k .7rite ψ Z k ( t ) = ∞ X j =1 c ( k ) j ( it ) j j ! , where (cid:0) c ( k ) j (cid:1) j is the sequence of the cumulants of the B (cid:0) , k (cid:1) distribution. Hence ψ ′ Z k ( t ) = ∞ X j =1 ic ( k ) j ( it ) j − ( j − . (8)Denote by ψ Π k ( t ) the second characteristic function of the Poisson law with parameter k ., i.e. ψ Π k ( t ) = e it − k , ψ ′ Π k ( t ) = ik e it = ik ∞ X j =1 ( it ) j − ( j − . (9)Hence, by (7) and (9), γ m,n ( t ) = P nk = m +1 ki φ ′ Zk ( tk ) φ Zk ( tk ) − P nk = m +1 e itk n − m = P nk = m +1 ki (cid:0) φ ′ Zk ( tk ) φ Zk ( tk ) − ik e itk (cid:1) n − m = P nk = m +1 ki (cid:0) ψ ′ Z k ( tk ) − ψ ′ Π k ( tk ) (cid:1) n − m . Since, by (8) and (9), ψ ′ Z k ( t ) − ψ ′ Π k ( t ) = ∞ X j =1 (cid:16) ic ( k ) j − ik (cid:17) ( it ) j − ( j − , we get γ m,n ( t ) = P nk = m +1 P ∞ j =1 (cid:16) kc ( k ) j − (cid:17) ( itk ) j − ( j − n − m = ∞ X j =1 ( it ) j − ( j − n P nk = m +1 k j − ( kc ( k ) j − n − m o . (10)Putting α ( m,n ) j = P nk = m +1 k j − ( kc ( k ) j − n − m , we obtain the formula γ m,n ( t ) = ∞ X j =1 ( it ) j − ( j − α ( m,n ) j . (11)Let B (1 , p ) be the Bernoullian law with parameter p ∈ (0 ,
1) an c j ( p ) the j − th cumulant of B (1 , p ).In Section 7 we give an explicit form for the quantity c j ( p ) p − p = k , this quantity is precisely the expression kc ( k ) j − kc ( k ) j −
1, and in turn of γ m,n (see (11)). 8he following result specifies Proposition 1 of [6] quantitatively in terms of the characteristicfunctions. Proposition 3.5
There exists an absolute constant C such that for all integers n > m > and allreal numbers t , (cid:12)(cid:12)(cid:12)(cid:12) φ Tnmn − m ( t ) − exp n Z e itu − u du o(cid:12)(cid:12)(cid:12)(cid:12) f m,n ( t ) , where f m,n ( t ) = exp ( Ct (cid:16) log nm ( n − m ) + m + 2 n − m (cid:17)) − . Proof.
First (cid:12)(cid:12)(cid:12)(cid:12) exp n Z e itu − u du o − φ Tnmn − m ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) φ Tnmn − m ( t ) (cid:12)(cid:12)(cid:12)(cid:12) · (cid:12)(cid:12)(cid:12) exp n Z e itu − u du − n X k = m +1 log (cid:16) e it kn − m − k (cid:17)o − (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12) exp n n X k = m +1 h Z k − mn − mk − − mn − m e itu − u du − log (cid:16) e it kn − m − k (cid:17)io − (cid:12)(cid:12)(cid:12) exp ( n X k = m +1 (cid:12)(cid:12)(cid:12) Z k − mn − mk − − mn − m e itu − u du − log (cid:16) e it kn − m − k (cid:17)(cid:12)(cid:12)(cid:12)) − exp ( n X k = m +1 (cid:12)(cid:12)(cid:12) e it kn − m − k − log (cid:16) e it kn − m − k (cid:17)(cid:12)(cid:12)(cid:12)| {z } (a) ++ n X k = m +1 (cid:12)(cid:12)(cid:12) Z k − mn − mk − − mn − m e itu − u du − e it kn − m − k (cid:12)(cid:12)(cid:12)| {z } (b) ) − . We shall prove that n X k = m +1 (cid:12)(cid:12)(cid:12) log (cid:16) e it kn − m − k (cid:17) − e it kn − m − k (cid:12)(cid:12)(cid:12) Ct (cid:16) log nm ( n − m ) + 1 n − m (cid:17) (a)and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z k − mn − mk − − mn − m e itu − u du − e it kn − m − k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( m + 1) t n − m ) , ∀ k ∈ ( m, n ] . (b)These inequalities give (cid:12)(cid:12)(cid:12)(cid:12) exp n Z e itu − u o du − φ Tnmn − m ( t ) (cid:12)(cid:12)(cid:12)(cid:12) exp ( Ct (cid:16) log nm ( n − m ) + 1 n − m + n X k = m +1 ( m + 1)2( n − m ) (cid:17)) − exp ( Ct (cid:16) log nm ( n − m ) + m + 2 n − m (cid:17)) − f m,n ( t ) . (a) The inequality (cid:12)(cid:12) e ix − (cid:12)(cid:12) = 2 (cid:12)(cid:12)(cid:12) sin x (cid:12)(cid:12)(cid:12) ∧ | x | x = tkn − m gives (cid:12)(cid:12)(cid:12) e it kn − m − k (cid:12)(cid:12)(cid:12) k ∧ | t | n − m . (12)From | log(1 + u ) − u | ∞ X j =2 | u | j j = | u | ∞ X j =0 | u | j j + 2 , | u | < , applied to u = e it kn − m − k (with k > m + 1; notice that (cid:12)(cid:12) e it kn − m − k (cid:12)(cid:12) < k > m + 1 >
3) we get n X k = m +1 (cid:12)(cid:12)(cid:12) log (cid:16) e it kn − m − k (cid:17) − e it kn − m − k (cid:12)(cid:12)(cid:12) n max m +1 k n (cid:12)(cid:12)(cid:12) e it kn − m − k (cid:12)(cid:12)(cid:12) o ∞ X j =0 j + 2 n X k = m +1 (cid:12)(cid:12)(cid:12) e it kn − m − k (cid:12)(cid:12)(cid:12) j |{z} by (12) t ( n − m ) (cid:16)
12 ( n − m ) + 23 n X k = m +1 k + ∞ X j =2 j + 2 n X k = m +1 (cid:16) k (cid:17) j (cid:17) C · t ( n − m ) (cid:16) ( n − m ) + log nm + ∞ X j =2 j j + 2 Z ∞ x j d x (cid:17) C · t ( n − m ) (cid:16) ( n − m ) + log nm + 1 (cid:17) Ct (cid:16) log nm ( n − m ) + 1 n − m (cid:17) . (b) Putting η t ( x ) = e itx − x , x > , − π t π, we can write (cid:12)(cid:12)(cid:12) Z k − mn − mk − − mn − m e itu − u du − e it kn − m − k (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z k − mn − mk − − mn − m e itu − u du − e it kn − m − n − m ) kn − m (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z k − mn − mk − − mn − m η t ( u ) du − η t (cid:16) kn − m (cid:17) n − m (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z k − mn − mk − − mn − m n η t ( u ) − η t (cid:16) kn − m (cid:17)o du (cid:12)(cid:12)(cid:12) n − m sup u ∈ [ k − − mn − m , k − mn − m ] (cid:12)(cid:12)(cid:12) η t ( u ) − η t (cid:16) kn − m (cid:17)(cid:12)(cid:12)(cid:12) n − m · n sup u ∈ [ k − − mn − m , k − mn − m ] (cid:12)(cid:12)(cid:12) u − kn − m (cid:12)(cid:12)(cid:12)o · n sup x ∈ R (cid:12)(cid:12) η ′ t ( x ) (cid:12)(cid:12)o = m + 1( n − m ) · sup x ∈ R (cid:12)(cid:12) η ′ t ( x ) (cid:12)(cid:12) ( m + 1) t n − m ) , since sup x ∈ R (cid:12)(cid:12) η ′ t ( x ) (cid:12)(cid:12) t , as we are going to prove. First (cid:12)(cid:12) η ′ t ( x ) (cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) itxe itx − e itx + 1 x (cid:12)(cid:12)(cid:12) = δ ( tx ) x , δ ( u ) = 2(1 − u sin u − cos u ) + u . Put now H ( u ) = u − δ ( u ). We have H ′ ( u ) = u − u (1 − cos u ) = 4 u (cid:16) u − sin u (cid:17) > , u > H is non–decreasing for u >
0, and from the fact that H (0) = 0, we deduce that H ( u ) > u >
0, hence also for every u ∈ R since H ( − u ) = H ( u ). In other words δ ( u ) u and asa consequence (cid:12)(cid:12) η ′ t ( x ) (cid:12)(cid:12) = δ ( tx ) x t . (cid:3) The following result specifies Proposition 1 of [6] quantitatively in terms of distribution functions.
