key: cord-0775646-wq5okxcr
authors: Hüsler, J.; Temido, M. G.; Valente-Freitas, A.
title: On the Maximum of a Bivariate INMA Model with Integer Innovations
date: 2022-02-15
journal: Methodol Comput Appl Probab
DOI: 10.1007/s11009-021-09920-3
sha: efa8eaecc4a88cb06f67daa16bcaae1bfcbd8296
doc_id: 775646
cord_uid: wq5okxcr

We study the limiting behaviour of the maximum of a bivariate (finite or infinite) moving average model, based on discrete random variables. We assume that the bivariate distribution of the iid innovations belong to the Anderson’s class (Anderson, 1970). The innovations have an impact on the random variables of the INMA model by binomial thinning. We show that the limiting distribution of the bivariate maximum is also of Anderson’s class, and that the components of the bivariate maximum are asymptotically independent.

studied the limiting distribution of the maximum term M n = max(X 1 , ⋯ , X n ) of stationary sequences {X j } defined by non-negative integer-valued moving average (INMA) sequences of the form

where the innovation sequence {V i } is an iid sequence of non-negative integer-valued random variables (rvs) with exponential type tails of the form where ∈ ℝ, > 0, L(n) is slowly varying at +∞ and i • denotes binomial thinning with probability i ∈ [0, 1] . Hall (2003) proved that {X j } satisfies Leadbetter's conditions D(x + b n ) and D � (x + b n ) , for a suitable real sequence b n , and then for all real x and max ∶= max{ i , i ∈ ℤ} . Note that max plays an important role in this result. This is an extension of Theorem 2 of Anderson (1970) , where it is proved that for sequences of iid rvs with an integer-valued distribution function (df) F with infinite right endpoint, the limit is equivalent to for all real x.

The class of dfs satisfying (1), which is a particular case of (2) (see, e.g., Hall and Temido (2007) ) is called Anderson's class.

In this paper we extend the result of Hall (2003) for the bivariate case of an INMA model. Concretely, we study the limiting distribution of the maximum term of stationary sequences {(X j , Y j )} where the two marginals are defined by non-negative integer-valued moving average sequences of the general form where X j and Y j are defined as above with respect to a two-dimensional iid innovation sequence {V i , W i } . The binomial thinning operator • , due to Steutel and van Harn (1979) , is defined by •Z = ∑ Z s=1 B s ( ), ∈ [0, 1], where {B s ( )} is an iid sequence of Bernoulli rvs independent of the positive integer rv Z. The possible class of bivariate discrete distributions F V,W (see (4)) includes also the bivariate geometric models.

We assume that X = •V and Y = •W are conditionally independent given (V, W), because the binomial thinning with • and • are independent, X and Y are binomial rv's. with parameters (V, ) respectively (W, ) , i.e. for all events A and B and for all possible values of v and w. We assume that i , i ∈ [0, 1] and (1) 1 − F V (n) ∼ n L(n)(1 + ) −n , n → +∞, lim sup n→+∞ P(M n ≤ x + b n ) ≤ exp(−(1 + ∕ max ) −x ) lim inf n→+∞ P(M n ≤ x + b n ) ≥ exp(−(1 + ∕ max ) −(x−1) ),

(3) i , i = O |i| − , |i| → +∞, 1 , 1 + 2 } and > 1 − 1 2 , some real constants i and slowly varying functions L i , i = 1, 2, 3, 4 , and where (v, w) is a positive bounded (say by ) function which converges to a positive constant L as v, w → ∞ . That (v, w) converges to L is for simplicity. It has no impact on the results if the limit L would depend on v < w, v = w or v > w . By [x] we denote the greatest integer not greater than x.

