{'fraction_non_alphanumeric': 0.054786318389342115, 'fraction_numerical': 0.015118628845146321, 'mean_word_length': 3.9240832872673668, 'pattern_counts': {'":': 0, '<': 16, '<?xml version=': 0, '>': 21, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 13, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Extremal linkage networks

We demonstrate how sophisticated graph properties, such as small distances and scale-free degree distributions, arise naturally from a reinforcement mechanism on layered graphs. Every node is assigned an a-priori i.i.d. fitness with max-stable distribution. The fitness determines the node attractiveness w.r.t. incoming edges as well as the spatial range for outgoing edges. For max-stable fitness distributions, we thus obtain a complex spatial network, which we coin extremal linkage network.


Motivation
Many real-world networks share a number of stylized facts, including • scale-free: the degree sequence resembles a power-law distribution; • small world: the graph distances between different network nodes are typically short, say o(N δ ) for every δ > 0 (where N is the total number of nodes in the network); • geometric clustering: nodes that are close to each other (in a certain geometric sense) have higher chance of being connected; • hierarchies: nodes with high degree have high probability of being connected even if they are far from each other. A variety of mathematical network models have been proposed addressing all or at least some of the features above, and these appear in contexts from a wide range of domains. It is nevertheless not clear, why real-world networks share these ubiquitous features.
The preferential attachment model, introduced by Barabási and Albert [2], establishes reinforcement mechanisms as an attempt to explain the universality of these features from an algorithmic point of view. Indeed, preferential attachment graphs exhibit the scale-free and small-world properties [4]. In spatial versions, there is even geometric clustering and hierarchies [1,13].
Although the preferential attachment mechanism presents a compelling explanation for network formation, the ramifications of selecting nodes proportionally to their fitness can sometimes be prohibitively challenging to handle from a mathematical point of view. This raises the question, whether it is possible to recover many of the desirable features through a much simpler game. For instance, can we replace the proportional selection by simply picking the node with the maximum fitness in a spatial neighborhood? Although at first sight, this may seem like an entirely different story, it actually approximates the strong reinforcement regime, where the selection occurs according to power-weighted fitnesses with a large exponent. Naturally, this maximality-based selection scheme opens the door towards connections with extreme value theory, and we explore this path in detail in the present work.
To put this abstract blueprint on a concrete footing, we focus on an activity-based reinforcement model inspired from synaptic plasticity in neuroscience. We first outline the model motivation and definition and then showcase how extreme value theory enters the stage in the form of the max-stability of the Fréchet distribution. To this end, we are considering a caricature model for a neural network: There is a set of neurons, each of them equipped with one axon and a number of dendrites connected to axons of other neurons. Pairs of axons and dendrites may form synapses, i.e., functional connections between neurons. However, not all geometric connections necessarily also form functional connections. The resulting network can be interpreted as a directed graph with neurons as nodes and synapses as edges (directed from dendrite to axon).
A number of experiments revealed that the resulting neural network is rather sparse and very well connected, that is, any pair of neurons is connected through a short chain of neural connections reminiscent of the "small-world property". These features allow for very fast and efficient signal processing. The challenge is to explain the mechanism behind the formation of such sophisticated neural networks. Kalisman, Silberberg, and Markram [14] use experimental evidence to advocate a tabula rasa approach: In an early stage, there is a (theoretical) all-toall geometrical connectivity. Stimulation and transmission of signals enhance certain touches to ultimately form functional connections, which results in a network with rather few actual synapses. This describes brain plasticity at an early development stage.
It is clear that the actual formation of the brain involves much more complex processes that are beyond the scope of a rigorous treatment. Yet, we aim at clarifying which network characteristics can be explained by a simple reinforcement scheme, and which cannot. The mathematical question that we put forward is: Can a simple reinforcement mechanism give rise to complex network properties such as the scale-free and small-world properties?
A naive modelling using so-called (W, A)-reinforcement models (or 'WARM') suggests a negative answer. In this model involving Pólya urns with graph-based competition, there are only two regimes: In the strong reinforcement regime, the process is supported on small isolated islands [9,10,12], whereas in the weak reinforcement regime the support is on the entire graph [5,11]. There is thus no regime in which a subgraph with suitable properties emerges.
The situation changes dramatically when looking at a slightly different setup. In an earlier work [8], we investigated a WARM-type model on a layered network. On this layered network, we proved rigorously that sufficiently strong reinforcement is responsible for logarithmic distances, and thus the small-world property applies for the resulting random graph.
However, one may criticize that many of the final findings in [8] were already hard-wired exogenously into the model from the beginning. In particular, each layer is assigned a (fixed) scope, which grows exponentially in the number of layers, and this scope determines how far a vertex can connect. This scope misses physical plausibility, and in the present manuscript we rectify this shortcoming by modelling the reinforcement scheme endogenously. Indeed, we assign each vertex an a-priori fitness, which has two functions: first it measures the node attractiveness in comparison with the fitnesses of neighboring nodes; secondly, it encodes how far a connection from this vertex may reach. Instead of activity-based reinforcement, like in [8], we simplify the interaction by drawing one directed edge from vertex (i, h) ∈ Z × N 0 with fitness F i,h > 0 (where i is the spatial location and h is the layer) to the vertex with highest fitness in the set This resembles a strong reinforcement regime in case the fitnesses have a max-stable distribution, which we henceforth assume. We prove the scale-free and small-world property for such networks under suitable parameters. Since the max-stable fitness distribution is instrumental both for modelling the reinforcement and also for obtaining the desired behavior, we coin this network model extremal linkage networks.

