Assortativity and clustering of sparse random intersection graphs

We consider sparse random intersection graphs with the property that the clustering coefficient does not vanish as the number of nodes tends to infinity. We find explicit asymptotic expressions for the correlation coefficient of degrees of adjacent nodes (called the assortativity coefficient), the expected number of common neighbours of adjacent nodes, and the expected degree of a neighbour of a node of a given degree k. These expressions are written in terms of the asymptotic degree distribution and, alternatively, in terms of the parameters defining the underlying random graph model.


Introduction
Assortativity and clustering coefficients are commonly used characteristics describing statistical dependency of adjacency relations in real networks ( [18], [2], [20]). The assortativity coefficient of a simple graph is the Pearson correlation coefficient between degrees of the endpoints of a randomly chosen edge. The clustering coefficient is the conditional probability that three randomly chosen vertices make up a triangle, given that the first two are neighbours of the third one. It is known that many real networks have non-negligible assortativity and clustering coefficients, and a social network typically has a positive assortativity coefficient ( [18], [21]). Furthermore, Newman et al. [21] remark that the clustering property (the property that the clustering coefficient attains a non-negligible value) of some social networks could be explained by the presence of a bipartite graph structure. For example, in the actor network two actors are adjacent whenever they have acted in the same film. Similarly, in the collaboration network authors are declared adjacent whenever they have coauthored a paper. These networks exploit the underlying bipartite graph structure: actors are linked to films, and authors to papers. Such networks are sometimes called affiliation networks. In this paper we study assortativity coefficient and its relation to the clustering coefficient in a theoretical model of an affiliation network, the so called random intersection graph. In a random intersection graph nodes are prescribed attributes and two nodes are declared adjacent whenever they share a certain number of attributes ( [11], [15], see also [1], [13]). An attractive property of random intersection graphs is that they include power law degree distributions and have tunable clustering coefficient see [5], [6], [8], [12]. In the present paper we show that the assortativity coefficient of a random intersection graph is non-negative. It is positive in the case where the vertex degree distribution has a finite third moment and the clustering coefficient is positive. In this case we show explicit asymptotic expressions for the assortativity coefficient in terms of moments of the degree distribution as well as in terms of the parameters defining the random graph. Furthermore, we evaluate the average degree of a neighbour of a vertex of degree k, k = 1, 2, . . . , (called neighbour connectivity, see [16], [23]), and express it in terms of a related clustering characteristic, see (3) below. Let us rigorously define the network characteristics studied in this paper. Let G = (V, E) be a finite graph on the vertex set V and with the edge set E. The number of neighbours of a vertex v is denoted d(v). The number of common neighbours of vertices v i and v j is denoted d(v i , v j ). We are interested in the correlation between degrees d(v i ) and d(v j ) and the average value of d(v i , v j ) for adjacent pairs v i ∼ v j (here and below '∼' denotes the adjacency relation of G). We are also interested in the average values of d(v i ) and d(v i , v j ) under the additional condition that the vertex v j has degree d(v j ) = k. In order to rigorously define the averaging operation we introduce the random pair of vertices (v * 1 , v * 2 ) drawn uniformly at random from the set of ordered pairs of distinct vertices. By we denote the average value of measurements f (v i , v j ) evaluated at each ordered pair (v i , v j ), i = j. Here N = |V| denotes the total number of vertices. By we denote the average value over ordered pairs of adjacent vertices. Here p e * = P(v * 1 ∼ v * 2 ) denotes the edge probability and I {v i ∼v j } = 1, for v i ∼ v j , and 0 otherwise. Furthermore, =k} , denotes the average value over ordered pairs of adjacent vertices, where the second vertex is of degree k.
The average values of d(v i )d(v j ) and d(v i , v j ) on adjacent pairs v i ∼ v j are now defined as follows We also define the average values and the correlation coefficient called the assortativity coefficient of G, see [18], [19].
In the present paper we assume that our graph is an instance of a random graph. We consider two random intersection graph models: active intersection graph and passive intersection graph introduced in [10] (we refer to Sections 2 and 3 below for a detailed description). Let G denote an instance of a random intersection graph on N vertices. Here and below the number of vertices is non random. An argument bearing on the law of large numbers suggests that, for large N , we may approximate the characteristics b(G), b k (G), h(G) and h k (G) defined for a given instance G, by the corresponding conditional expectations where now the expected values are taken with respect to the random instance G and the random pair . The main results of this paper are explicit asymptotic expressions as N → +∞ for the correlation coefficient r, the neighbour connectivity b k , and expected number of common neighbours h k defined in (1). As a corollary we obtain that the random intersection graphs have tunable assortativity coefficient r ≥ 0. Another interesting property is expressed by the identity as N → +∞ (2) saying that the average value of the difference d( That is, a neigbour v j of v i may affect the average degree d(v i ) only by increasing/decreasing the average number of common measures the probability of an edge between two neighbours of a vertex of degree k. In particular, we have The remaining part of the paper is organized as follows. In Section 2 we introduce the active random graph and present results for this model. The passive model is considered in Section 3. Section 4 contains proofs.