The marginal tails of F V,W are of the form:

for v, w → +∞ . Hence, both marginal dfs belong to the Anderson's class with From (4), we can derive the probability function (pf) of (V, W). Because the proofs of the following propositions are technical, we move them to Appendix Proofs. Example 2.1 The Bivariate Geometric (BG) distribution is a particular case of the model (4) with margins (5). Consider the bivariate Bernoulli random vector (B 1 , B 2 ) with P(B 1 = k, B 2 = ) = p k , (k, ) ∈ {0, 1} 2 , and success marginal probabilities p +1 = p 01 + p 11 and p 1+ = p 10 + p 11 . Due to Mitov and Nadarajah (2005) , using the construction of a BG, the pf and the df of a random vector (V, W) with BG distribution are given, respectively, by for v, w ∈ ℕ 0 , and for v, w ∈ ℝ + 0 , assuming that 0 < p 0+ , p +0 < 1 . Hence, this df satisfies (4) with the constants 1 , 2 given by

and the index associated to the dependence structure of (B 1 , B 2 ) is

The slowly varying functions are constants and i = 0 , for i = 1, 2, 3, 4 . The independence case occurs when = 1 . For dependence cases, we can have 0 < < 1 or > 1 . (6).

The marginal df of V and W are obviously which means V and W are geometrically distributed rvs with parameter p 1+ and p +1 , respectively.

In order to characterize the df of (X, Y) = ( •V, •W) we start by establishing the relationship between the probability generating function (pgf) of (V, W) and (X, Y), defined e.g. for (V, W) as which exists for (s 1 , s 2 ) in the following region R (given in Lemma 2.1).

Taking into account Proposition 2.1, the series G V,W (s 1 , s 2 ) converges obviously for any s i ≤ 1 . Even for some s i > 1 the series converges because of the assumption (4). By this assumption, we have E(s V 1 ) < +∞ if s 1 < 1 + 1 and E(s W 2 ) < +∞ if s 2 < 1 + 2 . The following lemma gives a condition such that the series G V,W (s 1 , s 2 ) exists.

Its more technical proof is given also in the appendix. As consequence of this lemma, the pgf G V,W (s 1 , s 2 ) exists for s 1 , s 2 > 1 , if s i ≤ 1 + i , i = 1, 2 in case ≤ 1 , and if s 1 ≤ 1 + 1 and s 2 ≤ 1 + 2 in case of > 1 . In the following, we use these convenient conditions for the convergence of G V,W . Now the relationship of the two pgf is the following. It holds as long as the pgf's exist. For our derivations it is convenient to use in the following the given domain R . The proof of this relationship is also given in the appendix.

The pgf of (X, Y) = ( •V, •W) is given in terms of the pgf of (V, W):

for all (s 1 , s 2 ) such that ( s 1 + 1 − , s 2 + 1 − ) ∈ R.

We want to derive an exact relationship of the two distributions F V,W and F X,Y with the help of a suitable transformation, as a modified pgf or a Mellin transform. We define the (bivariate) modified pgf or tail generating function (Sagitov (2017) 

□ and analogously for X, Y. The relationship between Q V,W and G V,W is given in the following proposition. 

and the bound of (v, w).

Note that 1 < 2 < 2 . We observe that the stationary bivariate INMA model X j , Y j introduced in our work is an extension of the BINAR model of Pedeli and Karlis (2011) defined by with an iid innovations sequence {(R 1j , R 2j )} . In their paper it is stated that it has also the representation

We consider now the stationary bivariate (4). We establish first the tail behaviour of X j , Y j . The maximal values of i and i are most important as in the univariate case. Therefore we write max = max{ i ∶ |i| ≥ 0} and max = max{ i ∶ |i| ≥ 0} . We assume that they are unique. It may happen in the bivariate case that max and max occurs at the same index or at different ones. We consider both cases. Furthermore, we use that which holds because of (3). Suppose first that max and max are occuring at different indexes i 0 and i 1 , respectively. We write for any j and

and Y j = T 1 + T 2 + T 3 = T 1 + T . Note that S, S i , T and T i depend on j.

For the proof of the main proposition of this section we need the following lemma.

E(1 + h) V = 1 + hE(V)(1 + o h (1)), as h → 0 + . b

The proof of this lemma is given in the appendix. We deal now with the limiting behaviour of the tail of (X j , Y j ) . Besides of the univariate tail distributions we derive only an appropriate positive upper bound H * (x, y) for the joint tail which is sufficient for the asymptotic limit distribution of the maxima. We will see that we get asymptotic independence of the components of the bivariate maxima (M (1) n , M (2) n ) , since this normalized H * (x, y) is vanishing, not contributing to the limit.

