The convergence rate of the Gibbs sampler for generalized 1− D Ising model
aa r X i v : . [ m a t h . P R ] F e b THE CONVERGENCE RATE OF THE GIBBS SAMPLER FORGENERALIZED − D ISING MODEL
AMINE HELALI
Abstract.
The rate of convergence of the Gibbs sampler for the generalized one-dimensional Ising model is determined by the second largest eigenvalue of its transitionmatrix in absolute value denoted by β ∗ . In this paper we generalize a bound for β ∗ fromShiu and Chen (2015) for the one-dimensional Ising model with two states to a multi-ple state situation. The method is based on Diaconis and Stroock bound for reversibleMarkov processes. The new bound presented in this paper improves Ingrassia’s (1994) result. Introduction
The Ising model is a crude model for ferromagnetism. It is the simplest model of statisticalmechanics and it has been applied in many other fields like chemistry, molecular biology andimage analysis. The distribution of the one-dimensional Ising model with three states is: π ( x ) = 1 Z T exp n T n − X k =1 (cid:0) { x k = x k +1 } − { x k = x k +1 } ) o ∀ x = ( x , · · · , x n (cid:1) ∈ χ , where χ = { c (1) , c (2) , c (3) } n is the state space, T is the temperature and Z T = X x ∈ χ exp n T n − X k =1 (cid:0) { x k = x k +1 } − { x k = x k +1 } (cid:1)o is the normalizing constant. Monte Carlo Markov chain M CM C method is a very usefultechnique to draw samples from the Ising model. Suppose that the transition probability P ( x, y ) for an irreducible Markov chain has π as its invariant measure. The pair ( P, π ) issaid to be reversible if it verifies the detailed balance equation: Q ( x, y ) = π ( x ) P ( x, y ) = π ( y ) P ( y, x ) = Q ( y, x ) ∀ x, y ∈ χ. The Gibbs sampler introduced by Geman and Geman and the Metropolis-Hastings algorithmintroduced by Metropolis et al. and Hastings (see [2], [6] and [3]) are the most popular MonteCarlo Markov chain methods. The matrix P satisfies detailed balance and thus is symmetricwith respect to the scalar product introduced by the measure π . Therefore its eigenvaluesare real and can be arranged as follows: β > β ≥ β ≥ · · · ≥ β | χ |− > − . Let β ∗ = max { β , | β | χ |− |} . By using the total variance distance, the second largest eigen-value in absolute value determines the convergence rate of the Markov chain. Ingrassia givesa lower bound for β | χ |− and an upper bound for β (see [5]). Mathematics Subject Classification.
Primary: 60J22, Secondary: 60F99, 60J10.
Key words and phrases.
Markov chain Monte Carlo, Rate of convergence, Gibbs sampler, Ising model.
