Borodin-P\'ech\'e fluctuations of the free energy in directed random polymer models

We consider two directed polymer models in the Kardar-Parisi-Zhang (KPZ) universality class: the O'Connell-Yor semi-discrete directed polymer with boundary sources and the continuum directed random polymer with (m,n)-spiked boundary perturbations. The free energy of the continuum polymer is the Hopf-Cole solution of the KPZ equation with the corresponding (m,n)-spiked initial condition. This new initial condition is constructed using two semi-discrete polymer models with independent bulk randomness and coupled boundary sources. We prove that the limiting fluctuations of the free energies rescaled by the 1/3rd power of time in both polymer models converge to the Borodin-Peche type deformations of the GUE Tracy-Widom distribution.

By the presence of the non-linear term, the equation is not rigorously well-posed and serious work is required to make sense of the solution directly [12]. A natural way to give a solution to the equation formally is via the stochastic heat equation (SHE) with multiplicative noise ∂ T Z(T, X) = 1 2 ∂ 2 X Z(T, X) + Z(T, X)ξ(T, X), Z(0, X) = Z 0 (X). (1. 2) The latter equation is well-posed and F (T, X) = ln Z(T, X) with initial condition F (0, X) = ln Z(0, X) defines a formal solution to (1.1) which is the Hopf-Cole solution of the KPZ equation. See [8] for a review on the KPZ equation and its universality class which is the family of models with the same scaling and asymptotic behaviour as the solution of the KPZ equation. where the expectation E is taken over the law of a Brownian motion B which is running backwards from time T and position X and where : exp : is the Wick exponential. The representation (1.3) defines the partition function of the continuum directed random polymer (CDRP) as it is the total weight of Brownian paths where the weight is proportional to the exponential function of the integral of the disorder along the path. The logarithm of the partition function F (T, X) = ln Z(T, X) is called the free energy of the CDRP. The present paper describes limiting fluctuations in two directed polymer models. Directed polymers are well-studied objects in the KPZ universality class of models in the recent mathematics and physics literature. The reason for the special interest is that certain models possess exact solvable properties, i.e. explicit formulas can be derived for some of their important observables. The first directed polymer model with exact solvability is the O'Connell-Yor semidiscrete polymer [18,16]. Exactly solvable polymers on the square lattice are the log-gamma polymer [19,9,20], the strict-weak polymer [10,17], the beta polymer [3] and the inverse beta polymer [21]. Methods to obtain exact solvability include explicit stationary measure, Bethe Ansatz integrability and the (geometric) Robinson-Schensted-Knuth (RSK) correspondence.
In [5], the O'Connell-Yor model was considered with boundary perturbations. The large time limit of the free energy was proved to be the Baik-Ben Arous-Péché (BBP) distribution [2] which is the perturbed version of the GUE Tracy-Widom distribution. A similar limit distribution was obtained for the CDRP with m-spiked boundary perturbation in [5].
The results of the present paper generalize those of [5] in the following sense. We investigate the large scale behavior of the free energy of two directed polymer models. The first model is the O'Connell-Yor semi-discrete random polymer with log-gamma boundary sources [6] which is the mixture of the O'Connell-Yor semi-discrete polymer with boundary perturbations considered in [5] and the log-gamma discrete directed polymer. As the limit distribution of the free energy, we obtain the single time version of the Borodin-Péché distribution which is a generalization of the BBP distribution. The Borodin-Péché distribution was first described in its multi-time version in last passage percolation with defective rows and columns and in a single time version in a random matrix model in [7]. To deduce the limit theorem for the O'Connell-Yor model with log-gamma boundary sources, we use explicit Fredholm determinant expressions from [6] for the Laplace transform of the partition function of the polymer mixture model.
The second model considered in the present paper is the CDRP which can be obtained as the limit of the O'Connell-Yor semi-discrete polymer under the intermediate disorder scaling [11,15]. Extending the investigations of the CDRP with m-spiked boundary perturbation in [5], we introduce the (m, n)-spiked boundary perturbation. The m-spiked boundary perturbation is non-zero for the positive values of the space variable, the (m, n)-spiked boundary perturbation can be seen as its two-sided version with the appropriate coupling of the two sides. We prove Borodin-Péché limit distribution for the free energy of the CDRP with (m, n)-spiked boundary perturbation based on explicit Fredholm determinant formulas from [6].