Model and results
We define a random network on an infinite set of layers, each consisting of N ≥ 1 nodes. The node i ∈ {0, . . . , N − 1} in layer h ∈ Z has a fitness F i,h , where we assume the family {F i,h } i∈{0,...,N −1},h∈Z to be independent and identically distributed (i.i.d.). We say that (j, h+1) Then, the number of nodes on layer h + 1 that are visible for the ith node in layer h, which we call the scope of (i, h), is given by Now, (i, h) connects to precisely one visible node (j, h + 1) in layer h + 1, namely the one of maximum fitness. In other words, We illustrate this extremal linkage network in Figure 2. To ease notation, we write (i, h) → (j, h + 1) and think of the directed edges as arrows. In this work, we identify the asymptotic degree and distance distribution as N → ∞.
2.1. Degree distribution. First, we study the typical indegree We assume that the fitnesses are i.i.d. copies of a random variable F with tails for some δ > 0. Here and throughout we write f (s) ≍ g(s) whenever there exist constants c 1 , c 2 > 0 such that c 1 ≤ f (s)/g(s) ≤ c 2 uniform in s. As N → ∞, the typical indegree converges in distribution to a non-degenerate random variable D ∞ with the following tail behavior.
Theorem 1 (Degree distribution). The typical indegree D N converges in distribution to a nondegenerate random variable D ∞ . Moreover,

2.2.
Distances. Since from each node, we draw precisely one arrow to the next layer, the shortest distance between two nodes at layer 0 equals the coalescence time of the corresponding trajectories. We let H N denote the first layer where the trajectories emanating from two random nodes from the initial layer coalesce.
Theorem 2 (Distances). If the fitness distribution is given by (3), then the typical distance H N is almost surely finite for every N ≥ 1. Moreover, (1) if δ = 1, then asymptotically almost surely, Our results show that scale-free and small-world behavior is present if δ < 1. A softer version of small-world behavior (with logarithmic distances) is present in the border case δ = 1. Geometric clustering is incorporated through the network mechanism.
The proof for δ > 2 is based on the central limit theorem, while the case δ ∈ (1, 2) is based on a corresponding stable limit theorem. It is plausible that an analog result is true for the case δ = 2, however, logarithmic corrections might appear. We did not pursue this further, because from a modeling perspective our results are most interesting for δ ≤ 1.

Organization.
We prove the asymptotic degree distribution of Theorem 1 in Section 3. For the distance result in Theorem 2, we give separate proofs for the lower and upper bound in Sections 4 and 5, respectively.