This paper deals with the Gibbs sampler for the one-dimensional Ising model with multi-ple states. It chooses a random coordinate which is updated according to the conditionalprobability given the other coordinates. The resulting Markov chain is reversible and theassociated transition matrix has the form: P ( x, y ) = n π ( y i | x ) if x j = y j for all j = i − n n X i =1 X y i ∈{ c (1) ,c (2) ,c (3) } π ( y i | x ) if x = y elsewhere π ( y i | x ) = π ( x , · · · , x i − , y i , x i +1 , · · · , x n ) X l =1 π ( x , · · · , x i − , c ( l ) , x i +1 , · · · , x n ) . Diaconis and Stroock (see [1]) give a bound for the total variation distance to equilibriumin terms of β ∗ . We recall this result in the following theorem: Theorem 1 (Diaconis and Stroock 1991) . If P is a reversible Markov chain with uniqueinvariant measure π and P is irreducible then for all x ∈ χ and k ∈ N : || P k ( x, . ) − π || var = (cid:16) X y ∈ χ | P k ( x, y ) − π ( y ) | (cid:17) ≤ − π ( x ) π ( x ) ( β ∗ ) k . Moreover, Diaconis and Stroock (see [1]) develop a method to calculate an upper bound forthe second largest eigenvalue β for a reversible Markov chain using geometric quantities suchas the maximum degree, diameter and covering number of the associated graph. Considerthe graph G ( P ) = ( χ, E ) where χ is the vertex set and E = { ( x, y ) | P ( x, y ) > } is the edgeset. For each pair of distinct points x, y ∈ χ we choose a path γ xy from x to y , such thateach edge appears at most once in a given path. The fact that P is irreducible guaranteesthat such paths exist. Let Γ be the collection of all such paths γ xy (one for each pair). Thegeometric bound given by Diaconis and Stroock is, β ≤ − κ (1)with κ = max e ∈ E Q ( e ) − X γ xy ∋ e | γ xy | π ( x ) π ( y ) (2)where | γ xy | designates the length of the path γ xy (see [1]).In their paper Shiu and Chen (see [7]) present a method to explicitly compute the bound ofDiaconis and Stroock (see [1]) for the Gibbs sampler for a two state one-dimensional Isingmodel.In this paper we generalize the result of Shiu and Chen to the case of the one-dimensionalIsing model with three and more states (see [7]).Our method is based on the idea from [7] which consists of defining suitable paths γ xy linking each pair ( x, y ) from the state space χ and then to explicitly compute κ defined inequation (2) with some suitable symmetry argument. In the discussion section of the paperwe compare our bound to results from the literature. It turns out that the result generalizesthe bound given in [7] to the case of the Ising model with three states (see Theorem ) andalso to multiple states (see Theorem ). It also improves the bound presented by Ingrassiain [5]. HE GIBBS SAMPLER FOR THE 1-D ISING MODEL 3 Main result
Selection of paths.
To be able to use the result of Diaconis and Stroock (see [1]) andto calculate the bound of the second largest eigenvalue, we have to fix a collection of pathsconnecting any configuration x ∈ χ to any configuration y ∈ χ . To get a small upper boundfor β , we seek a small value for κ and we should therefore use short paths γ xy to link x with y . Moreover, we have to keep the number of paths passing through a given edge low.For a pair of distinct configurations x, y ∈ χ there exist some increasing sequence d , · · · , d m such that x i = y i for i ∈ { d , · · · , d m } and x i = y i otherwise. In the same way as Shiu andChen (see [7]) we define a path linking a given pair ( x, y ) as follows: ( x , · · · , x n ) = ( y , · · · , y d − , x d , x d +1 , · · · , x d − , x d , x d +1 , · · · , x n ) → ( y , · · · , y d − , y d , x d +1 , · · · , x d − , x d , x d +1 , · · · , x n )= ( y , · · · , y d − , y d , y d +1 , · · · , y d − , x d , x d +1 , · · · , x n ) → ( y , · · · , y d − , y d , y d +1 , · · · , y d − , y d , x d +1 , · · · , x n ) ... → ( y , · · · , y n ) . We turn now to give an upper bound for the value of κ defined in equation (2).1.2. Geometric bound of the second largest eigenvalue.