Active intersection graph
Let s > 0. Vertices v 1 , . . . , v n of an active intersection graph are represented by subsets D 1 , . . . , D n of a given ground set W = {w 1 , . . . , w m }. Elements of W are called attributes or keys. Vertices v i and v j are declared adjacent if they share at least s common attributes, i.e., we have |D i ∩ D j | ≥ s.
In the active random intersection graph G s (n, m, P ) every vertex v i ∈ V = {v 1 , . . . , v n } selects its attribute set D i independently at random ( [11]) and all attributes have equal chances to belong to D i , for each i = 1, . . . , n. We assume, in addition, that independent random sets D 1 , . . . , D n have the same probability distribution. Then, we have for each A ⊂ W , where P is the common probability distribution of the sizes X i = |D i |, 1 ≤ i ≤ n of selected sets. We remark that X i , 1 ≤ i ≤ n are independent random variables. We are interested in the asymptotics of the assortativity coefficient r and moments (1) in the case where G s (n, m, P ) is sparse and n, m are large. We address this question by considering a sequence of random graphs {G s (n, m, P )} n , where the integer s is fixed and where m = m n and P = P n depend on n. We remark that subsets of W of size s plays a special role, we call them joints: two vertices are adjacent if their attribute sets share at least one joint. Our conditions on P are formulated in terms of the number of joints X i s available to the typical vertex v i . We denote a k = E X 1 s k . It is convenient to assume that as n → ∞ the rescaled number of joints Z 1 = m s −1/2 n 1/2 X 1 s converges in distribution. We also introduce the k-th moment condition (i) Z 1 converges in distribution to some random variable Z; (ii-k) 0 < EZ k < ∞ and lim n→∞ EZ k 1 = EZ k . We remark that the distribution of Z, denoted P Z , determines the asymptotic degree distribution of the sequence {G s (n, m, P )} n (see [5], [6], [8], [25]). We have, under conditions (i), (ii-1) that Here we denote z k = EZ k . Let d * be a random variable with the probability distribution P(d * = k) = p k , k = 0, 1, . . . . We call d * the asymptotic degree. It follows from (5) that the asymptotic degree distribution is a Poisson mixture, i.e., the Poisson distribution with a random (intensity) parameter z 1 Z. For example, in the case where P Z is degenerate, i.e., P(Z = z 1 ) = 1, we obtain the Poisson asymptotic degree distribution. Furthermore, the asymptotic degree has a power law when P Z does. We denote Another important characteristic of the sequence {G s (n, m, P )} n is the asymptotic ratio β = lim m→∞ m s /n. Together with P Z it determines the first order asymptotics of the clustering [6], [8]. Under conditions (i), (ii-2), and m s n −1 → β ∈ (0, +∞) we have Furthermore, we have α = o(1) in the case where m s n −1 → +∞. We remark that α = o(1) also in the case where the second moment condition (ii-2) fails and we have EZ 2 = +∞, see [6]. To summarize, the clustering coefficient α does not vanish as n, m → ∞ whenever the asymptotic degree distribution (equivalently P Z ) has finite second moment and 0 < β < ∞. Our Theorem 1, see also Remark 1, establishes similar properties of the assortativity coefficient r: it remains bounded away from zero whenever the asymptotic degree distribution (equivalently P Z ) has finite third moment and 0 < β < ∞. Theorem 1. Let s > 0 be an integer. Let m, n → ∞. Assume that (i) and (7) are satisfied. In the case where (ii-3) holds we have In the case where (ii-2) holds and EZ 3 = ∞ we have r = o(1).
We note that the inequality a 1 a 3 ≥ a 2 2 , which follows from Hölder's inequality, implies that the ratio in the right hand side of (9) is positive. Remark 1. In the case where (i), (ii-2) hold and m s n −1 → +∞ we have r = o(1). Our next result Theorem 2 shows a first order asymptotics of the neighbour connectivity b k and the expected number of common neighbours h k . Theorem 2. Let s ≥ 1 and k ≥ 0 be integers. Let m, n → ∞. Assume that (i), (ii-2) and (7) and Here a 1 = (βδ 1 ) 1/2 + o(1) and a 2 = βδ 2 /δ 1 + o(1).
We remark that the distribution of the random graph G s (n, m, P ) is invariant under permutation of its vertices (we refer to this property as the symmetry property in what follows). Therefore, . In particular, the increment b k+1 − b shows how the degree of v 2 affects the average degree of its neighbour v 1 . By (11), (13) In Examples 1 and 2 below we evaluate this quantity for a power law asymptotic degree distribution and the Poisson asymptotic degree distribution.
Example 1. Assume that the asymptotic degree distribution has a power law, i.e., for some c > 0 and γ > 3 we have p k = (c + o(1))k −γ as k → +∞. Then Hence, for large k, we obtain as n, Example 2. Assume that the asymptotic degree distribution is Poisson with mean λ > 0, i.e., and, for large k, we obtain as n, m → +∞ that Our interpretation of (14) is as follows. We assume, for simplicity, that s = 1. We say that an attribute w ∈ W realises the link v i ∼ v j , whenever w ∈ D i ∩ D j . We note that in a sparse intersection graph G 1 (n, m, P ) each link is realised by a single attribute with a high probability. We also remark that in the case of the Poisson asymptotic degree distribution, the sizes of the random sets, defining intersection graph, are strongly concentrated about their mean value a 1 . Now, by the symmetry property, every element of the attribute set D 2 of vertex v 2 realises about k/|D 2 | ≈ k/a 1 links to some neighbours of v 2 other than v 1 . In particular, the attribute responsible for the link v 1 ∼ v 2 attracts to v 1 some k/a 1 neighbours of v 2 . Hence, b k+1 − b ≈ a −1 1 k ≈ (βλ) −1/2 k. Finally, we remark that (11), (12), and (13) imply (2).