Degrees
As a first step, we describe the limiting distribution D ∞ . This description rests on the observation that, as N → ∞, the N -torus Z/N Z converges locally to Z. More precisely, consider an i.i.d. family {F i,h } (i,h)∈Z×Z of Fréchet random variables with distribution as in (2). Then, we let denote the locations of the first points to the left and to the right of (0, 0) with a higher fitness and put for the number of connections to F 0,0 coming from the left and the right. Then, a coupling argument shows that D N converges in distribution to Equipped with this knowledge, we now establish the tail behavior of D ∞ .
Proof of Theorem 1. We only establish the upper bound because the arguments for the lower bound are similar. Moreover, by symmetry, Then, conditioned on R, also D R,+ ∞ is a Poisson random variable and has parameter λ(R) := If δ > 1, then lim r→∞ λ(r) < ∞, thereby leading to the asserted Poisson tails.
For δ ≤ 1, we leverage the bound where by the Poisson concentration inequality [16, Lemma 1.2], the second summand decays exponentially fast in s. Moreover, the event {R ≥ m} encodes that the fitness at the origin is largest among the fitnesses of the first m nodes. That is, Then, since fitnesses are identically distributed, so that P(R ≥ m) = 1/m. We conclude the proof by noting that log(λ −1 (r)) ≍ r for δ = 1 and log(λ −1 (r)) ≍ log(r) for δ < 1.

Distances -Lower bounds
To prove the lower bounds in Theorem 2, we relate the graph distance H N to the coalescence of two walkers. By symmetry, we may assume one of the randomly chosen nodes in the initial layer to be at position 0. Next, let X L,N h ∈ {0, . . . , N − 1} denote the position after h ≥ 0 steps of a walker starting from 0 and following the arrows. More precisely, we put recursively Additionally, denote the walker started from a uniformly chosen random position in layer 0.
To prove the lower bound of Theorem 2, note that for every h ≥ 0 and η > 0, For δ ≥ 1, we establish highly accurate upper bounds on the growth of the fitnesses F L,N h as a function of the layer h. Consequently, we also obtain bounds on the location after h steps, where the addition of h on the right hand side arises from rounding. Using the same fitnesses attached to the nodes yields a natural coupling between the model on a finite torus and the limit model on the integers Z. We write {F L h } h≥0 for the fitnesses in this limit model. As long as ϕ(F L,N i ) ≤ N , the wrapping around the torus is not observable so that, for η ∈ (0, 1), and only if in the coupled limit model Hence, it suffices to study the limit model.

4.1.
Proof for δ = 1. To bound the right-hand side in (6), we show that the log-fitnesses concentrate sharply around the current layer h.

Lemma 3 (Fluctuations of log-fitnesses). For δ = 1 and G
Before proving Lemma 3, we explain how to derive from it the lower bound on the distances.
Proof of Theorem 2; lower bound; δ = 1. Let I ≥ 0 be an almost surely finite random variable such that, by Lemma 3, G L h − h ≤ h 2/3 holds for all h ≥ I. Then, inserting into (6), The probability on the right hand side vanishes as N → ∞. Further, we fix ε ′ > 0 and let h = log µ (N )(1 − ε ′ ), then the indicator vanishes as well in the limit N → ∞ for arbitrary η > 0. Inserting this into (5) proves the claim.
The key towards obtaining the bounds on the scopes in Lemma 3 is the max-stability of the Fréchet distribution: if F 1 , . . . , F m are i.i.d. Fréchet random variables with tail index δ = 1, then max{F 1 , . . . , F m } has the same distribution as mF , where F is again a Fréchet random variable with tail index 1. Moreover, G i = log µ (2F i ) follows a Gumbel distribution.
In particular, writing m = ϕ(F L h ) (recall (1) for the definition of ϕ) and F = F h+1 , we represent F L h+1 recursively as so that where Starting from this observation, we now prove Lemma 3.
Proof of Lemma 3. In order to develop an intuition for the proof, we first establish the asserted lower bound on the growth. That is, Indeed, applying the bound (8), Moreover, now the lower bound on the growth of G L h implies that the error terms of the form h −2/3 j≤h log µ (ρ h ) in (8) tend to 0 as h → ∞, thereby concluding the proof.