In what follows we willessentially follow the arguments from Shiu and Chen (see [7]) to find an upper bound for κ = max e ∈ E Q ( e ) − X γ xy ∋ e | γ xy | π ( x ) π ( y ) . Let e = ( e − , e + ) be some edge from E where e − and e + are two configurations from χ whichdiffer by only one coordinate. Without loss of generality we consider the case where the site i choosen to be updated passes from the color c (1) to the color c (2) . Then the configurations e − and e + must have the following form: e − = ( z , z , · · · , z i − , c (1) , z i +1 , · · · , z n ) and e + = ( z , z , · · · , z i − , c (2) , z i +1 , · · · , z n ) . The transition probability for a transition from e − to e + can then be computed. A shortcomputation shows for i = 1 : P ( e − , e + ) = 1 n π ( c (2) , z , · · · , z n ) X j =1 π ( c ( j ) , z , · · · , z n )= 1 n exp n T (cid:0) { c (2) = z } − { c (2) = z } + n − X k =2 { z k = z k +1 } − { z k = z k +1 } (cid:1)o X j =1 exp n T (cid:0) { c ( j ) = z } − { c ( j ) = z } (cid:1) + n − X k =2 { z k = z k +1 } − { z k = z k +1 } (cid:1)o = 1 n exp n T (cid:0) { c (2) = z } − { c (2) = z } (cid:1)o X j =1 exp n T (cid:0) { c ( j ) = z } − { c ( j ) = z } (cid:1)o , (3) AMINE HELALI similarly for i = n : P ( e − , e + ) = 1 n exp n T (cid:0) { z n − = c (2) } − { z n − = c (2) } (cid:1)o X j =1 exp n T (cid:0) { z n − = c ( j ) } − { z n − = c ( j ) } (cid:1)o (4)and for i ∈ { , · · · , n − } : P ( e − , e + ) = 1 n exp n T (cid:0) { z i − = c (2) } − { z i − = c (2) } + { c (2) = z i +1 } − { c (2) = z i +1 } (cid:1)o X j =1 exp n T (cid:0) { z i − = c ( j ) } − { z i − = c ( j ) } + { c ( j ) = z i +1 } − { c ( j ) = z i +1 } (cid:1)o . (5)Then we turn to compute an upper bound of κ defined in equation (2) for each class ofedges. The main conclusion of this paper is given in the following theorem: Theorem 2.
The second largest eigenvalue eigenvalue of the Gibbs sampler for the one-dimensional Ising model with three states satisfies : β < − × n − e − T e − T (6)The proof of this theorem is given in section . The above theorem can be generalized tothe case of multiple colors where the state space is χ = { c (1) , · · · , c ( N ) } n as follows: Theorem 3.
The second largest eigenvalue of the Gibbs sampler for the one-dimensionalIsing model with N states satisfies: β < − N × n − e − T N − e − T . (7)We give a sketch of the proof of this theorem in section .To be able to quantify the convergence rate with Theorem given by Diaconis and Stroock(see [1]) we must control the smallest eigenvalue in order to bound the second largest eigen-value in absolute value. This question is addressed in the following subsection:1.3. Bound for the absolute value of the second largest eigenvalue.
A theoremproved by Ingrassia (see [5], Theorem . ) gives the following lower bound for the smallesteigenvalue: β | χ |− ≥ − C − e ∆ T . For the one-dimensional Ising model with three states, C = 3 and ∆ = 2 . This yields forany natural number n > / √ | β | χ |− | ≤ | − e T | = 1 −
21 + 2 e T < − e − T < − n − e − T e − T + 1 . In the general case where χ = { c (1) , · · · , c ( N ) } n , the parameter C is equal to N and Ingras-sia’s bound behaves for any natural number n > N/ √ as follows: | β | χ |− | ≤ | − N − e T | = 1 −
21 + ( N − e T < − N e − T < − N × n − e − T N − e − T . The previous considerations prove the following corollary:
HE GIBBS SAMPLER FOR THE 1-D ISING MODEL 5
Corollary 1.
The upper bounds for β given in theorem and theorem are also upperbounds for the absolute value of all eigenvalues { β , · · · , β | χ |− } of the Gibbs sampler for theone-dimensional Ising model with three states and more respectively. Discussion
Ingrassia (see [5]) gives the following upper bound for the second largest eigenvalue of theGibbs sampler: β ≤ − Z T b Γ γ Γ C | S | e − mT . In this expression Z T is the normalizing constant, S is the lattice of sites, Γ is the collectionof paths, γ Γ is the maximum length of each path γ xy ∈ Γ , b Γ is the maximum number ofpaths containing any edge of Γ , C is the number of configurations that differ by only one siteand m is the least total elevation gain of the Hamiltonian function in the sense as describedby Holley and Stroock (see [4]).In our case, we have: | S | = n , γ Γ = n , b Γ = 3 n − , C = 3 , Z T ≤ e − T ) n − and m = 2 .It gives that: β ≤ − n − e − T ! n − e − T . This upper bound differs from the result introduced in Theorem by the multiplicativefactor θ = e T + 2 e − T e − T ! n − To get improvement we need that θ < which means e T +2 e − T (cid:18) e − T (cid:19) n − < . An elementary computation leads to: n > log exp ( T ) +2 exp ( − T ) ! log ( − T ) ! + 1 . ( ∗ ) For a choice of temperature T near to zero we can find an integer n sufficiently large whichverifies ( ∗ ) (it is natural in the case of the Gibbs sampler where n ∼ ) . Remark 1.