For the asymptotic behaviour of the tail of the stationary distribution of the sequence {(X j , Y j )} , we write simply (X, Y) for any (X j , Y j ) . As mentioned we deal with the two cases that max and max are occurring at different indexes or at the same one. We start with the first case and the above defined S, S i , T, T i .

For this derivation, we use , ∈ (0, 1) and > 0 such that 1 max < < 1 * , with * = max{ i , i ≠ i 0 } , and 2 given in (9), and Proposition 3.1 If (V, W) satisfies (4) and max and max are unique and taken at different indexes, then (i) for the marginal dfs and (ii) for the joint df with , , satisfying (12) and (13)

We deal with the three terms in (16), separately.

For the first sum of (17), we get by applying Proposition 2.6 with = max for the marginal distribution

by dominated convergence. For the second sum in (17), we get for x large using the Markov inequality, since E(1 + ) S is finite for < 1 ∕ * . Since (1 + ) > 1 + 1 max , we get by Theorem 4 of Hall (2003) and thus together

With the same arguments we characterize the tail 1 − F Y . Hence, the statements on the marginal dfs are shown.

(ii) Now we deal with the third term in (16). Note that (S 1 , T 2 ) , (S 2 , T 1 ) and (S 3 , T 3 ) in the representation of X and Y are independent. For any ∈ (0, 1) and > 0 satisfying (12), we use that (18) and (19) imply and The probability in the third term of (16) is split into four summands with < 1 satisfying (12), x = [ x] and y = [y − (log y) 2 ] . We get for x and y large, (20). For the first sum S 1 ( x , y ) of (22) we use Proposition 2.6 and obtain with < 1 such that (13) holds,

Hence the sum is bounded above by since the last pgf exists due to Lemma 3.1 and (13) Note that

The expectations exist by assumption (4) since 1 + i 1 max < 1 + 1 , and also (13), by using the arguments of Lemma 3.1. We consider now the approximation of the second sum S 2 ( x , y ) in (22). We have with some positive constant C By the arguments used to approximate P(X > x) = P(S 1 + S 2 + S 3 > x) in (i), we also obtain with some generic constant C. Hence, it implies together with (23) For the third sum S 3 ( x , y ) in (22), we get analogously to the derivation of the second sum

Combining now the bounds of the four terms S i ( x , y ) , we get the upper bound for H * (x, y) which shows our statement. ◻ Suppose now the case that the unique max and max are taken at the same index i 0 , say. Write for any j and

for Proposition 3.1. Observe that (S 1 , T 1 ) and (S,T) are independent. Then the (23)

.

corresponding statement of Proposition 3.1 holds for this case (letting i 0 = max ) which is given in Proposition 3.2. We omit the proof since it is very similar to the given one with a few obvious changes. 

Proof The convergence for the marginal distributions holds by applying Proposition 3.1 or 3.2 with the chosen normalization sequences. Since u n (x) and v n (y) are similar in type, we only show the derivation of the first marginal. Because the normalization u n (x) is not always an integer, we have to consider lim sup and lim inf . Let us deal with the lim sup case. Note that and For the normalization we get

The derivation of the lim inf is similar using [u n (x)] ≤ u n (x). Now for the joint distribution we use the bounds of H * (u n , v n ) of the two propositions. First we consider the case of Proposition 3.1 with max and max at different indexes. We have to derive the limits of three boundary terms of H * (u n , v n ) given in Proposition 3.1 multiplied by n. The last of these terms tends to 0 because (21) holds and due to the fact that from (14), we get which is bounded.

The first of the three boundary terms of H * (u n , v n ) is smaller than

≲ n × (d 1 log n) 1 1 + 1 max −x+1−d 1 (log n+ 1 log log n+log L * * 1 (log n)+ 1 log d 1 )

because ∕B > 0 with B given by (13). The second boundary term of H * (u n , v n ) is smaller than since 1 − > 0 and where C 1 represents a generic positive constant. Thus the limiting distribution is proved in case of Proposition 3.1. Now let us consider the changes of the proof for the case of Proposition 3.2. Again we have to deal with the three boundary terms of H * (u n , v n ) where the last two are as in Proposition 3.1. In the first of these terms we have similarly since d 2 log 1 + 2 max > 0 . Thus the statements are shown. ◻