4.2.
Proof for δ < 1. In the heavy-tailed setting, we need a strong tightness property for the log-fitnesses.
Lemma 4 (Tightness for heavy tails). For δ < 1, First, we explain how Lemma 4 enters the proof of the lower bound.
Proof of Theorem 2; lower bound; δ < 1. Let I be a strictly positive random variable such that δ h G L h ≤ I holds almost surely for all h ≥ 0. Hence, writing h = log 1/δ log µ (N )− K and inserting the bound from Lemma 4 into the representation from (6) gives that We conclude the proof by noting that the right-hand side tends to 0 if we first take N sufficiently large and then let K tend to ∞.
It remains to show Lemma 4. The proof mimics the arguments presented in Lemma 3. Therefore, we present in detail only those arguments that are substantially different. The key identity now reads so that, as in (8), where Proof of Lemma 4. First, we iterate (10) to get that for every h 2 ≥ h 1 ≥ 1, Now, the key step is to show that Then, almost surely, Moreover, by the Borel-Cantelli lemma also tends to 0 almost surely as h → ∞. Hence, δ h G L h converges to an almost surely finite limit. It remains to show (12). To achieve this goal, we assert that there exists an almost surely finite random variable I such that Once (13) is established, we obtain that so that taking the limit as h → ∞ concludes the proof.
To prove (13), we may first apply the Borel-Cantelli lemma to see that the sum j≥0 δ j G j+1 converges almost surely. Hence, In particular, the Kolmogorov 0-1-law yields almost surely finite random variable I such that Since G L i ≥ G i for every i ≥ 1, this observation concludes the proof of (13).

4.3.
Proof for δ > 2. In the light-tailed setting, we show that the suitably rescaled walker {X L h } h≥0 satisfies the invariance principle.
Throughout we write ht for ⌊ht⌋. Before establishing Lemma 5, we explain how to conclude the proof of the lower bound. Mind that, for the lower bound, a central limit theorem suffices. However, for the proof of the upper bound in the next section, we need a full functional CLT.
Proof of Theorem 2; lower bound; δ > 2. The invariance principle in the form of Lemma 5 for t = 1 gives as asserted.
In order to prove Lemma 5, we rely on the general martingale functional CLT from [15, Theorem D.6.4]. To cast this problem in the setting of the present context, we let for some constant 0 < γ 2 < ∞. (M2). Almost surely, for every ε > 0, Note that, when fixing any ε 0 > 0, condition (M2) follows from the following Lyapunov-type condition.
(M2'). Almost surely, Before establishing the invariance principle for the random walk {X L h } h≥0 , we first show that the underlying Markov chain of fitnesses {F L h } h≥0 satisfies a Foster-Lyapunov drift condition, thereby forming the basis for a Markov-chain LLN.
Proof of Lemma 5.
w(r) = ϕ(r) 2 − 1 12 denotes the variance of the uniform distribution on ϕ(r) consecutive integers. Then, condition (M1) becomes This is a prototypical Markov-chain LLN that follows from the drift condition (14)

4.4.
Proof for δ ∈ (1, 2). Finally, we deal with the stable case, i.e., 1 < δ < 2. In the lighttailed setting, a key ingredient was the invariance principle in the form of Lemma 5, which stated that the rescaled walker {h −1/2 X L ht } t∈[0,1] converges to Brownian motion as h → ∞. Now, we need a stable analog of this result. More precisely, we establish convergence to a symmetric δ-stable process with Lévy measure for some c(δ) > 0.
Before establishing Lemma 7, we elucidate how it gives the tightness of (H N /N δ ) away from 0. Essentially, this relies on the same line of arguments that we have seen in Section 4.3.
In order to prove Lemma 7, we proceed as in [7] and apply the versatile functional limit theorem [6,Theorem 4.1].
We now formulate a version of [6,Theorem 4.1], where we adapted (actually simplified) the conditions to our needs: Let {Z j,h } 1≤j≤h be a triangular array of centered random variables and let {F j,h } 1≤j≤h be a triangular array of σ-algebras such that Z j,h is F j,h measurable. Now, assume the following conditions: (D1) There exists a symmetric measure ν such that for all x > 0 and t ≤ 1, in probability as h → ∞.
Then, j≤ht Z j,h t≤1 converges in distribution as h → ∞ to a symmetric stable process with Lévy measure ν.
We now use this criterion to prove convergence with the δ-stable Lévy measure ν as in (16). To this end, we set and let We postpone the proof of the lemma and first show how it implies the proof of Lemma 7.
Verification of condition (D1). To verify condition (D1), we need to compute the conditional expectation P(Z h > x | F j−1,h ). Now, similar to the proof for δ > 2, the key insight is that X L j is distributed uniformly in the scope of size ϕ(F L j−1 ). As an initial observation, we note that max j≤h F L j /h ∈ o h (1) with high probability. Indeed, by the Markov inequality, for any β ∈ (1, δ), so that similarly to the arguments in Section 4.3, we may invoke the Markov ergodic theorem, [15,Theorem 14.0.1]. We also recall from (9) Hence, by part (1) of Lemma 8, Finally, as in the computations in Section 4.3, we deduce that there is an LLN, so that 1 h j≤h F L j−2 converges weakly.
Verification of condition (D2). We use (17) to get We observed already before that max j≤h F L j /h ∈ o h (1) with high probability. Hence, max which, together with (D1), implies the desired claim.
Verification of condition (D3). Finally, we show that First, conditioned on the event {|Z j,h | ≤ η}, the increment Z j,h is uniformly distributed in an interval of length η ∧ F L j−1 h −1/δ . Hence, it suffices to show that lim sup tends to 0 as η → ∞. By part (2) of Lemma 8, the conditional expectation inside the probability becomes (2 − δ)hx δ We may once more cite the Markov LLN for the weak convergence of 1 h j≤h F L j−2 to deduce that (18) tends to 0 as η → ∞.
It remains to establish the limits in Lemma 8.