The application of Ingrassia’s bound to the one-dimensional Ising model withmultiple states gives: β ≤ − n − N − e − T N ! n − e − T which differs from the result introduced in Theorem by the factor ˜ θ defined as follows: ˜ θ = e T + ( N − e − T N N − e − T N ! n − . As previous, an elementary computation leads to: n > log exp ( T ) +( N −
1) exp ( − T ) N ! log N N −
1) exp ( − T ) ! + 1 . ( ∗∗ ) For a choice of temperature T near to zero we can find an integer n sufficiently large whichverifies ( ∗∗ ) (it is natural in the case of the Gibbs sampler where n ∼ ) . AMINE HELALI Proofs of the main results
Proof of theorem . We have two principle cases:a) If i = { , n } : According to the selection of the paths in Section . , if a path γ xy passingthrough the edge e = ( e − , e + ) connects x with y , then these extremities must have thefollowing form: x = ( x , x , · · · , x i − , c (1) , z i +1 , · · · , z n ) and y = ( z , z , · · · , z i − , c (2) , y i +1 , · · · , y n ) . This yields: π ( x ) = 1 Z T exp (cid:26) T (cid:16) i − X k ′ =1 ( { x k ′ = x k ′ +1 } − { x k ′ = x k ′ +1 } ) + n − X k ′ = i +1 ( { z k ′ = z k ′ +1 } − { z k ′ = z k ′ +1 } )+ ( { x i − = c (1) } − { x i − = c (1) } + { c (1) = z i +1 } − { c (1) = z i +1 } ) (cid:17)(cid:27) . Moreover, the probabilities π ( y ) and π ( e − ) can be expressed similarly. It follows that: Q ( e ) − π ( x ) π ( y ) = π ( x ) π ( y ) π ( e − ) P ( e − , e + )= nZ T (cid:26) n T (cid:0) − { z i − = c (2) } + { z i − = c (2) } − { c (2) = z i +1 } + { c (2) = z i +1 } (cid:1)o × X j =1 , exp n T (cid:0) { z i − = c ( j ) } − { z i − = c ( j ) } + { c ( j ) = z i +1 } − { c ( j ) = z i +1 } (cid:1)o(cid:27) × exp n T (cid:16) i − X k ′ =1 ( { x k ′ = x k ′ +1 } − { x k ′ = x k ′ +1 } ) + n − X k ′ = i +1 ( { y k ′ = y k ′ +1 } − { y k ′ = y k ′ +1 } ) (cid:17)o × exp n T (cid:0) { x i − = c (1) } − { x i − = c (1) } + { z i − = c (2) } − { z i − = c (2) } + { c (2) = y i +1 } − { c (2) = y i +1 } (cid:1)o exp n T (cid:0) { z i − = c (1) } − { z i − = c (1) } (cid:1)o . We introduce the notation ( x, c ( l ) , y ) := ( x , x , · · · , x i − , c ( l ) , y i +1 , · · · , y n − , y n ) (8)for l ∈ { , , } . The previous expression becomes: Q ( e ) − π ( x ) π ( y ) = nπ ( x, c (1) , y ) exp n T (cid:0) − { c (1) = y i +1 } + { c (1) = y i +1 } + { z i − = c (2) } − { z i − = c (2) } + { c (2) = y