Passive intersection graph
A collection D 1 , . . . , D n of subsets of a finite set W = {w 1 , . . . , w m } defines the passive adjacency relation between elements of W : w i and w j are declared adjacent if w i , w j ∈ D k for some D k . In this way we obtain a graph on the vertex set W , which we call the passive intersection graph, see [11]. We assume that D 1 , D 2 , . . . , D n are independent random subsets of W having the same probability distribution (4). In particular, their sizes X i = |D i |, 1 ≤ i ≤ n are independent random variables with the common distribution P . The passive random intersection graph defined by the collection D 1 , . . . , D n is denoted G * 1 (n, m, P ). We shall consider a sequence of passive graphs {G * 1 (n, m, P )} n , where P = P n and m = m n depend on n = 1, 2, . . . . We remark that, in the case where β n = mn −1 is bounded and it is bounded away from zero as n, m → +∞, the vertex degree distribution can be approximated by a compound Poisson distribution ( [6], [14]). More precisely, assuming that β n → β ∈ (0, +∞); (iii) X 1 converges in distribution to a random variable Z; (iv) EZ 4/3 < ∞ and lim m→∞ EX it is shown in [6] that d(w 1 ) converges in distribution to the compound Poisson random variable d * * := Λ j=1Z j . HereZ 1 ,Z 2 ,. . . are independent random variables with the distribution in the case where EZ > 0. In the case where EZ = 0 we put P(Z 1 = 0) = 1. The random variable Λ is independent of the sequenceZ 1 ,Z 2 ,. . . and has Poisson distribution with mean EΛ = β −1 EZ. We note that the asymptotic degree d * * has a power law whenever Z has a power law. Furthermore, we have Ed i * * < ∞ ⇔ EZ i+1 < ∞, i = 1, 2, . . . . In Theorems 3, 4 below we express the moments b, h, b k , h k and the assortativity coefficient In the case where β n → +∞ we have r = 1 − o(1). In the case where β n → 0 and nβ 3 n → +∞ we have r = o(1).
Our last result Theorem 4 shows a first order asymptotics of the neighbour connectivity b k and the expected number of common neighbours h k in the passive random intersection graph.