Proof of Lemma 8.
Part (1). Integration with respect to the Fréchet density yields that For large x, the exponential factor approaches 1 and Part (2). We split the expectation depending on which of the two contributions in the minimum becomes relevant. First, as in part (1), Hence, it remains to determine lim a→∞ a 1−2/δ E F 2/δ ½{F ≤ aη δ } .
By l'Hôpital's rule, we see that Combining the latter with the result in (19) concludes the proof.

Distances -Upper bound
For the lower bound on the distances in Section 4, it was sufficient to control the deviation of a single walker. Establishing the upper bound is substantially more involved, as we need to understand the joint movements of the left and the right walker. To lighten notation, we omit the torus size N in the quantities X L,N , X R,N , F L,N and F R,N . 5.1. Proof for δ < 1. We begin by discussing the heavy-tailed setting, as the argument is particularly short. The reason herefore lies in the rapid growth of the fitnesses following from recursion (9). In particular, the upper bound N for the fitnesses becomes absorbing: after reaching it, it remains there for a long period of time.
Lemma 9 (Absorbing upper fitness bound). For δ < 1, Proof. Although the recursion leading to the bound (9) refers to the limiting model where the torus is replaced by the integers, we obtain a finite-volume version by the same arguments together with an additional truncation: Thus, P ϕ(F L h ) ≥ N > ϕ(F L h+1 ) ≤ P(F h+1 < N δ−1 ) = exp − N 1−δ , so that invoking the union bound over h ≤ N concludes the proof.
Equipped with this auxiliary result, we now establish the tightness asserted in Theorem 2.
Proof of Theorem 2; upper bound; δ < 1. First, the walkers coalesce with certainty once the scopes reach N . Hence, by Lemma 9, it suffices to show that Now, as long as ϕ(F L h ) ≤ N , the fitness in the torus model coincides with the one in the infinite limit. Next, by Lemma 4, the fitnesses in the limit model grow as µ Zδ −h for a positive random variable Z. We conclude by noting that lim K→∞ sup N ≥1 P µ Zδ −(log 1/δ log µ (N)+K) < N = lim K→∞ P Zδ −K < 1 = 0.
Lemma 10 (Lower bound on fitnesses). For δ = 1, The second ingredient is a specifically constructed coupling between ( , the latter moving w.r.t. two independent copies of the fitnesses (F i,h ) i,h . We write for the events that the coupling succeeds, respectively fails at level h. Moreover, let denote the σ-algebra of information on the coupled walkers up to level h. Note that if coalescence occurs at level h, whereas F L,i h = F R,i h almost surely, by absolute continuity of the fitnesses. Therefore, The crux of the coupling is that, whenever it fails, the walkers coalesce with probability at least 1/4.