The rest of the paper is organized as follows. We introduce the O'Connell-Yor semi-discrete directed random polymer with log-gamma boundary sources and the CDRP with (m, n)-spiked boundary perturbation in Section 2. Our main results, Theorem 2.3 and Theorem 2.5 are also stated in this section. We prove Theorem 2.3 in Section 3 and Theorem 2.5 in Section 4. Figure 1: The O'Connell-Yor semi-discrete directed polymer with log-gamma boundary sources. The thick solid line is a possible path φ from (−n, 1) to (τ, N). The random variables ω −k,l have log-gamma distribution with parameter α k − a l and the Brownian motions B 1 , . . . , B N have drifts a 1 , . . . , a N .

Models and main results
We present the two models considered in this paper: the O'Connell-Yor semi-discrete directed polymer with log-gamma boundary sources and the continuum directed random polymer (CDRP) with (m, n)-spiked boundary perturbation. These models were defined in [6], but the (m, n)-spiked boundary perturbation is new. We consider a slightly different scaling of the boundary perturbations as in [6] yielding our main results which are stated in this section.

O'Connell-Yor semi-discrete directed polymer with log-gamma boundary sources
The O'Connell-Yor semi-discrete directed polymer with log-gamma boundary sources is the mixture of the semi-discrete polymer model introduced by O'Connell and Yor [18] and the discrete one by Seppäläinen [19]. By log-gamma distribution with parameter θ > 0 we mean the distribution of the random variable − ln X where X has gamma distribution with parameter θ, i.e. when X has density x θ−1 e −x /Γ(θ) for x > 0. Fix N ≥ 1 and n ≥ 0. Let a = (a 1 , . . . , a N ) ∈ R N and α = (α 1 , . . . , α n ) ∈ R n + be such that α k − a l > 0 for all 1 ≤ l ≤ N and 1 ≤ k ≤ n. In the polymer model that we introduce, the horizontal axis is discrete on the left of 0 and continuous on the right of 0 while the vertical axis is discrete. For all 1 ≤ k ≤ n and 1 ≤ l ≤ N, let ω −k,l be independent log-gamma random variables with parameter α k − a l . For all 1 ≤ l ≤ N, let B l be independent Brownian motions with drift a l which are also independent of the log-gamma variables. The ω −k,l can be thought of as sitting at the lattice points (−k, l) while B l can be thought of as sitting along the horizontal ray from (0, l) as shown on Figure 1.
To an up-right path described above, we associate an energy which aggregates the randomness along the path, hence itself is random depending on ω i,j and B 1 , . . . , B N . The polymer measure on a path φ is proportional to its Boltzmann weight given by e E(φ) . The normalizing constant or polymer partition function for the O'Connell-Yor semidiscrete directed polymer with log-gamma boundary sources is the integral of the Boltzmann weight over the background measure on the path space φ, i.e.
where dφ sd represents the Lebesgue measure on the simplex 0 ≤ s k < s k+1 < · · · < s N −1 ≤ τ with which φ sd is identified. The free energy of the O'Connell-Yor semi-discrete directed polymer with log-gamma boundary sources is given by 3) The distribution of the partition function Z a,α (τ, N) of the O'Connell-Yor semi-discrete directed polymer with log-gamma boundary sources was characterized in [6] as follows.
For 1 ≤ k ≤ n and 1 ≤ l ≤ N let ω −k,l be independent log-gamma random variables with parameter α k − a l and for all 1 ≤ l ≤ N let B l be independent Brownian motions with drift a l . Then for all u ∈ C with positive real part where the operator K u is defined in terms of its integral kernel The contours C a;α;ϕ and D v are given in Definition 2.2 below where ϕ ∈ (0, π/4) is arbitrary.
The contour is oriented so as to have increasing imaginary part. For every v ∈ C a;α;ϕ , we choose R = − Re(v) + η, d > 0, and define a contour D v as follows. D v goes by straight lines from R − i∞, to R − id, to 1/2 − id, to 1/2 + id, to R + id, to R + i∞. The parameter d is taken small enough so that v + D v does not intersect C a;α;ϕ . See Figure 2 for an illustration.