Proofs
Proofs for active and passive graphs are given in Section 4.1 and Section 4.2 respectively. We note that the probability distributions of G s (n, m, P ) and G * 1 (n, m, P ) are invariant under permutations of the vertex sets. Therefore, for either of these models we have Here ω 1 = ω 2 are arbitrary fixed vertices and E 12 denotes the conditional expectation given the event ω 1 ∼ ω 2 . In the proofP andẼ (respectively,P * andẼ * ) denote the conditional probability and expectation given X 1 , . . . , X n (respectively, D 1 , D 2 , X 1 , . . . , X n ). Limits are taken as n and m = m n tend to infinity. We use the shorthand notation f k (λ) = e −λ λ k /k! for the Poisson probability.

Active graph
Before the proof we introduce some more notation. Then we state and prove auxiliary lemmas. Afterwards we prove Theorem 1, Remark 1 and Theorem 2.
The conditional expectation given D 1 , D 2 is denoted E * . The conditional expectation given the and introduce events Observe that E ij is the event that v i and v j are adjacent in G s (n, m, P ). We denote We remark that the distributions of The following inequality is referred to as LeCam's lemma, see e.g., [26].
Lemma 1. Let S = I 1 + I 2 + · · · + I n be the sum of independent random indicators with probabilities P(I i = 1) = p i . Let Λ be Poisson random variable with mean p 1 + · · · + p n . The total variation distance between the distributions P S and P Λ of S and Λ sup A⊂{0,1,2... }

|P(S ∈
. . , m} such that D 1 (respectively D 2 ) is uniformly distributed in the class of subsets of W of size k 1 (respectively k 2 ). The probabilities p ′ := P(|D 1 ∩ D 2 | = s) and Here we denote p * For any 0 ≤ u ≤ 3 and any sequence A n → +∞ as n → ∞ we have Proof of Lemma 3. The uniform integrability property (26) of the sequence {Z 3 n1 } n is a simple consequence of (i) and (ii-3), see, e.g., Remark 1 in [5]. The first and second identity of (27) follows from (ii-3) and (26) respectively. Finally, (28) follows from (26) and (27).
We recall that Y i and δ ij are defined in (23).
Proof of Lemma 4. The right hand side of (29), (30) and inequality (31) are immediate consequences of (25). In order to show the left hand side inequality of (29) and (30) we apply the left hand side inequality of (25). We only prove (29). We have, see (23), In order to show (32) we apply the right-hand side inequality of (25) and writẽ Invoking the inequalities Xt s+1 m s+1 2 ) we obtain (32).
Lemma 5. Assume that conditions of Theorem 2 are satisfied. Let k ≥ 0 be an integer. For We have, by Lemma 1, Lemma 6. Let m, n → ∞. Assume (i), (ii-3) and (7) hold. Then Proof of Lemma 6. Proof of (37). In order to prove (37) we write and invoke the identities Note that (43) follows from (30) and (28). Let us prove (42). To this aim we write , and show that Let us prove (44). Assuming that E 12 holds we can write d ′ i = n t=3 I E it , i = 1, 2, and To show the first identity of (44) we write Eκ 1 = EI E ′ 12 S 1 + EI E ′ 12 S 2 =: I 1 + I 2 and evaluate We first evaluate I 1 . Given t ≥ 3, consider events Assuming that E ′ 12 holds we have that ¿From (48) and (32) we obtain, by the symmetry property, where ). Next, we evaluateP(E ′ 12 ) and P(E ′ 12 ) = EP(E ′ 12 ) using (29): Combining these relations with (49) we obtain the first relation of (46). Let us we evaluate I 2 . We writẽ and apply (29) to each probability in the right-hand side. We obtaiñ (1), see (28). Now, by the symmetry property, we obtain from (51) the second relation of (46) To prove the second bound of (44) we write, see (45),κ 2 = I E ′′ 12 (S 1 + S 2 ) and show that Here x 2s+1 , x s+1 , x s = O(1), by (27). Let us prove (52). We have, see (29), Furthermore, by the symmetry property and (31), we obtain Since the expected value in the right hand side does not exceed x 2s+1 x s+1 x s , we obtain the first bound of (52). In order to prove the second bound we write, cf. (50), In the last step we used (29) and (31). Now, by the symmetry property, we obtain Proof of (38). We write, by the symmetry property, and evaluate using (29), (30) Invoking this relation and (43) in (54) we obtain (38).