i +1 } − { c (2) = y i +1 } − { z i − = c (1) } + { z i − = c (1) } } (cid:1)o + nπ ( x, c (2) , y ) exp n T (cid:0) − { x i − = c (2) } + { x i − = c (2) } − { z i − = c (1) } + { z i − = c (1) } + { x i − = c (1) } − { x i − = c (1) } − { c (2) = z i +1 } + { c (2) = z i +1 } (cid:1)o X j =1 , exp n T (cid:0) { z i − = c ( j ) } − { z i − = c ( j ) } + { c ( j ) = z i +1 } − { c ( j ) = z i +1 } (cid:1)o . = nαπ ( x, c (1) , y ) exp n T (cid:0) − { c (1) = y i +1 } + { c (1) = y i +1 } + { c (2) = y i +1 } − { c (2) = y i +1 } (cid:1)o + nβπ ( x, c (2) , y ) exp n T (cid:0) − { x i − = c (2) } + { x i − = c (2) } + { x i − = c (1) } − { x i − = c (1) } (cid:1)o . (9)where HE GIBBS SAMPLER FOR THE 1-D ISING MODEL 7 i) α = exp n T (cid:0) { z i − = c (2) } − { z i − = c (2) } − { z i − = c (1) } + { z i − = c (1) } } (cid:1)o . ii) β = exp n T (cid:0) − { z i − = c (1) } + { z i − = c (1) } − { c (2) = z i +1 } + { c (2) = z i +1 } (cid:1)o X j =1 , exp n T (cid:0) { z i − = c ( j ) } − { z i − = c ( j ) } + { c ( j ) = z i +1 } − { c ( j ) = z i +1 } (cid:1)o . From the notation in equation (8) we have [ ( x,y ): γ xy ∋ e n ( x, c (1) , y ) , ( x, c (2) , y ) , ( x, c (3) , y ) o := χ . This yields Q ( e ) − X ( x,y ): γ xy ∋ e | γ xy | π ( x ) π ( y ) ≤ αn X ( x,y ): γ xy ∋ e π ( x, c (1) , y ) exp n T (cid:0) − { c (1) = y i +1 } + { c (1) = y i +1 } + { c (2) = y i +1 } − { c (2) = y i +1 } (cid:1)o + βn X ( x,y ) ,γ xy ∋ e π ( x, c (2) , y ) × exp n T (cid:0) − { x i − = c (2) } + { x i − = c (2) } + { x i − = c (1) } − { x i − = c (1) } (cid:1)o = αn X w ∈ χ : w i = c (1) π ( w ) exp n T (cid:0) − { c (1) = w i +1 } + { c (1) = w i +1 } + { c (2) = w i +1 } − { c (2) = w i +1 } (cid:1)o + βn X w ∈ χ : w i = c (2) π ( w ) exp n T (cid:0) − { w i − = c (2) } + { w i − = c (2) } + { w i − = c (1) } − { w i − = c (1) } (cid:1)o = αn A + βn B. We now turn to the computation of the two terms A and B on the right side of the previousequation separately:In order to compute A we generalize some symmetry argument from Shiu and Chen (see[7]) to the three state case. In this situation we define three spaces W ( k ) = { w ∈ χ, w i = c (1) , w i +1 = c ( k ) } for k ∈ { , , } . In order to compute their π measure we will establishsome equations between those numbers π ( W (1) ) , π ( W (2) ) and π ( W (3) ) .We now establish some identification between the