Our contribution on the O'Connell-Yor semi-discrete directed polymer with log-gamma boundary sources is that we prove a Borodin-Péché scaling limit of its free energy. To define the limiting distribution, fix two integers m and n. Let b = (b 1 , . . . , b m ) ∈ R m and β = (β 1 , . . . , β n ) ∈ R n be two sets of parameters and assume that Figure 2: Left: the contour C a;α;ϕ (dashed) where the black dots symbolize the set of singularities which is a natural constraint, since otherwise the corresponding polymer models are not well defined, see Theorem 2.3 and 2.5 below for the parameter scaling. The Borodin-Péché distribution with parameters b and β [7] is defined as with the kernel where the integration contours γ and Γ are given as follows. Let c > 0 be arbitrary. Then γ is −c + iR modified in a neighbourhood of the real axis so that it crosses the axis between max 1≤l≤m b l and min 1≤k≤n β k . The contour Γ is c + iR modified in a neighbourhood of the real axis so that it crosses the real axis between max 1≤l≤m b l and min 1≤k≤n β k and it does not intersect γ. We mention that for n = 0, the Borodin-Péché distribution reduces to the BBP distribution and for n = m = 0 to the GUE Tracy-Widom distribution.
To state our main theorem on the scaling limit of the O'Connell-Yor semi-discrete directed polymer with log-gamma boundary sources, we will use the following parametrization. Let Ψ(z) = d dz ln Γ(z) be the digamma function. For a given θ ∈ R + , define We may alternatively parameterize θ ∈ R + in terms of κ ∈ R + as holds where F a,α is the free energy of the O'Connell-Yor semi-discrete directed random polymer given in (2.3) and F BP,b,β is the Borodin-Péché distribution function defined in (2.7).
2.2 Continuum directed random polymer (CDRP) with (m, n)-spiked boundary perturbation The partition function Z(T, X) of the continuum directed random polymer with boundary perturbation Z 0 (X) is given by the solution to the stochastic heat equation with multiplicative noise (1.2) with initial condition Z 0 (X). The initial data Z 0 (X) may be random but it is assumed to be independent of the space-time white noise. By the Feynman-Kac representation (1.3), Z(T, X) is indeed a partition function of a directed polymer model, since Brownian paths are reweighted in a way that the weight of a path is proportional to the Wick exponential of the randomness integrated along the path. The normalizing constant which is the partition function Z(T, X) is the integral of weights over the space of all possible paths. Note that Z(T, X) itself is random as the randomness of the space-time white noise remains in the formula (1.3).
By the work of Mueller [14], as long as Z 0 (X) is almost surely positive, Z(T, X) is positive for all T > 0 and X ∈ R almost surely. Hence we can take its logarithm and define the free energy for the continuum directed random polymer with boundary perturbation ln Z 0 (X) by F (T, X) = ln(Z(T, X)) to be the Hopf-Cole solution of the KPZ equation (1.1) with initial condition F 0 (X) = ln Z 0 (X).
Let us now introduce the CDRP with (m, n)-spiked boundary perturbation and let us construct the corresponding (m, n)-spiked initial condition for the stochastic heat equation. For fixed integers m and n, let b = (b 1 , . . . , b m ) ∈ R m and β = (β 1 , . . . , β n ) ∈ R n be such that (2.6) holds. Let B 1 , B 2 , . . . , B m be independent Brownian motions with drifts b 1 , b 2 , . . . , b m , and let B 1 , B 2 , . . . , B n be independent Brownian motions with drifts β 1 , β 2 , . . . , β n . Furthermore, let ω −k,l be independent log-gamma random variables with parameter β k − b l for 1 ≤ l ≤ m and 1 ≤ k ≤ n. Assume that the two families of Brownian motions and the log-gamma random variables are independent of each other. For X ≥ 0, let the semi-discrete partition function Z b,β (X, m) be constructed as in (2.2) using the Brownian motions B 1 , B 2 , . . . , B m and the loggamma random variables.