Furthermore, (85) follows from the bounds EI
. We show these bound using (29). For 1 ≤ j ≤ 3 the proof is obvious. For j = 4 we need to show that . For this purpose we write (using the inequality I 1 Y 1 ≤ I 1 m s/4 ) and note that the expected values of both summands in the right hand side tend to zero as n → +∞. Finally, (86) follows from (29) and implies directly (87). Now we derive (81) from (87). We observe that (here we use the fact that the weak convergence of distributions (i) implies the convergence of expectations of smooth functions). Furthermore, by (5), Ef k (z 1 Z) = p k . Hence, (87) implies Proof of (80).
We obtain (80) in several steps. We show that We note that (88) is obtained by replacing I E 23 by the product I E ′ 23 I C in the formula defining τ 3 . In order to bound the error of this replacement we apply the inequality and invoke the bound , see the proof of (84) above. We remark that the left hand side inequality of (93) is obvious. The right hand side inequality holds because the event E 23 implies (E ′ 23 ∩ C) ∪ B 3 . In (89) we replacep by H. To prove (89) we show that We remark that the first and third relations follow from the simple bounds, see (29), (30), . In order to show the second relation of (94) we split and observe thatp 1 is the probability that the random subset D 3 ∩ D 2 (of size s) of D 2 does not match the subset D 1 ∩ D 2 (we note that |D 1 ∩ D 2 | = s, since the event E ′ 12 holds). Hence, Finally,p 2 is the probability that the random subset D 3 of W \ (D 1 \ D 2 ) intersects with D 2 in exactly s elements. Taking into account that the event E ′ 12 holds we obtain (see (29), (33)) Here we denotem 1 := m ′ s and m ′ = |W \ (D 1 \ D 2 )| = m − (X 1 − s). We remark that on the event {X 1 < m 1/4 } we have m ′ = m − O(m 3/4 ). Hence, for large m, (97) implies Now, collecting (96), (98), and the identityp 1 = 1 − Y −1 2 in (95) we obtain the inequalities that imply the second relation of (94).

Passive graph
Before the proof we introduce some more notation. Then we present auxiliary lemmas. Afterwards we prove Theorems 3, 4. By E ij we denote the conditional expectation given the event E ij = {w i ∼ w j }. Furthermore, we denote For w ∈ W , we denote I i (w) = I {w∈D i } and I i (w) = 1 − I i (w), and introduce random variables We say that two vertices w i , w j ∈ W are linked by D k if w i , w j ∈ D k . In particular, a set D k defines X k 2 links between its elements. We note that L t = L(w t ) counts the number of links incident to w t . Similarly, Q t = Q(w t ) counts the number of different parallel links incident to w t (a parallel link between w ′ and w ′′ is realized by a pair of sets D i , D j such that w ′ , w ′′ ∈ D i ∩D j ). Furthermore, S 1 counts the number of links connecting w 1 and w 2 and S 2 counts the number of different pairs of links connecting w 1 and w 2 . We denote the degree d t = d(w t ) and introduce Lemma 7. The factorial moments δ * i = E(d * * ) i and u i = E(Z) i satisfy the identities Proof of Lemma 7. We only show the third identity of (100). The proof of the first and second identities is similar, but simpler. We color z = z 1 + · · · + z r distinct balls using r different colors so that z i balls receive i-th color.
Here the first sum counts triples of the same color, the second sum counts triples having two different colors, etc. We apply (101) to the random variable d * * 3 , where d * * =Z 1 + · · · +Z Λ . We obtain, by the symmetry property,