We consider now the stationary sequence {(X j , Y j )} . From extreme value theory it is known that the behaviour of their extremes is as in the case of an iid sequence {(X j , Y j )} if the following two conditions hold: a mixing condition, called D(u n , v n ) , and a local dependence condition, called D � (u n , v n ) . In our bivariate extreme value case we consider the conditions D(u n , v n ) and D � (u n , v n ) of Hüsler (1990) (see also Hsing (1989) and Falk et al. (1990) ). The condition D(u n , v n ) is a long range mixing one for extremes and means that extreme values occurring in largely separated (by n ) intervals of positive integers are asymptotically independent. The condition D � (u n , v n ) considers the local dependence of extremes and excludes asymptotically the occurrences of local clusters of extreme or large values in each

individual margin of {(X j , Y j )} as well as jointly in the two components. We write u n , v n for short because x, y do not play a role in the following proofs.

1 ≤ i 1 < ... < i p < j 1 < ... < j q ≤ n, for which j 1 − i p > n , we have for some n, n with lim n→+∞ n, n = 0 , for some integer sequence n = o(n).

We use the following D � (u n , v n ) condition.

In the following we use the sequences {s n }, { n } and n, n such that Such a sequence {s n } in (26) exists always. Take e.g. for the given n and n, n in condition D(u n , v n ) the sequence s n = min( √ n∕ n , 1∕ √ n, n ) → +∞ . In our proof we use simpler sequences.

Write M (1) n = max{X 1 , ⋯ , X n } and M (2) n = max{Y 1 , ⋯ , Y n } . For the stationary sequence {(X j , Y j )} satisfying D(u n , v n ) and D � (u n , v n ) , the limiting behaviour of the bivariate maxima M (1) n , M (2) n , under linear normalization, is given in Theorem 3.1, as if the sequence {(X j , Y j )} would be a sequence of independent (X j , Y j ).

In Theorem 3.1 we derived upper and lower bounds of the limiting distribution of the maximum term of non-negative integer-valued moving average sequences which leads to a "quasi max-stable" limiting behavior of the bivariate maximum in the sense of Anderson's type. So the main result of the maximum of this bivariate discrete random sequence is the following. 

{(X j , Y j )} defined by | | P p ⋂ s=1 {X i s ≤ u n , Y i s ≤ v n }, q ⋂ t=1 {X j t ≤ u n , Y j t ≤ v n } −P p ⋂ s=1 {X i s ≤ u n , Y i s ≤ v n } P q ⋂ t=1 {X j t ≤ u n , Y j t ≤ v n } | | ≤ n, n , n [n∕s n ] ∑ j=1 P X 0 > u n , X j > u n + P X 0 > u n , Y j > v n + P Y 0 > v n , Y j > v n + P Y 0 > v n , X j > u n → 0, n → +∞.(X j , Y j ) = +∞ ∑ i=−∞ i •V j−i , +∞ ∑ i=−∞ i •W j−i .

and max and max are unique. Then, for all real x and y and where u n (x) and v n (y) are defined by (24) and (25).

To prove this theorem, it remains to show that the conditions D(u n , v n ) and D � (u n , v n ) hold with u n and v n given by (24) and (25).

where < 1 . We select later. We use the following notation:

and Note that and are independent. a) We have as upper bound

We split furthermore this upper bound.

The last four terms in (29) tend to 0 as it is proved in Hall (2003) depending on n . We show it for one term.

for some generic constant C and { k } satisfying (3) with > 2 . Selecting > 1∕( − 1) , this bound tends to 0. The sum of the bounds of the last four terms in (29) gives the bound n, n = Cn 1− n , which tends to 0. b) In the same way we establish the lower bound of (28). In fact, using again the independence mentioned in (27), we get (29)

using (29) and (30). Hence the condition D(u n , v n ) holds.

In the proof of D � (u n , v n ) , we need also that s n n, n → 0 . With s n = n we select such that s n n 1− n = n 1+ − ( −1) → 0 , which holds for 1 + < ( − 1).