Proposition 3.6
There exists an absolute positive constant C such that, for all positive integers n , m , with n > m > , sup x ∈ R (cid:12)(cid:12)(cid:12) P (cid:16) T nm n − m x (cid:17) − D ( x ) (cid:12)(cid:12)(cid:12) Cg m,n , where g m,n = exp C n log nm ( n − m ) + m + 2 n − m o log nm ! − nm . Proof.
In view of Theorem 2 p. 109 in [9], if τ is an arbitrary positive number, then for b > π sup x ∈ R (cid:12)(cid:12)(cid:12) P (cid:16) T nm n − m x (cid:17) − D ( x ) (cid:12)(cid:12)(cid:12) b Z τ − τ (cid:12)(cid:12) exp n R e itu − u du o − φ Tnmn − m ( t ) (cid:12)(cid:12) | t | d t + r ( b ) τ sup x ∈ R | D ′ ( x ) | where r ( b ) is a positive constant depending on b only. Hencesup x ∈ R (cid:12)(cid:12)(cid:12) P (cid:16) T nm n − m x (cid:17) − D ( x ) (cid:12)(cid:12)(cid:12) C n Z τ − τ (cid:12)(cid:12) exp n R e itu − u du o − φ Tnmn − m ( t ) (cid:12)(cid:12) | t | d t + 1 τ sup x ∈ R | D ′ ( x ) | o C n Z τ − τ f m,n ( t ) | t | d t + 1 τ o , by Proposition 3.5. Now, for every positive constant A we havesup x τ e Ax − x = e Aτ − τ . Applying this with A = C (cid:16) log nm ( n − m ) + m +2 n − m (cid:17) we obtain Z τ − τ f m,n ( t ) | t | d t τ e C (cid:16) log nm ( n − m )2 + m +2 n − m (cid:17) τ − τ = 2 (cid:16) e C (cid:16) log nm ( n − m )2 + m +2 n − m (cid:17) τ − (cid:17) . Hence, for every τ ,sup x ∈ R (cid:12)(cid:12)(cid:12) P (cid:16) T nm n − m x (cid:17) − D ( x ) (cid:12)(cid:12)(cid:12) C ( exp C n log nm ( n − m ) + m + 2 n − m o τ ! − τ ) . (13)11aking τ = log nm we getexp C n log nm ( n − m ) + m + 2 n − m o τ ! − τ = exp C n log nm ( n − m ) + m + 2 n − m o log nm ! − nm = g m,n , so that, from (13) sup x ∈ R (cid:12)(cid:12)(cid:12) P (cid:16) T nm n − m x (cid:17) − D ( x ) (cid:12)(cid:12)(cid:12) Cg m,n , as claimed. (cid:3) This section is devoted to a convergence result for the characteristic functions of T n n that has beenused before in the proof of Proposition 3.2; it gives also a weak form of the Local Limit Theorem(see Remark 4.2). Proposition 4.1
We have (a) Z + ∞−∞ (cid:12)(cid:12) φ ( t ) (cid:12)(cid:12) dt < + ∞ ;(b) Z πn − πn (cid:12)(cid:12)(cid:12) φ Tnn ( u ) (cid:12)(cid:12)(cid:12) du → Z ∞−∞ | φ ( u ) | du, n → ∞ . Proof. (a) By symmetry ( t (cid:12)(cid:12) φ ( t ) (cid:12)(cid:12) is an even function), it is sufficient to prove that Z + ∞ (cid:12)(cid:12) φ ( t ) (cid:12)(cid:12) dt < + ∞ . Theorem 2 p. 11 of [9] assures that there exist positive constants δ and C such that | φ ( t ) | − Ct , | t | < δ. This implies that Z δ | φ ( t ) | dt Z δ (1 − Ct ) dt = C. Let’s turn to R + ∞ δ | φ ( t ) | dt. First observe that (cid:12)(cid:12) φ ( t ) (cid:12)(cid:12) = φ ( t ) φ ( − t ) = exp n Z e itu − u du o · exp n Z e − itu − u du o = exp n − Z − cos tuu du o . (14)12ow, for every ǫ ∈ (0 , t ) Z − cos tuu du = Z t − cos zz dz > Z tǫ − cos zz dz = h z − sin zz i tǫ + Z tǫ z − sin zz dz = log tǫ − sin tt + sin ǫǫ − Z tǫ sin zz dz > log tǫ + C (the constant C might be negative here, but this is irrelevant as it appears clearly from thesequel). Hence, taking ǫ = δ , Z + ∞ δ | φ ( t ) | dt Z + ∞ δ exp n − (cid:16) log tδ + C (cid:17)o dt C Z + ∞ δ t dt = C. (b) By part (a), the Proposition will be proved if we show that Z πn − πn n(cid:12)(cid:12)(cid:12) φ Tnn ( u ) (cid:12)(cid:12)(cid:12) − | φ ( u ) | o du → , n → ∞ . (15)Recall that, by Proposition 3.1, (cid:12)(cid:12)(cid:12) φ Tnn (cid:12)(cid:12)(cid:12) converges to | φ | pointwise and uniformly on everybounded interval. Hence, for any positive A , Z A − A n(cid:12)(cid:12)(cid:12) φ Tnn ( u ) (cid:12)(cid:12)(cid:12) − | φ ( u ) | o du → , n → ∞ . Thus we are left with the proof of Z { A | t | nπ } n(cid:12)(cid:12)(cid:12) φ Tnn ( u ) (cid:12)(cid:12)(cid:12) − | φ ( u ) | o du → , n → ∞ . (16)We split the first member of (16) as follows: for a fixed ǫ ∈ (0 , Z { A | t | nπ } = Z { A | t | ǫπ √ n } + Z { ǫπ √ n | t | nπ } = I + I . We consider the two summands I and I separately.( I ) Notice that (cid:12)(cid:12)(cid:12) φ T n ( t ) (cid:12)(cid:12)(cid:12) = n Y k =1 (cid:12)(cid:12)(cid:12) e ikt − k (cid:12)(cid:12)(cid:12) = n Y k =1 n(cid:16) − k + 1 k cos kt (cid:17) + (cid:16) k sin kt (cid:17) o = n Y k =1 n k − k (cid:16) cos kt − (cid:17)o = exp n n X k =1 log h k − k (cid:16) cos kt − (cid:17)io . (17)Hence | I | Z { A | t | ǫπ √ n } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ Tnn ( t ) (cid:12)(cid:12)(cid:12) − exp n n X k =1 k − k (cid:16) cos ktn − (cid:17)o(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ++ Z { A | t | ǫπ √ n } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp n n X k =1 k − k (cid:16) cos ktn − (cid:17)o − exp n n X k =1 k (cid:16) cos ktn − (cid:17)o(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt ++ Z { A | t | ǫπ √ n } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp n n X k =1 k (cid:16) cos ktn − (cid:17)o − | φ ( t ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = I + I + I . We consider the three summand I , I and I separately.13 I ) First observe that, by relation (17), I = Z { A | t | ǫπ √ n } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ Tnn ( t ) (cid:12)(cid:12)(cid:12) − exp n n X k =1 k − k (cid:16) cos ktn − (cid:17)o(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z { A | t | ǫπ √ n } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp n n X k =1 log h k − k (cid:16) cos ktn − (cid:17)io − exp n n X k =1 k − k (cid:16) cos ktn − (cid:17)o(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt Z { A | t | ǫπ √ n } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp ( n X k =1 n log h k − k (cid:16) cos ktn − (cid:17)i − k − k (cid:16) cos ktn − (cid:17)o) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt, since 0 exp n n X k =1 k − k (cid:16) cos ktn − (cid:17)o . Now using the developmentlog(1 + z ) − z = X j > ( − j j z j , | z | < z = k − k (cid:0) cos ktn − (cid:1) (which, for sufficiently large n , is strictly less than 1 inmodulus for every k > h k − k (cid:16) cos ktn − (cid:17)i − k − k (cid:16) cos ktn − (cid:17) = X j > ( − j j ( k − j j · k j (cid:16) cos ktn − (cid:17) j = X j > j ( k − j jk j (cid:16) − cos ktn (cid:17) j > . It follows that Z { A | t | ǫπ √ n } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp ( n X k =1 n log h k − k (cid:16) cos ktn − (cid:17)i − k − k (cid:16) cos ktn − (cid:17)o) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z { A | t | ǫπ √ n } exp n n X k =1 X j > j ( k − j jk j (cid:16) − cos ktn (cid:17) j o − ! dt. − cos z z the above can be bounded by Z { A | t | ǫπ √ n } exp n n X k =1 X j > j ( k − j jk j (cid:16) ktn (cid:17) j o − ! dt = Z { A | t | ǫπ √ n } exp n n X k =1 X j > j ( k − j j (cid:16) tn (cid:17) j o − ! dt = Z { A | t | ǫπ √ n } exp n X j > j (cid:16) √ tn (cid:17) j n X k =1 ( k − j o − ! dt Z { A | t | ǫπ √ n } exp n X j > j (cid:16) √ tn (cid:17) j (cid:0) Z n x j dx (cid:1)o − ! dt Z { A | t | ǫπ √ n } exp n X j > j (cid:16) √ tn (cid:17) j (cid:16) n j +1 j + 1 (cid:17)o − ! dt = Z { A | t | ǫπ √ n } exp n X j > j ( j + 1) · ( √ t ) j n j − o − ! dt = Z { A | t | ǫπ √ n } exp n C t n X j > j + 3)( j + 2) · ( √ t ) j n j o − ! dt. Now, for | t | ǫπ √ n we have also | t | π √ n (recall that ǫ < n not dependent on ǫ such that, for n > n ( √ t ) n Cn . Hence, for n > n , X j > j + 3)( j + 2) · ( √ t ) j n j X j > j + 3)( j + 2) · j = C, and we get Z { A | t | ǫπ √ n } exp n C t n X j > j + 3)( j + 2) · · ( √ t ) j n j o − ! dt Z { A | t | ǫπ √ n } exp n C t n o − ! dt exp n C ( ǫπ √ n ) n o − ! ǫπ √ n = C · e C √ n − C √ n ǫ < Cǫ . ( I ) Here we observe that 0 exp n n X k =1 k (cid:16) cos ktn − (cid:17)o , I = Z { A | t | ǫπ √ n } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp n n X k =1 k − k (cid:16) cos ktn − (cid:17)o − exp n n X k =1 k (cid:16) cos ktn − (cid:17)o(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt Z { A | t | ǫπ √ n } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp n n X k =1 (cid:16) k − k − k (cid:17)(cid:16) cos ktn − (cid:17)o − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z { A | t | ǫπ √ n } exp n n X k =1 k (cid:16) − cos ktn (cid:17)o − ! dt Z { A | t | ǫπ √ n } exp n n X k =1 k (cid:16) ktn (cid:17) o − ! dt = Z { A | t | ǫπ √ n } (cid:16) e t n − (cid:17) dt (cid:16) e ǫπ √ n )2 n − (cid:17) ǫπ √ n = e ǫπ )2 n / − ǫπ ) n / · ǫπ ) n / ǫπ √ n → , n → ∞ . ( I ) Recalling the explicit form of | φ ( t ) | given in equation (14), we have I = Z { A | t | ǫπ √ n } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp n n X k =1 k (cid:16) cos ktn − (cid:17)o − | φ ( t ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt Z { A | t | ǫπ √ n } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp n n X k =1 k (cid:16) cos ktn − (cid:17)o − exp n Z cos tu − u du o(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt, and, observing again that0 exp n n X k =1 k (cid:16) cos ktn − (cid:17)o , we get I Z { A | t | ǫπ √ n } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp n Z cos tu − u du − n X k =1 nk (cid:16) cos ktn − (cid:17) n o − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt = Z { A | t | ǫπ √ n } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) exp n n X k =1 Z knk − n (cid:2) γ t ( u ) − γ t (cid:16) kn (cid:17)(cid:3) du o − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dt Z { A | t | ǫπ √ n } exp n n X k =1 Z knk − n (cid:12)(cid:12) γ t (cid:16) kn (cid:17) − γ t ( u ) (cid:12)(cid:12) du o − ! dt, where we have put γ t ( u ) = cos tu − u and have used the inequality | e x − | e | x | − . It is not difficult to see that sup u ∈ R (cid:12)(cid:12) γ ′ t ( u ) (cid:12)(cid:12) = Ct ;in fact γ ′ t ( u ) = η ( tu ) t , with η ( z ) = 1 − z sin z − cos zz , z ∈ R | η ( z ) | = C < + ∞ is a simple exercise. Thus, by Lagrange’sTheorem, Z knk − n (cid:12)(cid:12) γ t (cid:16) kn (cid:17) − γ t ( u ) (cid:12)(cid:12) du Ct Z knk − n (cid:12)(cid:12)(cid:12) kn − u (cid:12)(cid:12)(cid:12) du C t n . (18)Using (18) in the last bound for I we find I Z { A | t | ǫπ √ n } exp n n X k =1 Z knk − n (cid:12)(cid:12) γ t (cid:16) kn (cid:17) − γ t ( u ) (cid:12)(cid:12) du o − ! dt Z { A | t | ǫπ √ n } exp n C t n o − ! dt exp n C ( ǫπ √ n ) n o − ! ǫπ √ n = C e Cn / − Cn / Cn / → , n → ∞ . ( I ) We recall that I = Z { ǫπ √ n | t | nπ } n(cid:12)(cid:12)(cid:12) φ Tnn ( u ) (cid:12)(cid:12)(cid:12) − | φ ( u ) | o du Z { ǫπ √ n | t | nπ } (cid:12)(cid:12)(cid:12) φ Tnn ( u ) (cid:12)(cid:12)(cid:12) du + Z { ǫπ √ n | t | nπ } | φ ( u ) | du. The second summand above goes to 0 as n → ∞ since | φ ( u ) | is integrable on R (recallpoint (a) of this proposition); hence