elements from W (1) and the elements of W (2) respective W (3) .For any vertex ξ ∈ W (1) , there exist a unique vertex ξ ∈ W (2) such that: • If k < i , ξ k = ξ k . • If k > i + 1 , then: – If ξ k = c (1) then ξ k = c (2) . – If ξ k = c (2) then ξ k = c (1) . – If ξ k = c (3) then ξ k = c (3) .Similarly, for any vertex ξ ∈ W (1) , there exist a unique vertex ξ ∈ W (3) such that : • If k < i , ξ k = ξ k . • If k > i + 1 , then: – If ξ k = c (1) then ξ k = c (3) . – If ξ k = c (3) then ξ k = c (1) . – If ξ k = c (2) then ξ k = c (2) .Those relations yield that π ( ξ ) = e T π ( ξ ) = e T π ( ξ ) . Therefore, we obtain: X w ∈ W (1) π ( w ) = e T X w ∈ W (2) π ( w ) = e T X w ∈ W (3) π ( w ) . (10) AMINE HELALI
On the other hand we have also: X w ∈ W (1) π ( w ) + X w ∈ W (2) π ( w ) + X w ∈ W (3) π ( w ) = 13 . (11)From equations (10) and (11) we deduce: X w ∈ W (1) π ( w ) = 13(1 + 2 e − T ) , X w ∈ W (2) π ( w ) = e − T e − T ) and X w ∈ W (3) π ( w ) = e − T e − T ) . The subdivision of the sum in A to three sums over the sets W (1) , W (2) and W (3) gives A = X w ∈ χ : w i = c (1) π ( w ) exp n T (cid:0) { c (2) = w i +1 } − { c (2) = w i +1 } − { c (1) = w i +1 } + { c (1) = w i +1 } (cid:1)o = 13 . (12)With the same tricks, we obtain the same result for BB = X w ∈ χ : w i = c (2) π ( w ) exp n T (cid:0) { w i − = c (1) } − { w i − = c (1) } − { w i − = c (2) } + { w i − = c (2) } (cid:1)o = 13 . (13)Using equations (12) and (13) we obtain: Q ( e ) − X ( x,y ): γ xy ∋ e | γ xy | π ( x ) π ( y ) ≤ αn A + βn B = n α + β ) . The sites z i − and z i +1 take the values c (1) , c (2) or c (3) . The worst value of α + β is obtainedwhen z i − = c (3) = z i +1 . In this case we have: Q ( e ) − X ( x,y ): γ xy ∋ e | γ xy | π ( x ) π ( y ) ≤ n e T ) . (14)b) If i = 1 then the configurations x and e − coincide, from equations (2) and (3) we obtain: κ = max e ∈ E Q ( e ) − X γ xy ∋ e | γ xy | π ( x ) π ( y ) ≤ n X γ xy ∋ e π ( x ) π ( y ) n π ( e − ) P ( e − , e + ) ≤ n (1 + 2 e T ) X y ∈ χ : y = c (2) π ( y ) = n e T ) . (15)For i = n , then the configurations y and e + coincide and some computation gives a similarresult as in equation (15) .By regrouping the results in equations (14) and (15) we obtain an upper bound for theconstant κ defined in equation (2) as follows: κ = max e ∈ E Q ( e ) − X γ xy ∋ e | γ xy | π ( x ) π ( y ) ≤ n e T ) . (16) Remark 2.