Similarly, we construct another semi-discrete partition function which is coupled to the previous one. Let the possible paths φ be composed of a discrete up-right part φ d : (−n, 1) ր (k − n − 1, m) and of a semi-discrete part φ sd . For X ≥ 0, let the semi-discrete part φ sd : The energy of a such a path is instead of (2.1) defined by Then a partition function analogously to (2.2) is given by 14) The Brownian motions B k can be thought of as sitting on the vertical rays starting at (k − n − 1, m) for 1 ≤ k ≤ n which makes the definitions (2.13)-(2.14) natural. for the CDRP, i.e. the (m, n)-spiked initial condition for the stochastic heat equation. It is realized by two semi-discrete polymer partition functions with log-gamma boundary sources where the log-gamma random variables are sampled jointly. For X > 0, Z b,β (X, m) appears on the horizontal half-line starting at (0, m + 1) whereas Z β,b (−X, n) for X ≤ 0 appears on the vertical half-line starting at the same point.
For b ∈ R m and β ∈ R n , let define the (m, n)-spiked boundary perturbation for the CDRP, see Figure 3. Let Z b,β (T, X) denote the partition function of the CDRP with (m, n)-spiked boundary perturbation which is the solution of the stochastic heat equation (1.2) with initial condition given by (2.15). Let F b,β (T, X) = ln(Z b,β (T, X)) denote the free energy of the CDRP with (m, n)-spiked boundary perturbation. Note that for n = 0, the boundary perturbation (2.15) reduces to the m-spiked boundary perturbation considered in [5]. According to the next theorem, the CDRP with (m, n)-spiked boundary perturbations is the limit of the O'Connell-Yor semi-discrete directed polymer with log-gamma boundary sources under the intermediate disorder scaling. This result was first announced in [11] for the O'Connell-Yor semi-discrete directed polymer with boundary perturbations and it was proved for the unperturbed multi-layer case in [15]. We state the theorem in the form which is the most suitable for the present setup, i.e. with perturbed boundaries. as N goes to infinity where Z b,β (T, X) is the CDRP with (m, n)-spiked boundary perturbation given in (2.15).
The main contribution of this work gives the large time limit of the CDRP free energy with (m, n)-spiked boundary perturbation.
Theorem 2.5. Let b = (b 1 , . . . , b m ) ∈ R m and β = (β 1 , . . . β n ) ∈ R n be such that b l < β k for all 1 ≤ l ≤ m and 1 ≤ k ≤ n. Let σ = (2/T ) 1/3 be scaled with the time parameter and let Y ∈ R and r ∈ R be arbitrary. Then for the free energy of the CDRP with (m, n)-spiked boundary perturbation of parameters σb and σβ at rescaled position X = 2 1/3 Y T 2/3 , holds where F BP,b+Y,β+Y is the Borodin-Péché distribution function given by (2.7) with parameter vectors shifted coordinatewise.

Scaling limit for the O'Connell-Yor semi-discrete polymer
We prove Theorem 2.3 in this section which is a modification of the proof of Theorem 1.3 in [5]. We mention that in Theorem 1.3 in [5] which is the n = 0 case of Theorem 2.3, the factor c κ is missing in the scaling of parameters (2.11). To keep our discussion self-contained, we recall the main steps of the proof and extend it to the present setup. Let us scale u = u(N, r, κ) = exp(−Nf κ − rc κ N 1/3 ) (3.1) and set τ = κN. After the change of variablesz = s + v in (2.5) and by using Euler's reflection formula 1/(Γ(−s)Γ(1 + s)) = −π/ sin(πs), where G(z) = ln Γ(z) − κz 2 /2 + f κ z. The integration contour Cz in (3.2) was defined in [5] in the absence of boundary parameters to be where B v+q denotes a small circle around v + q and clockwise oriented. r ∈ N 0 is chosen such . This choice of r and p(v) is needed to keep a uniformly positive distance from the poles coming from the sine in the denominator in (3.2), see Section 5.1 in [5] for the precise definition. It is also argued in [5] that kernel K u has enough decay along the contour in (3.3) which corresponds to the ϕ = π/4 case for the C a;α;ϕ contour in Theorem 2.1.
In the present setup when there are boundary parameters a l and α k scaled according to (2.11), the contour Cz is defined to be the contour (3.3) with a local modification in an N −1/3 neighbourhood of θ κ in a way that it crosses the real axis between the a l and the α k singularities. By the Cauchy theorem, the contour v + D v forz seen on the left of Figure 2 can be replaced by Cz without changing the kernel K u in (3.2).