we have to prove that Z { ǫπ √ n | t | nπ } (cid:12)(cid:12)(cid:12) φ Tnn ( u ) (cid:12)(cid:12)(cid:12) du → , n → ∞ . By relation (17), we have, for every k ∈ N and n > k (cid:12)(cid:12)(cid:12) φ T n ( u ) (cid:12)(cid:12)(cid:12) = exp n k − X k =1 log h k − k (cid:16) cos kt − (cid:17)io · exp n n X k = k log h k − k (cid:16) cos kt − (cid:17)io exp n n X k = k log h k − k (cid:16) cos kt − (cid:17)io . Now, using the relation (cid:12)(cid:12) log(1 − z ) + z (cid:12)(cid:12) | z | , | z | < , z ∈ C with z = − k − k (cid:0) cos kt − (cid:1) and choosing k such that (cid:12)(cid:12) k − k (cid:0) cos kt − (cid:1)(cid:12)(cid:12) < for k > k and every t , we findlog h k − k (cid:16) cos kt − (cid:17)i − k − k (cid:16) − cos kt (cid:17) + 4( k − k (cid:16) − cos kt (cid:17) . − cos x n n X k = k log h k − k (cid:16) cos kt − (cid:17)io exp n − n X k = k k − k (cid:16) − cos kt (cid:17) + n X k = k k − k (cid:16) − cos kt (cid:17) o exp n − n X k = k k − k + 2 n X k = k k − k cos kt + n X k = k k − k o exp n − n X k = k k + 2 n X k = k cos ktk + C o exp n − Z nk x dx + 2 n X k = k cos ktk + C o = Cn exp n n X k = k cos ktk o . By [14], p. 191 we know thatsup n > (cid:12)(cid:12)(cid:12) n X k =1 cos ktk (cid:12)(cid:12)(cid:12) log 1 t + C. Hence sup n > k (cid:12)(cid:12)(cid:12) n X k = k cos ktk (cid:12)(cid:12)(cid:12) sup n > (cid:12)(cid:12)(cid:12) n X k =1 cos ktk (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) k − X k =1 cos ktk (cid:12)(cid:12)(cid:12) log 1 t + C. It follows that Cn exp n n X k = k cos ktk o Cn exp n (cid:16) log 1 t + C (cid:17)o = Cn t , so that Z { ǫπ √ n | t | nπ } (cid:12)(cid:12)(cid:12) φ Tnn ( u ) (cid:12)(cid:12)(cid:12) du = n Z π ǫn / (cid:12)(cid:12)(cid:12) φ T n ( t ) (cid:12)(cid:12)(cid:12) dt n Z π ǫn / Cn t dt C n / n = Cn / → , n → ∞ . Now (15) is proved by letting ǫ → I . (cid:3) Remark 4.2
The relation (15) yields a weak form of Local Limit Theorem (see Corollary 2.1).Let c T n be the n -th partial sum c T n = n X k =1 k c Z k , where ( c Z n ) n > is an independent copy of ( Z n ) n > . Denote by d s the symmetrized Dickman density,which has characteristic function (cid:12)(cid:12) φ (cid:12)(cid:12) . Then, by the inversion formula, (15) and Proposition 4.1,2 π (cid:8) nP ( T n − c T n = κ n ) − d s ( n − κ n ) (cid:9) = Z nπ − nπ e − itn − κ n n(cid:12)(cid:12)(cid:12) φ Tnn ( t ) (cid:12)(cid:12)(cid:12) − | φ ( t ) | o dt + Z {| t | >nπ } e − itn − κ n | φ ( t ) | dt → , n → ∞ . The correlation inequality
In this section we present a correlation inequality for the sequence of random variables ( Y n ) n > ,where Y n = n { T n = κ n } . (19) Theorem 5.1 (Basic correlation inequality).
Let x > be given and let κ = ( κ n ) be any fixedsequence of integers with lim n →∞ κ n n = x . Then, for every x > and for n > m > | Cov ( Y m , Y n ) | C (cid:26) nn − m χ ( κ ,x ) m,n + mn − m + χ ( κ ,x )2 ,n + 1 n (cid:27) . where C is a positive constant (depending on x ) and χ ( κ ,x ) m,n = n − mκ n − κ m · log nm √ n − m + n − mκ n − κ m · g m,n + x (cid:12)(cid:12)(cid:12) n − mκ n − κ m − x (cid:12)(cid:12)(cid:12) + m + 1 κ n − κ m . Proof.
Cov ( Y m , Y n ) = nm (cid:8) P ( T m = κ m , T n = κ n ) − P ( T m = κ m ) P ( T n = κ n ) (cid:9) = nm (cid:8) P ( T m = κ m , T nm = κ n − κ m ) − P ( T m = κ m ) P ( T n = κ n ) (cid:9) = (cid:8) mP ( T m = κ m ) (cid:9)(cid:8) nP ( T nm = κ n − κ m ) − nP ( T n = κ n ) (cid:9) . Hence, by the local Theorem (Corollary 2.1), we have (cid:12)(cid:12)
Cov ( Y m , Y n ) (cid:12)(cid:12) C (cid:12)(cid:12) nP ( T nm = κ n − κ m ) − nP ( T n = κ n ) (cid:12)(cid:12) C (cid:16)(cid:12)(cid:12)(cid:12) nn − m (cid:8) ( n − m ) P ( T nm = κ n − κ m ) − D ′ ( x ) (cid:9)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:16) nn − m − (cid:17) D ′ ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12) nP ( T n = κ n ) − D ′ ( x ) (cid:12)(cid:12)(cid:17) C (cid:16)(cid:12)(cid:12)(cid:12) nn − m (cid:8) ( n − m ) P ( T nm = κ n − κ m ) − D ′ ( x ) (cid:9)(cid:12)(cid:12)(cid:12) + mn − m + (cid:12)(cid:12) nP ( T n = κ n ) − D ′ ( x ) (cid:12)(cid:12)(cid:17) = C (cid:16) nn − m Γ + mn − m + ∆ (cid:17) , (20)where we have put for simplicityΓ = | ( n − m ) P ( T nm = κ n − κ m ) − D ′ ( x ) | , ∆ = | nP ( T n = κ n ) − D ′ ( x ) (cid:12)(cid:12) . The aim is to obtain bounds for Γ and ∆. (a) Γ . Set Φ = P (cid:0) ( κ n − κ m ) − n T nm ( κ n − κ m ) − ( m + 1) (cid:1) ;Ψ = (cid:12)(cid:12)(cid:12) ( κ n − κ m ) P ( T n = κ n − κ m ) − Φ (cid:12)(cid:12)(cid:12) . Γ n − mκ n − κ m Ψ + (cid:12)(cid:12)(cid:12) n − mκ n − κ m Φ − D ′ ( x ) (cid:12)(cid:12)(cid:12) = n − mκ n − κ m Ψ + (cid:12)(cid:12)(cid:12) n − mκ n − κ m Φ − D ( x ) − D ( x − x (cid:12)(cid:12)(cid:12) , by (1). From Proposition 3.2 we know thatΨ C log nm √ n − m . (21)19oreover, puttingΛ = (cid:12)(cid:12)(cid:12) P (cid:16) T nm n − m ( κ n − κ m ) − ( m + 1) n − m (cid:17) − D (cid:16) ( κ n − κ m ) − ( m + 1) n − m (cid:17)(cid:12)(cid:12)(cid:12) , Θ = (cid:12)(cid:12)(cid:12) P (cid:16) T nm n − m ( κ n − κ m ) − ( n + 1) n − m (cid:17) − D (cid:16) ( κ n − κ m ) − ( n + 1) n − m (cid:17)(cid:12)(cid:12)(cid:12) , Σ = (cid:12)(cid:12)(cid:12) n − mκ n − κ m D (cid:16) ( κ n − κ m ) − ( m + 1) n − m (cid:17) − D ( x ) x (cid:12)(cid:12)(cid:12) , Ω = (cid:12)(cid:12)(cid:12) n − mκ n − κ m D (cid:16) ( κ n − κ m ) − ( n + 1) n − m (cid:17) − D ( x − x (cid:12)(cid:12)(cid:12) , it is easily checked that (cid:12)(cid:12)(cid:12) n − mκ n − κ m Φ − D ( x ) − D ( x − x (cid:12)(cid:12)(cid:12) n − mκ n − κ m (cid:0) Λ + Θ (cid:1) + Σ + Ω . We know from Proposition 3.6 thatsup x ∈ R (cid:12)(cid:12)(cid:12) P (cid:16) T nm n − m x (cid:17) − D ( x ) (cid:12)(cid:12)(cid:12) Cg m,n . Hence Λ + Θ Cg m,n . (22)Moreover Σ n − mκ n − κ m (cid:12)(cid:12)(cid:12) D (cid:16) ( κ n − κ m ) − ( m + 1) n − m (cid:17) − D ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) n − mκ n − κ m − x (cid:12)(cid:12)(cid:12) D ( x ) n − mκ n − κ m (cid:12)(cid:12)(cid:12) D (cid:16) ( κ n − κ m ) − ( m + 1) n − m (cid:17) − D ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) n − mκ n − κ m − x (cid:12)(cid:12)(cid:12) , and, by Lagrange Theorem, there exists ξ n such that n − mκ n − κ m (cid:12)(cid:12)(cid:12) D (cid:16) ( κ n − κ m ) − ( m + 1) n − m (cid:17) − D ( x ) (cid:12)(cid:12)(cid:12) n − mκ n − κ m (cid:12)(cid:12)(cid:12) ( κ n − κ m ) − ( m + 1) n − m − x (cid:12)(cid:12)(cid:12) D ′ ( ξ n ) n − mκ n − κ m (cid:12)(cid:12)(cid:12) ( κ n − κ m ) − ( m + 1) n − m − x (cid:12)(cid:12)(cid:12) x (cid:12)(cid:12)(cid:12) n − mκ n − κ m − x (cid:12)(cid:12)(cid:12) + m + 1 κ n − κ m , (23)since sup x> D ′ ( x ) = 1. For Ω we get exactly the same bound as in (23).In conclusion, from (21), (22) and (23) we have obtainedΓ = (cid:12)(cid:12) ( n − m ) P ( T nm = κ n − κ m ) − D ′ ( x ) (cid:12)(cid:12) Cχ ( κ ,x ) m,n . (24) (b) ∆ . Recall that ∆ = (cid:12)(cid:12) nP ( T n = κ n ) − D ′ ( x ) (cid:12)(cid:12) . Notice that we cannot apply (24) directly since we have proved it for m > U = Z + 2 Z ,∆ = (cid:12)(cid:12)(cid:12) X j =0 P ( U = j ) (cid:16) nP ( T n = κ n − j ) − D ′ ( x ) (cid:17)(cid:12)(cid:12)(cid:12) sup j (cid:12)(cid:12)(cid:12) nP ( T n = κ n − j ) − D ′ ( x ) (cid:12)(cid:12)(cid:12) = sup j (cid:12)(cid:12)(cid:12) nn − (cid:8) ( n − P ( T n = κ n − j ) − D ′ ( x ) (cid:9) + 2 n − D ′ ( x ) (cid:12)(cid:12)(cid:12) nn − n sup j | ( n − P ( T n = κ n − j ) − D ′ ( x ) | o + 2 n − Cnn − j χ ( κ ( j ) ,x )2 ,n + 2 n − C (cid:16) χ ( κ ,x )2 ,n + 1 n (cid:17) , (25)20pplying (24) (with m = 2) for the sequence κ ( j ) = ( κ ( j ) n ) n defined as κ ( j ) n = κ n − j andnoticing that χ ( κ ( j ) ,x )2 ,n = χ ( κ ,x )2 ,n .The two relations (24) and (25), inserted into (20), conclude the proof. (cid:3) As we pointed out in the Introduction, the Almost Sure Limit Theorem that we are going to provein the present section (i.e. Theorem 6.8) is in the spirit of Theorem 1 of T. Mori’s paper [8]; insection 6 it will be applied to the sequence ( Y n ) defined in (19): notice that Mori’s result is notapplicable in the context of (19), due to the fact that it requires that (cid:12)(cid:12) Cov ( Y m , Y n ) (cid:12)(cid:12) h (cid:0) nm (cid:1) for all1 m n (for a suitable function h ); for m = n this inequality becomes V arY m h (1) = C ,i.e. the sequence ( V arY m ) m > must be bounded; unfortunately this is not true in our setting (seeLemma 7.2). Theorem 6.1
Let ( U n ) n > a sequence of centered random variables. Assume that there exist twonumbers α > and σ > , a non–negative function f ( u, z ) defined on the set { u > , z > σ } , anon–negative double–indexed sequence g defined on the set { ( m, n ) ∈ N : σm n } such that ( i ) sup n > σm g ( m, n ) = C < + ∞ ; (ii) uniformly in u > the functions v f (cid:0) u, vu (cid:1) are ultimately non–increasing (i.e. there exists m such that v f (cid:0) u, vu (cid:1) is non–increasing on ( m , + ∞ ) and for every u > );(iii) the functions z φ ( z ) = sup u > f ( u, z ) z , u F ( u ) = Z uσ φ ( z ) dz are defined on [ σ, + ∞ ) ; ( iv ) | Cov ( U m , U n ) | C ( m for m = n m < n σm ; (v) there exists m such that, for n > m > m | Cov ( U m , U n ) | g ( m, n ) 1 m α f (cid:16) m, nm (cid:17) . Denote V n = σ n X k = σ n − +1 U k k . Then, for every n and every sufficiently large mE (cid:2) ( m + n X i = m +1 V i ) (cid:3) C (cid:18) n + 1 σ α ( m + n ) Z σ n σ F ( u ) u α − du (cid:19) . roof. Since E h(cid:0) m + n X i = m +1 V i (cid:1) i = m + n X i = m +1 E [ V i ] + 2 X m +1 ≤ i Theorem 6.2 Let ( U n ) n > be a sequence of centered random variables. Let N be an integer andassume that there exist a number σ > and for each j = 1 , , . . . , N numbers α j > a non–negativefunction f j ( u, z ) defined on the set { u > , z > σ } , a non–negative double–indexed sequence g j defined on the set { ( m, n ) ∈ N : σm n } such that ( i ) sup n > σm g j ( m, n ) = C < + ∞ ; (ii) uniformly in u > the functions u f j (cid:0) u, vu (cid:1) are ultimately non–increasing (i.e. there exists m such that v f j (cid:0) u, vu (cid:1) are non–increasing on ( m , + ∞ ) , for each j = 1 , . . . , N and for every u > );(iii) for each j = 1 , , . . . , N the functions z φ j ( z ) = sup u > f j ( u, z ) z , u F j ( u ) = Z uσ φ j ( z ) dz re defined on [ σ, + ∞ ) ; ( iv ) | Cov ( U m , U n ) | C ( m for m = n m < n σm ; (v) there exists m such that, for n > m > m | Cov ( U m , U n ) | N X j =1 g j ( m, n ) 1 m α j f j (cid:16) m, nm (cid:17) . Denote V n = σ n X k = σ n − +1 U k k . Then, for every n and every sufficiently large mE (cid:2) ( m + n X i = m +1 V i ) (cid:3) C n + N X j =1 σ α j ( m + n ) Z σ n σ F j ( u ) u α j − du . Corollary 6.3 In the setting of Theorem 6.1, assume in addition that α = 0 and there exists β > such that F ( x ) C (log x ) β − for every x > σ . Then, for every sufficiently large m , E h(cid:0) m + n X i = m +1 V i (cid:1) i C (cid:16) ( m + n ) β − m β (cid:17) . (38) Proof. Putting α = 0 in the claim of Theorem 6.1 we obtain E h(cid:0) m + n X i = m +1 V i (cid:1) i C (cid:16) n + Z σ n σ F ( u ) u du (cid:17) C (cid:16) n + Z σ n σ (log u ) β − u du (cid:17) = C (cid:16) n + h (log u ) β i σ n σ (cid:17) C (cid:16) n + n β (cid:17) Cn β . On the other hand, the function z n ( z + n ) β − z β o being increasing (its derivative is β ( z + n ) β − − βz β − > n β (1 + n ) β − ( m + n ) β − m β . (cid:3) Corollary 6.4 In the setting of Theorem 6.1, assume in addition that α > and there exists β > such that F ( x ) C (log x ) β for every x > σ . Then, for every sufficiently large m , E h(cid:0) m + n X i = m +1 V i (cid:1) i C (cid:16) ( m + n ) β − m β (cid:17) . (39) Proof. In this case Theorem 6.1 gives E h(cid:0) m + n X i = m +1 V i (cid:1) i C (cid:16) n + 1 σ α ( m + n ) Z σ n σ F ( u ) u α − du (cid:17) C (cid:16) n + 1 σ αn Z σ n σ (log u ) β u α − du (cid:17) C (cid:16) n + 1 σ αn (cid:0) log( σ n ) (cid:1) β Z σ n σ u α − du (cid:17) = C (cid:16) n + (cid:0) log( σ n ) (cid:1) β ( σ αn − σ α ) ασ αn (cid:17) C (cid:16) n + n β (cid:17) Cn β . The rest of the proof is identical to Corollary 6.3. (cid:3) orollary 6.5 In the setting of Theorem 6.2, assume that there exists β > such that P Nj =1 F j ( x ) C (log x ) β for every x > σ . Then(i) for every sufficiently large m and for every n , E h(cid:0) m + n X i = m +1 V i (cid:1) i C (cid:16) ( m + n ) β − m β (cid:17) . (ii) for every δ > , n X i =1 V i = O ( n β/ (log n ) δ ) , P − a.s. Proof. Point (i) follows from Corollaries 6.3 and 6.4. Point (ii) is a consequence of the well knownGaal–Koksma Strong Law of Large Numbers (see [10], p. 134); here is the precise statement: Theorem 6.6 Let ( V n ) n > be a sequence of centered random variables with finite variance. Supposethat there exists a constant β > such that, for all integers m ≥ , n > , E h(cid:0) m + n X i = m +1 V i (cid:1) i ≤ C (cid:0) ( m + n ) β − m β (cid:1) , (40) for a suitable constant C independent of m and n . Then, for every δ > , n X i =1 V i = O ( n β/ (log n ) δ ) , P − a.s. Remark 6.7 It is not difficult to see that Theorem 6.6 is in force even if the bound (40) holdsonly for all integers m ≥ h , n > 0, where h is an integer strictly greater than 0. A rigorous proofof this statement can be found in the appendix of [3]. From now on, this slight generalization willbe tacitly used. Theorem 6.8 ( General ASLT ) Let ( Y n ) n > be a sequence of non–negative (resp. non–positive)random variables with lim n →∞ E [ Y n ] = ℓ > resp. ℓ < and such that the sequence ( U n ) n > defined by U n = Y n − E [ Y n ] verifies the assumptions of Theorem6.2. Assume that there exists β > such that P Nj =1 F j ( x ) C (log x ) β for every x > σ . Then lim n →∞ n n X k =1 Y k k = ℓ, a.s. Proof. By point (ii) of Corollary 6.5, for every δ > P ni =1 V i n = O ( n β/ (log n ) δ ) n −→ n →∞ . (41)Since n X i =1 V i = n X i =1 σ i X k = σ i − +1 U k k = σ n X k =2 U k k = σ n X k =2 Y k k − σ n X k =2 E [ Y k ] k n log σ σ n X k =2 E [ Y k ] k −→ n →∞ ℓ, the relation (41) is equivalent to 1 n log σ σ n X k =2 Y k k −→ n →∞ ℓ ;By the same argument as in [3], pp. 789–790, this in turn implies that1log n n X k =1 Y k k −→ n →∞ ℓ, i.e. the claim. (cid:3) Let x > κ = ( κ n ) n > be a strictly increasing sequence of integers with κ n ∼ xn ,fixed throughout the sequel. Let ( Y n ) n > be the sequence defined in (19); the main result of thissection and of the paper (Theorem 2.2) is an ASLLT for the sequence ( Y n ) n > . Before proving it,we need some Lemmas.For every ǫ ∈ (0 , x ) we set σ = σ ǫ = 1 + x (1 − ǫ ) x (1 + ǫ ) = 1 + 21 + ǫ (cid:16) x − ǫ (cid:17) > . (42) Lemma 7.1 Let ǫ ∈ (0 , x ) be fixed. Then there exists m = m ( ǫ ) such that, for σm > n > m >m , P (cid:0) T nm = κ n − κ m (cid:1) = 0 Proof. Let A = n \ k = m +1 { Z k = 0 } . Then P (cid:0) T nm = κ n − κ m (cid:1) = P (cid:0) { T nm = κ n − κ m } ∩ A (cid:1) + P (cid:0) { T nm = κ n − κ m } ∩ A c (cid:1) . (i) Let m be such that, for every m > m , xm (1 − ǫ ) < κ m < xm (1 + ǫ ) . Then, for σm > n > m > m , κ n − κ m < xn (1 + ǫ ) − xm (1 − ǫ ) = m n x nm (1 + ǫ ) − x (1 − ǫ ) o m n xσ (1 + ǫ ) − x (1 − ǫ ) o = m. (43)27ence { T nm = κ n − κ m } ∩ A c = { T nm = κ n − κ m } ∩ (cid:16) n [ k = m +1 { Z k = 1 } (cid:17) = n [ k = m +1 { T nm = κ n − κ m , Z k = 1 } ⊆ n [ k = m +1 { T nm = κ n − κ m , T nm > m + 1 } = { T nm = κ n − κ m , T nm > m + 1 } = ∅ , by (43).(ii) A ⊆ { T nm = 0 } , hence { T nm = κ n − κ m } ∩ A ⊆ { T nm = κ n − κ m , T nm = 0 } = ∅ , since κ n − κ m > (cid:3) Lemma 7.2 Let ǫ ∈ (0 , x ) be fixed. Then there exists m = m ( ǫ ) such that, for n > m > m , | Cov ( Y m , Y n ) | C ( m for m = n m < n σm, where C is a positive constant.Proof. (a) For m = n : Cov ( Y m , Y m ) = m (cid:8) P ( T m = κ m ) − P ( T m = κ m ) (cid:9) = (cid:8) mP ( T m = κ m ) (cid:9)(cid:8) m − P ( T n = κ m ) (cid:9) Cm, by the Local Theorem (Corollary 2.1).