The above computation was done for the situation where x i = c (1) and y i = c (2) .Obviously we obtain the same result in the other cases, ie.: x i = c (1) and y i = c (3) , etc · · · .Finally, from inequalities (1) and (16) we obtain an upper bound for β which finishesthe proof. HE GIBBS SAMPLER FOR THE 1-D ISING MODEL 9
Proof of theorem . We follow the same approach as in the case where χ = { c (1) , c (2) , c (3) } n .We define an edge e = ( e − , e + ) as in section . and then distinguish two cases:a) For i = { , n } , we pass in equations (3), (8) and (9) from the case of three colors where χ = { c (1) , c (2) , c (3) } n to the case of multiple colors where χ = { c (1) , · · · , c ( N ) } n . This yields: Q ( e ) − X ( x,y ): γ xy ∋ e | γ xy | π ( x ) π ( y ) ≤ αn X w ∈ χ : w i = c (1) π ( w ) exp n T (cid:0) − { c (1) = w i +1 } + { c (1) = w i +1 } + { c (2) = w i +1 } − { c (2) = w i +1 } (cid:1)o + βn X w ∈ χ : w i = c (2) π ( w ) exp n T (cid:0) − { w i − = c (2) } + { w i − = c (2) } + { w i − = c (1) } − { w i − = c (1) } (cid:1)o = αn A ′ + βn B ′ . wherei) α = exp n T (cid:0) { z i − = c (2) } − { z i − = c (2) } − { z i − = c (1) } + { z i − = c (1) } } (cid:1)o , ii) β = exp n T (cid:0) − { z i − = c (1) } + { z i − = c (1) } − { c (2) = z i +1 } + { c (2) = z i +1 } (cid:1)o × N X j =1 j =2 exp n T (cid:0) { z i − = c ( j ) } − { z i − = c ( j ) } + { c ( j ) = z i +1 } − { c ( j ) = z i +1 } (cid:1)o . To compute the term A ′ we define for k = { , · · · , N } the spaces W ( k ) = { w ∈ χ : w i = c (1) , w i +1 = c ( k ) } and we consider some symmetry arguments as above:For any vertex ξ ∈ W (1) there exist a unique ξ l ∈ W ( l ) where l ∈ { , · · · , N } such that: • If k < i , ξ k = ξ lk . • If k > i + 1 , then: – If ξ k = c (1) then ξ lk = c ( l ) . – If ξ k = c ( l ) then ξ lk = c (1) . – If ξ k = c (˜ l ) where ˜ l = { , l } then ξ lk = c (˜ l ) .Then equations (10) and (11) becomes in the case of N colors: X w ∈ W (1) π ( w ) = e T X w ∈ W (2) π ( w ) = · · · = e T X w ∈ W ( N ) π ( w ) , (17) X w ∈ W (1) π ( w ) + · · · + X w ∈ W ( N ) π ( w ) = 1 N . (18)From equations (17) and (18) and with the same tricks used to obtain equation (12) we get: A ′ = X w ∈ χ : w i = c (1) π ( w ) exp n T (cid:0) { c (2) = w i +1 } − { c (2) = w i +1 } − { c (1) = w i +1 } + { c (1) = w i +1 } (cid:1)o = 1 N . (19)With the same tricks, we obtain the same result for B ′ B ′ = X w ∈ χ : w i = c (2) π ( w ) exp n T (cid:0) { w i − = c (1) } − { w i − = c (1) } − { w i − = c (2) } + { w i − = c (2) } (cid:1)o = 1 N . (20)
The application of equations (19) and (20) gives: Q ( e ) − X ( x,y ): γ xy ∋ e | γ xy | π ( x ) π ( y ) ≤ αn A ′ + βn B ′ = n N ( α + β ) . The worst value of α + β is obtained when z i − = c ( l ) = z i +1 for l ∈ { , · · · , N } . In thiscase we have: Q ( e ) − X ( x,y ): γ xy ∋ e | γ xy | π ( x ) π ( y ) ≤ n N ( N − e T ) . (21)b) For the boundary cases, when i = 1 equation (15) becomes Q ( e ) − X γ xy ∋ e | γ xy | π ( x ) π ( y ) ≤ n N ( N − e T ) . (22)Also, we obtain a similar result for the case where i = n .Equation (21) and (22) together give an upper bound of κ defined in (2) as follow: κ = max e ∈ E Q ( e ) − X ( x,y ): γ xy ∋ e | γ xy | π ( x ) π ( y ) ≤ n N ( N − e T ) . (23)Finally, we apply the upper bound given in (23) in equation (1) to finish the proof. Acknowledgements
I would like to thank my supervisors Brice Franke and Mondher Damak for their helpand advice during this work and their availability for answering all my questions.This work was supported by the Tunisian-French cooperation (PHC-UTIQUE) projectCMCU2016 Number 16G1505.I also would like to thank the reviewer for the comments which greatly helped in amelioratingthe paper.
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