Lemma 11 (Coupling with independent walkers). There is a coupling between the true walkers
Finally, we show that for the independent walkers starting from fitnesses at least as large as N exp(− log µ (N ) 3/4 ), with high probability F L h = F R h = N for some h ≤ log µ (N ) 7/8 . Lemma 12 (Absence of long excursions). For δ = 1 and every ε > 0 there exists N * = N * (ε) such that if N ≥ N * , then holds almost surely on the event Before establishing Lemmas 10, 11 and 12, we complete the proof of Theorem 2.
Proof of Theorem 2, upper bound; δ = 1. By Lemma 10, we may assume that min{F L h , F R h } ≥ N exp(− log µ (N ) 3/4 ) for all τ 0 ≤ h ≤ 2 log µ (N ). Now, if the coupling fails without coalescence at time h ≥ τ 0 , then we restart it by initializing (X L,i h , X R,i h ) at time h + 1 with the values of the true system (X L h , X R h ). Proceeding recursively, this produces stopping times {τ * i } i≥1 encoding the sequence of coupling failures, where we impose that τ * 0 = τ 0 . To ease notation, we let the sequence τ * i be constant after the index i 0 where τ * i 0 = H N , i.e., where coalescence occurs. In particular, Now, fixing ε > 0 and K * with 4 −K * ≤ ε, applying Lemmas 11 and 12 concludes the proof.
Proof of Lemma 10. First, so that we may assume F h ≥ 1/(2 log log(N )) for all h ≤ 2 log µ (N ). By Lemma 3, the event sup h≤τ 0 ϕ(F L h ) ≥ N occurs with high probability. Hence, it remains to show that To that end, we fix h, h ′ ≤ 2 log µ (N ) and apply the union bound afterwards. First, by (7) in companion with (20), On the other hand, if h ′ ≥ log(N ) 2/3 , then using moderate deviations [17,Theorem 11.2] for as asserted.
To construct the coupling, we write  Figure 3. To recover the true process, each of the walkers selects the maximum fitness within its scope. Figure 3. An illustration of the coupling in the proof of Lemma 10.
To construct the coupling, we may w.l.o.g. assume that n L h ≤ n R h . Otherwise, reverse the roles of L and R in the following argument. In order to define the independent process (X L,i h+1 , F L,i h+1 ), (X R,i h+1 , F R,i h+1 ) , we need to remove the dependence coming from the observation that in the true process, both the left and the right walker query the fitnesses F n ∪ To this end, we introduce a further copy of n ∩ h independent uniform fitnesses F n ∪ h +1 , . . . , F n ∪ h +n ∩ h . Then, the left walker selects the maximum fitness among F 1 , . . . , F n L h and the right walker selects the maximum fitness among F n L h +1 , . . . , F n L h +n R h . Hence, under E succ h , we may express the probability of a coupling failure succinctly as the last step relying on the assumption that n L h ≤ n R h . Finally, we establish Lemma 12. This gives domination by a geometric random variable and thereby concludes the proof.

5.3.
Proof for δ ∈ (1, 2) ∪ (2, ∞). For δ > 2 we think of {X L h − X R h } h≥0 as a centered random walk whose step size admits a finite second moment, so that we obtain convergence to Brownian motion just as in Lemma 5. For δ ∈ (1, 2) we proceed similarly, but now with the stable limit law (Lemma 7) rather then the invariance principle of Lemma 5. Since {X L h } h≥0 and {X R h } h≥0 are independent, we conclude from Lemma 7 that also N −1/δ (X L N t − X R N t ) t≤1 converges in distribution to a δ-stable process.
Proof of Theorem 2; upper bound; δ ∈ (1, 2) ∪ (2, ∞). We write T N := {h ≥ 0 : X R h ≤ X L h } for the first time, where the left walker moves past the right one. Note that when the left walker moves past the right one, then necessarily the scopes must intersect and there is a positive probability of coalescence. Thus, it suffices to derive bounds on T N . To that end, note that the distance |X L 0 − X R 0 | of the initial locations is at most N . Now, we write T for the first time that Brownian motion, respectively a symmetric δ-stable process exceeds 1, so that lim sup N →∞ P(T N ≥ KN 2∧δ ) ≤ P(T ≥ K).
Finally, we leverage that hitting times are almost surely finite, which is true not only for Brownian motion but also for the symmetric δ-stable process because it is recurrent when δ ∈ (1, 2), see [3,Theorem I.17]. Therefore, the right-hand side tends to 0 as K → ∞.