(b) For m < n σm : let m be as in Lemma 7.1. Then, for σm > n > m > m , | Cov ( Y m , Y n ) | = mn (cid:12)(cid:12) P ( T m = κ m , T n = κ n ) − P ( T m = κ m ) P ( T n = κ n ) (cid:12)(cid:12) = (cid:8) mP ( T m = κ m ) (cid:9)(cid:12)(cid:12) nP (cid:0) T nm = κ n − κ m (cid:1) − nP ( T n = κ n ) (cid:12)(cid:12) = (cid:8) mP ( T m = κ m ) (cid:9)(cid:8) nP ( T n = κ n ) (cid:9) C, by the Local Theorem again and observing that P (cid:0) T nm = κ n − κ m (cid:1) = 0, by Lemma 7.1. (cid:3) Remark 7.3 Notice that(i) κ n = [ xn ] is strictly increasing if x ≥ κ n = xn we can take ǫ = 0 and m = 1. 28 emma 7.4 In the setting of Theorem 6.1, assume that f has the form f ( u, z ) = ψ ( uz ) where t ψ ( t ) is a continuous ultimately non–increasing function, i.e. there exists t such that t ψ ( t ) is non–increasing for t > t . Then F ( u ) C ( for u t R ut ψ ( z ) z dz for u > t . (44) Proof. It is easy to see thatsup x > f ( x, z ) = sup x > ψ ( xz ) = sup u > z ψ ( u ) max u ∈ [1 ,t ] ψ ( u ) =: M for z t = ψ ( z ) for z > t . Hence φ ( z ) ( Mz for z t ψ ( z ) z for z > t . and F ( u ) = Z uσ φ ( z ) dz = Z t σ φ ( z ) dz | {z } = C + Z ut φ ( z ) dz C ( u t R ut ψ ( z ) z dz for u > t . (cid:3) Remark 7.5 Of course, the preceding lemma has an obvious generalization in the setting of The-orem 6.2.We are ready to give the Proof of Theorem 2.2. Though with tedious and cumbersome calculations, it is easy to see thatthe correlation inequality of Theorem 5.1 takes a slightly more tractable form for sufficiently large m : precisely (we neglect the multiplicative constant C for easy writing): | Cov ( Y m , Y n ) | nκ n − κ m ( nm √ n − m + h exp (cid:16) C n log nm ( n − m ) + m log nm n − m o(cid:17) − nm i + (cid:12)(cid:12) ( xn − κ n ) − ( xm − κ m ) (cid:12)(cid:12) n − m + mn − m ) + mn − m + log n √ n + exp (cid:16) C n log nn + log nn o(cid:17) − n + (cid:12)(cid:12) ( xn − κ n ) − (2 x − κ ) (cid:12)(cid:12) n − n . (45)(In fact (look at the formula in the statement of Theorem 5.1) nn − m χ κ ,xm,n = nκ n − κ m (cid:16) log nm √ n − m + g m,n (cid:17)| {z } ( a ) + x (cid:12)(cid:12)(cid:12) n − mκ n − κ m − x (cid:12)(cid:12)(cid:12) · nn − m | {z } ( b ) + nn − m · m + 1 κ n − κ m | {z } ( c ) , and 29a) nκ n − κ m (cid:16) nm √ n − m + g m,n (cid:17) = nκ n − κ m " nm √ n − m + exp (cid:16) C n log nm ( n − m ) + m + 2 n − m o log nm (cid:17) − nm nκ n − κ m " nm √ n − m + exp (cid:16) C n log nm ( n − m ) + m · log nm n − m o(cid:17) − nm ;(b) x (cid:12)(cid:12)(cid:12) n − mκ n − κ m − x (cid:12)(cid:12)(cid:12) · nn − m = nκ n − κ m · (cid:12)(cid:12)(cid:12) ( nx − κ n ) − ( mx − κ m ) n − m (cid:12)(cid:12)(cid:12) ;(c) nn − m · m + 1 κ n − κ m nκ n − κ m · mn − m . Further, by (a), (b) and (c) above χ κ ,x ,n nκ n − κ " n √ n − (cid:16) C n log n ( n − + 2 · log n n − o(cid:17) − n ++ nn − · ( nx − κ n ) − (2 x − κ ) n − n − log n √ n + exp (cid:16) C n log nn + log nn o(cid:17) − n + (cid:12)(cid:12) ( xn − κ n ) − (2 x − κ ) (cid:12)(cid:12) n − n , for sufficiently large n ; recall that we are neglecting multiplicative constants).The statement of the Theorem is a consequence of the general ASLT (Theorem 6.8): we checkassumption (v) of Theorem 6.2 for each summand in the basic correlation inequality (in the form(45)) and use Corollary 6.3 or Corollary 6.4, as needed (it is easy to see that assumptions (i)–(iv)of Theorem 6.2 are in force for each summand in the basic correlation inequality, hence we omitthe details). Precisely (with the notations of Theorem 6.2 and with σ defined in (42)):(1) First summand: nκ n − κ m · nm √ n − m . Fix δ ∈ (cid:16) , σ − σ +1 (cid:17) , and let m be such that1 − δ < κ n n < δ, n > m . Then, for n > σm and m > m , κ n − κ m n = κ n n − κ m m · mn > (1 − δ ) − δσ > , (46)hence sup n>σm nκ n − κ m − δ ) − δσ . (47)30oreover we have yx √ y − x = √ x · yx √ yx − , hence g ( m, n ) = nκ n − κ m , α = 12 , f ( u, z ) = 1 + log z √ z − , φ ( z ) = 1 + log zz √ z − F ( u ) = Z uσ φ ( z ) dz = Z uσ zz √ z − dz C C (log u ) β , ∀ β > . (2) Second summand: nκ n − κ m · n exp C n log nm ( n − m ) + m log nm n − m o! − o . We have again g ( m, n ) = nκ n − κ m ; moreoverexp C n log yx ( y − x ) + x log yx y − x o! − C n log yx x ( yx − + x log yx x ( yx − o! − , so that f ( u, z ) = exp C n log zu ( z − + x log zu ( z − o! − , and α = 0; further φ ( z ) = sup u > f ( u, z ) z = 1 z ( exp C n log z ( z − + log zz − o! − ) . Put M = sup z > σ exp C n log z ( z − + log zz − o! − n log z ( z − + log zz − o Then φ ( z ) M n log z ( z − + log zz − o z and F ( u ) = Z uσ φ ( z ) dz M Z uσ n log z ( z − + log zz − o z dz C C (log u ) β , ∀ β > . (3) Third summand: nκ n − κ m · nm we have g ( m, n ) = nκ n − κ m , α = 0 and f ( u, z ) = 1log z ; φ ( z ) = 1 z log z , hence F ( u ) = Z uσ φ ( z ) dz = Z uσ z log z dz = h log log z i uσ log log u (log u ) β , ∀ β > . nκ n − κ m · (cid:12)(cid:12) ( xn − κ n ) − ( xm − κ m ) (cid:12)(cid:12) n − m Once more, g ( m, n ) = nκ n − κ m , α = 0. Let δ > m such that | κ n − nx | < δxn, n > m . Then, for n > m > m , (cid:12)(cid:12) ( xn − κ n ) − ( xm − κ m ) (cid:12)(cid:12) n − m < δx n + mn − m = δx nm + 1 nm − f (cid:0) u, z (cid:1) = δx z + 1 z − φ ( z ) = δx z + 1 z ( z − < Cz . Hence F ( u ) = Z uσ φ ( z ) dz < C Z uσ z dz < C log u C log β u, ∀ β > . (5) Fifth summand: nκ n − κ m · mn − m = nκ n − κ m · nm − . Once more, g ( m, n ) = nκ n − κ m , α = 0 and f ( u, z ) = 1 z − φ ( z ) = 1 z ( z − , and F ( u ) = Z uσ φ ( z ) dz = Z uσ z ( z − dz C (log u ) β , ∀ β > . (6) Sixth summand: mn − m = 1 nm − . Here g ( m, n ) = 1, α = 0 and f ( u, z ) = 1 z − . The argument is identical to the previous one.