PSfrag replacements
The function G has a double critical point at 3 . This suggests the rescaling around θ κ by N 1/3 , that is the change of variables Then the rescaled kernel is defined as Let the new contour C w be the local perturbation of {−|y| + iy, y ∈ R} in a constant neighbourhood of 0 in a way that it crosses the real axis between the b l and β k singularities as shown on Figure 4, also compare with Figure 2. Further, let C z be the local modification of 1 + iR in a neighbourhood of 0 so that it does not intersect C w and it crosses the real axis between the two families of singularities. Then one can replace C a;α;ϕ by C w and the integration path Φ −1 (Cz) in (3.5) by C z so that one has the equality of Fredholm determinants det (½ + K u ) L 2 (Ca;α;ϕ) = det (½ + K N ) L 2 (Cw) by the Cauchy theorem.
Based on the next two propositions and by using Lemma 3.3 below, Theorem 2.3 on the scaling limit for the O'Connell-Yor semi-discrete polymer can be verified.
Proposition 3.2. For any w, w ′ ∈ C w there exists a constant C ∈ (0, ∞) such that uniformly for all N large enough. lim x→∞ Θ n (x) = 0 for all n and Θ n (x) → ½ x≤0 as n → ∞ uniformly on R \ [−δ, δ] for all δ > 0. Consider a sequence of random variables X n and a continuous probability distribution function p(r) such that E [Θ n (X n − r)] → p(r) as n → ∞ for each r ∈ R. Then X n converges in distribution to the distribution given by p(r).
Proof of Theorem 2.3. By Hadamard's bound and by dominated convergence, Proposition 3.1 and 3.2 together imply that as N → ∞ where the first equality above follows from the same reformulation of Fredholm determinants as in Lemma 8.7 of [5]. Let us define a sequence of functions Θ N (x) = exp(− exp(c κ N 1/3 x)). Now by (3.1), We introduce the extra gamma factors  as long as w ∈ C w and z ∈ C z . Furthermore, let N be large enough to make the N −1/3 difference of Cz and the contour in (3.3) small. Then the small circles in Cz and in (3.3) can only be present, i.e. r > 0 can only happen for a v ∈ C a;α;ϕ if |v| > ε for some fixed ε > 0. In this case there is a C such that for any v ∈ C a;α;ϕ and q = 1, . . . , r, Proof. After substituting (3.4) and the scaling (2.11) into the definition (3.11), one can write . (3.14) First we show that for the numerator holds if z ∈ C z . To this end, we use the asymptotics  [1]. If z ∈ C z and |z| > δN 1/3 for some fixed δ > 0, then the real part of the argument of the gamma function in (3.15) goes to 0 as N → ∞, hence it is bounded. Consequently, (3.16) yields an exponential decay of |Γ (β k − z + o(1))c −1 κ N −1/3 | in |z| which we bound by a constant. By the asymptotics Γ(Z) ∼ 1/Z around Z = 0, one gets that the left-hand side of (3.15) can be upper bounded by CN 1/3 /|z| as long as |z| < δN 1/3 . This proves (3.15).
Next we prove for the denominator that for w ∈ C w with a constant c small enough. Equation 6.1.37 in [1] reads as For w ∈ C w and |w| > δN 1/3 , one can write w = −tN 1/3 ± itN 1/3 for some t > δ/ √ 2. Hence the left-hand side of (3.17) grows as Ce −t t t as t → ∞ which we can lower bound by a small constant as t > δ/ √ 2. If |w| < δN 1/3 , by the asymptotics Γ(Z) ∼ 1/Z around Z = 0 again, the left-hand side of (3.17) is lower bounded by cN 1/3 /|w|. This shows (3.17). Putting (3.15) and (3.17) together yields (3.12) with a large enough C.
The uniform lower bound on |v| follows from the choice of the contours. On one hand, r is chosen such that Re(v) + r ≤ θ κ + O(N −1/3 ). On the other hand, v ∈ C a;α;ϕ satisfies v = θ κ + O(N −1/3 ) + e i(π±ϕ) y for some y ∈ R + . These two properties imply the lower bound if the small circles are present.
To show (3.13) if the circles are present, observe that the ratio which we want to bound in absolute value simplifies as . (3.19) This is bounded by an absolute constant since |v| > ε also means that | Im(v)| is uniformly positive.