Now invoking the simple identities
we obtain the third identity of (100).
and invoke the expressions Now the identities of Lemma 8 complete the proof of (15). Let us prove (119). We first write, by the inclusion-exclusion, Then we derive from (121) the inequalities which, in combination with (120) and (121), imply the inequalities Finally, invoking the upper bounds for the expected values of the quantities in the right hand sides of (123) shown in Lemma 9, we obtain (119). Now we derive (16) from (15). Firstly, using the fact that (iii), (v) imply the convergence of moments E(X 1 ) i → E(Z) i , for i = 2, 3, 4, we replace the moments y i by u i = E(Z) i in (15). Secondly, we replace u i by their expressions via δ * i . For this purpose we solve for u 2 , u 3 , u 4 from (100) and invoke the identities For β n → +∞ relation (15) remains valid and it implies r = 1 + o(1). For β n → 0 the condition nβ 3 n → +∞ on the rate of decay of β n ensures that the remainder terms of (119) and Lemma 8 are negligibly small. In particular, we derive (15) using the same argument as above. Letting β n → 0 in (15) we obtain the bound r = o(1).
Proof of Theorem 4. Before the proof we introduce some notation. We denote Given w i , w j ∈ W we write d ij = d(w i , w j ). A common neighbour w of w i and w j is called black if {w, w i , w j } ⊂ D r for some 1 ≤ r ≤ n, otherwise it is called red. Let d ′ ij and d ′′ ij denote the numbers of black and red common neighbours, so that d ′ ij + d ′′ ij = d ij . Let w * be a vertex drawn uniformly at random from the set W ′ = W \ {w 1 }. By d ′ 1 * we denote the number of black common neighbours of w 1 and w * . By E 1 * we denote the event {w 1 ∼ w * }. We assume that w * is independent of the collection of random sets D 1 . . . , D n defining the adjacency relation of our graph.
In the proof we use the identity, which follows from (102), (119), We also use the identities, which follow from (100) and (124) We remark that (126) in combination with relations y i → u i as n, m → +∞, imply the right hand side relations of (18), (19) and (21). Now we prove the left hand side relations of (18), (19) and (21), and the relation (20). In order to show (18) we write b = p −1 e Ed 1 I E 12 and invoke identities (119), (103) and (125). Proof of (19). We write h = p −1 e Ed 12 I E 12 and evaluate Combining (125) with (127) we obtain (19). Let us show (127). Using the identity we write where R 1 = Ed ′′ 12 I E 12 and R 2 = EI L 1 d ′ 12 I E 12 . Next, we observe that EI L 1 d ′ 12 I E 12 = EI L 1 d ′ 1j I E 1j , for 2 ≤ j ≤ n, and write We explain the second identity of (130). We observe that H(m − 1) −1 is the conditional expectation of d ′ 1 * I E 1 * given D 1 , . . . , D n . Indeed, any pair of sets D i , D j containing w 1 intersects in the single point w 1 , since the event L 1 holds. Consequently, each D i containing w 1 produces X i − 2 black common neighbours provided that w * hits D i . Since the probability that w * hits D i equals (X i − 1)/(m − 1), the set D i contributes (on average) (m − 1) −1 I i (w 1 )(X i − 1) 2 black vertices to d ′ 1 * . Now, by the symmetry property, we write the right-hand side of (130) in the form where, R 3 = n m−1 EI L 1 I 1 (w 1 )(X 1 − 1) 2 . Finally, we observe that (127) follows from (129), (130), (131) and the bounds R i = O(n −2 ), i = 1, 2, 3, which are proved below. In order to bound R i , i = 1, 2, we use the inequalities and write R 2 ≤ EQ 1 L 1 S 1 and R 3 ≤ n(m − 1) −1 EQ 1 I 1 (w 1 )(X 1 − 1) 2 . Then we apply (111) and (114). In order to bound R 1 we observe, that the number of red common neighbours of w 1 , w 2 produced by the pair of sets D i , D j is a ij = I i (w 1 )I j (w 2 )I j (w 1 )I i (w 2 ) + I j (w 1 )I i (w 2 )I i (w 1 )I j (w 2 ) X ij .

and inequalities
Ea 23 = 2EI 2 (w 1 )I 3 (w 2 )I 3 (w 1 )I 2 (w 2 )X 23 ≤ 2 X 2 m Proof of (20). In the proof we use the fact that the random vector (H, L 1 ) converges in distribution to (d 2 * , d * * ) as n → +∞. We recall that H is described after (130). The proof of this fact is similar to that of the convergence in distribution of L 1 = 1≤i≤n I i (w 1 )(X i − 1) to the random variable d * * , see Theorems 5 and 7 of [6]. We note that the convergence in distribution of (H, L 1 ) implies the convergence in distribution of HI {L 1 =k} to d 2 * I {d * * =k} . Furthermore, since under condition (v) the first moment EH is uniformly bounded as n → +∞ and Ed 2 * < ∞, we obtain the convergence of moments In order to prove (20) we write h k = E(d 12 |w 1 ∼ w 2 , d 1 = k) = p −1 ke Ed 12 I E 12 I {d 1 =k} and show that We remark that (134) in combination with (135) and (136) implies (20). Let us show (135). In view of the identities p ke = P(w i ∼ w 1 , d 1 = k), 2 ≤ i ≤ n, we can write p ke = P(w * ∼ w 1 , d 1 = k) = P(w * ∼ w 1 |d 1 = k)P(d 1 = k).
It remains to prove (140). In the proof we use the shorthand notation Let us prove (140). Using the identity 1 = I L 1 + I L 1 we write τ = Ed 12 I E 12 I {d 2 =k} I L 1 + R 4 , R 4 = Ed 12 I E 12 I {d 2 =k} I L 1 .