(7) Seventh summand: log n √ n + exp (cid:16) C n log nn + log nn o(cid:17) − n C h log n √ n + exp (cid:16) C n log nn o(cid:17) − n i . Here g ≡ α = 0 and f ( u, z ) = log uz √ uz + exp (cid:16) C n log uzuz o(cid:17) − uz = ψ ( uz ) , ψ ( t ) = log t √ t + exp (cid:16) C n log tt o(cid:17) − t . We can apply Lemma 7.4, and we find F ( u ) = C + Z ut log tt √ t + 1 t n exp (cid:16) C n log tt o(cid:17) − o + 1 t log t ! dt C log log u (log u ) β , ∀ β > , for some suitable t > σ .(8) Eighth summand: (cid:12)(cid:12) ( xn − κ n ) − (2 x − κ ) (cid:12)(cid:12) n − C. Here g ≡ α = 0 and f ( u, z ) = C = ψ ( uz ) , with ψ ( t ) = C . We can apply Lemma 7.4, and we find F ( u ) = C + Z ut t dt C log u (log u ) β , ∀ β > , for some suitable t > σ. (9) Ninth summand: 1 n The argument is the same as in (7) and (8). (cid:3) In this section we prove the explicit formula announced in Remark 3.4. For every integer n andevery integer k with 0 k n put a k,n = k X j =0 ( − j +1 (cid:18) kj (cid:19) j n . Remark 8.1 (i) Notice that a ,n = 1 for every n .(ii) The Stirling number of second kind S ( n, k ) has the explicit expression S ( n, k ) = 1 k ! k X j =0 ( − k − j (cid:18) kj (cid:19) j n . Hence a k,n = k X j =0 ( − j +1 (cid:18) kj (cid:19) j n = ( − k +1 k X j =0 ( − j − k (cid:18) kj (cid:19) j n = ( − k +1 k X j =0 ( − k − j (cid:18) kj (cid:19) j n = ( − k +1 k ! S ( n, k ) . (ii) We also recall that S ( n, n ) = 1 , which implies that a n,n = ( − n +1 n ! by the above relation. B (1 , x ) be the Bernoullian law with parameter x ∈ (0 , c n ( x ) the n − th cumulantof B (1 , x ), i.e. the n − th coefficient in the development of the logarithm of its characteristic function φ ( t ): log φ ( t ) = log (cid:16) x ( e it − (cid:17) = ∞ X n =1 c n ( x ) ( it ) n n ! . Remark 8.2 (i) It is easily seen that c ( x ) = x. (ii) It is well known (see [7] ex. 6 p. 312 for instance) that the sequence of functions (cid:0) c n ( x ) (cid:1) n verifies the recurrence relation c n +1 ( x ) = x (1 − x ) c ′ n ( x ) (48) Proposition 8.3 For every n > we have c n ( x ) = x (1 − x ) n n − X k =1 a k,n − x k − o . (49) P roof. By (48), we must prove that, for every n > c ′ n ( x ) = n X k =1 a k,n x k − . (50)The proof is by induction.For n = 1 the statement follows from Remarks 8.1 (i) and 8.2 (i).Assume that (50) holds for the integer n − 1; hence, by (48), we have c n ( x ) = x (1 − x ) n n − X k =1 a k,n − x k − o c ′ n ( x ) = (1 − x ) n n − X k =1 a k,n − x k − o + ( x − x ) n n − X k =2 ( k − a k,n − x k − o = n − X k =1 a k,n − x k − − n − X k =1 a k,n − x k + n − X k =2 ( k − a k,n − x k − − n − X k =2 ( k − a k,n − x k = n − X k =1 a k,n − x k − − n X k =2 a k − ,n − x k − + n − X k =2 ( k − a k,n − x k − − n X k =3 ( k − a k − ,n − x k − = a ,n − + (cid:0) a ,n − − a ,n − + a ,n − (cid:1) x ++ n − X k =3 x k − (cid:16) a k,n − − a k − ,n − + ( k − a k,n − − ( k − a k − ,n − (cid:17) ++ (cid:0) − a n − ,n − − ( n − a n − ,n − (cid:1) x n − = 1 + (cid:0) a ,n − − a ,n − (cid:1) x + n − X k =3 x k − (cid:16) ka k,n − − ka k − ,n − (cid:17) + (cid:0) − na n − ,n − (cid:1) x n − = 1 + n − X k =2 x k − (cid:16) ka k,n − − ka k − ,n − (cid:17) + (cid:0) − n ( − n ( n − (cid:1) x n − = 1 + n − X k =2 x k − (cid:16) ka k,n − − ka k − ,n − (cid:17) + ( − n +1 n ! x n − = a ,n + n − X k =2 x k − (cid:16) ka k,n − − ka k − ,n − (cid:17) + a n,n x n − = n X k =1 a k,n x k − , since ka k,n − − ka k − ,n − = k n k X j =0 ( − j +1 (cid:18) kj (cid:19) j n − − k − X j =0 ( − j +1 (cid:18) k − j (cid:19) j n − o = k n k − X j =0 ( − j +1 j n − h(cid:18) kj (cid:19) − (cid:18) k − j (cid:19)i + ( − k +1 k n − o = k n k − X j =0 ( − j +1 j n − (cid:18) k − j (cid:19)h kk − j − i + ( − k +1 k n − o = k n k − X j =0 ( − j +1 j n − (cid:18) k − j (cid:19) jk − j + ( − k +1 k n − o = k n k − X j =0 ( − j +1 j n ( k − j !( k − j )! + ( − k +1 k n − o = k − X j =0 ( − j +1 j n (cid:18) kj (cid:19) + ( − k +1 k n = k X j =0 ( − j +1 j n (cid:18) kj (cid:19) = a k,n . This completes the proof. (cid:3) orollary 8.4 The following formula holds c n ( x ) x − n X k =2 a k,n k x k − . Proof. Write c n ( x ) x = (1 − x ) n n − X k =1 a k,n − x k − o = n − X k =1 a k,n − x k − − n − X k =1 a k,n − x k = 1 + n − X k =2 a k,n − x k − − n − X k =1 a k,n − x k = 1 + n − X k =1 a k +1 ,n − x k − n − X k =1 a k,n − x k = 1 + n − X k =1 (cid:0) a k +1 ,n − − a k,n − (cid:1) x k − a n − ,n − x n − = 1 + n − X k =1 a k +1 ,n k + 1 x k + ( − n − + ( n − x n − , since, from the last calculation above a k +1 ,n − − a k,n − = a k +1 ,n k + 1 . Using now Remark 8.1 (ii), we get c n ( x ) x = 1 + n − X k =1 a k +1 ,n k + 1 x k + ( − n +1 ( n − x n − = 1 + n − X k =2 a k,n k x k − + ( − n +1 n ! n x n − = 1 + n X k =2 a k,n k x k − . Thus we have obtained c n ( x ) x − n X k =2 a k,n k x k − , as claimed. (cid:3) References [1] Arratia, R., Barbour, A. 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