Proof of Proposition 3.1. Knowing the asymptotics Γ(z) ≃ 1/z near zero from [1], one can conclude from (3.14) that under the scaling (2.11), Q(w, z, α k ) → (w − β k )/(z − β k ) holds as N → ∞ for k = 1, . . . , n. Similarly, P (w, z, a l ) → (z − b l )/(w − b l ) for l = 1, . . . , m. As in the proof of Proposition 5.1 in [5], the Taylor expansion of the remaining factors in the integrand of K N in (3.5) yields that the integrand converges to the integrand of K BP,b,β in (3.7). The applicability of dominated convergence follows the same line as in [5]. By the fact that the integration contour of K N in z is steep descent for the function −G(Φ(z)), this term determines the decay of the integrand which is Gaussian in | Im(z)|. Since the Q factors are bounded in (3.12), the decay remains strong enough in the presence of the Q factors, hence the steps of the proof of Proposition 5.1 in [5] can be followed. In particular, the integral which defines K N restricted to the set | Im(z)| > δN 1/3 is O(e −c(δ)N ). On the other hand on | Im(z)| < δN 1/3 , one can replace the integrand of K N by the integrand of K BP,b,β with an overall error of order O(N −1/3 ). This verifies the convergence of the kernels (3.6).
Proof of Proposition 3.2. The exponential bounds obtained in the proof of Proposition 5.2 in [5] are not affected by the presence of extra polynomial factors which upper bound Q in (3.12)-(3.13). Hence (3.8) follows. 4 Large time limit of the CDRP with (m, n)-spiked boundary In this section, we prove Theorem 2.5 about the large time limit of the free energy Z b,β of the CDRP with (m, n)-spiked boundary perturbations. We start by giving a Fredholm determinant formula for its Laplace transform below in Proposition 4.1 based on Theorem 2.1. Let b = (b 1 , . . . , b m ) and β = (β 1 , . . . , β n ) be such that (2.6) holds. Define the kernel and the integration contour for w is from − 1 4σ − i∞ to − 1 4σ + i∞ and crosses the real axis between max 1≤l≤m b l /σ and min 1≤k≤n β k /σ. The other contour for z goes from 1 4σ − i∞ to 1 4σ + i∞, it also crosses the real axis between max 1≤l≤m b l /σ and min 1≤k≤n β k /σ and it does not intersect the contour for w.  .2). Then where Z b,β is the partition function of the CDRP with (m, n)-spiked boundary perturbations and K as N → ∞. By definition (2.2), the partition function Z a,α is positive, hence (4.5) implies the convergence of the Laplace transforms as N → ∞ where τ = √ T N + X and u is defined in (4.4). On the other hand, the same scaling of parameters is used on the right-hand side of (2.4). Then Theorem 6.3 of [6] is used to conclude the convergence of Fredholm determinants under the following scaling of the parameters. As in Theorem 2.4, one sets τ = √ T N + X, κ = τ /N and θ κ is given by (2.10). This means that θ κ = N/T − X/T + 1/2 + O(N −1/2 ). One sets u given by (4.4) and σ given by (4.2). For the boundary parameters a l and α k , instead of the scaling given in Theorem 2.4, one sets a l = θ κ + b l and α k = θ κ + β k according to Section 6 of [6]. This difference results in the shift by X/T in the rescaled boundary parameters b l and β k which completes the proof.
The following proposition is the key for the proof of Theorem 2.5.
(4.12) Then the first factor in the double integral in (4.12) converges to e −r(z−w) /(z − w) as σ → 0. For the product of the gamma ratios, w − β k z − β k (4.13) as σ → 0. Hence integrand in (4.12) converges to that of K BP,b,β (x + r, y + r) given in (2.8) as σ → 0. Since along the contours for w and z, the factors e z 3 /3−w 3 /3 in (4.12) have fast enough decay, we conclude that σb,σβ (x, y) = K BP,b,β (x + r, y + r). (4.14) To show that the convergence of the kernels (4.14) implies the convergence of Fredholm determinants (4.8), one uses dominated convergence. Lemma 4.3 applied to K (σ) σb,σβ provides a uniform upper bound in σ. Using this upper bound, the nth term in the Fredholm determinant expansion of the left-hand side of (4.8) is bounded by R + e −(β 1 −bm) n j=1 x j dx 1 . . . dx n = C 2n n n/2 (β 1 − b m ) n n! where we also used the Hadamard bound in the first inequality above. Since the right-hand side of (4.15) is summable, dominated convergence implies (4.8).