We have to consider first the sums on the terms P X 0 > u n , Y j > v n and on the terms

We show it for the sum of the first terms, since for the second one the proof follows in the same way. Let n = n with < 1 − , which implies that n = o(n∕s n ) = o(n 1− ) . For j < 2 n , we write Note that i 0 = max for some i 0 and j 0 = max for some j 0 . For one j we have i 0 + j = j 0 , i.e. j = j 0 − i 0 . Hence the maximum terms occur at the same index for V −i 0 and W −i 0 if j = j 0 − i 0 . If j 0 = i 0 , hence j = 0 , but this case does not occur in the sum. For all other j's the maxima is occurring at different indexes. We consider the bound established in Proposition 3.1 and 3.2 for H * .

For j = j 0 − i 0 , we showed in the proof of Theorem 3.1 that nH * (u n , v n ) → 0.

For j ≠ j 0 − i 0 , we have i 0 +j < max for the terms P(X 0 > u n , Y j > v n ) and deduce from Proposition 3.1 the following upper bound for H * (u n , v n ) with , ∈ (0, 1) defined in (12) and (13). Note that = (j) should be such that (13) is satisfied. It means that the term B in (13) depends on j, i.e. B = B j . Note that B j may be larger or smaller than 1, but is bounded

above by log(1 + 2 ∕ max )∕ log(1 + 2 ∕ max ) = B * . For B j > 1 , we select < 1 large such that (1 − )B * < 1 , which implies that (1 − )B j < 1 , thus we select (j) > (1 − )B j . In case B j ≤ 1 , we select also (j) > (1 − )B j . It implies that there exists an > 0 to select (j) for every j ≠ j 0 − i 0 such that a) Now the sum of the first term in the bound (31) of H * (u n , v n ) multiplied by n, for {j ≤ 2 n , j ≠ j 0 } , is bounded by if also is such that < 1 − . The sum of the second term in (31) multiplied by n, for {j ≤ 2 n , j ≠ j 0 } , tends to 0 because if also < 1 − . Hence we choose < min{1 − , 1 − , 1 − }.

It remains to deal with the sum of the third terms in (31) for {j ≤ 2 n , j ≠ j 0 } . We showed that P( (20) with (1 + ) > 1 + 1 max in (12). Let ̃> 1 such that (1 + ) ∕̃= 1 + 1 max . This sum on {j ≤ 2 n , j ≠ j 0 } multiplied with n is bounded by if also <̃− 1 and C is a generic positive constant.

Thus combining these three bounds it shows that

We consider now the sum on j with 2 n < j ≤ n∕s n and write

and Note that X ′ 0 and Y ′ j are independent. We have, for j > 2 n and some k > 1 (chosen later, not depending on n), Similar to Hall (2003) , the last two probabilities are sufficiently fast tending to 0. For, we have

We select h n such that h n 1− n = C > 0, for some constant C. For i ≤ − n − 1 and > 2 and some positive constant C * , it follows that by the assumption (3) on the sequence { i } . It implies again that where the expectations exist, and, due to Lemma 3.1,

by the choice of h n . Note that h n = C −1 n = Cn ( −1) → +∞ . Now, select k depending on , and such that n 2 ∕((1 + h n ) k s n ) ∼ n 2 ∕(C k n k ( −1) n ) = o(1) which holds for k > (2 − )∕( ( − 1)) . This choice implies that (n 2 ∕s n )P X �� 0 ≥ k → 0 . In the same way we can show that also n ∑ j≤n∕s n P

→ 0 for such a k, since also i ≤ C |i| − for |i| ≥ n and some constant C > 0. c) In order to deduce we use the same arguments as for P X 0 > u n , Y j > v n . In this case, since X ′ 0 and X ′ j are independent, we get for some positive k As above we can show that n ∑ j≤n∕s n P � X �� j ≥ k � → 0 and (n 2 ∕s n )P X �� 0 ≥ k → 0 . In the same way it follows also that Hence condition D � (u n , v n ) holds.

We investigate the convergence of the distribution of the bivariate maxima (M (1) n , M (2) n ) to the limiting distribution as given in Theorem 4.1. We notice that the thinning coefficients i and i have an impact on the norming values of the bivariate maxima, besides of the distribution of the (V i , W i ).

Let us consider the bivariate geometric distribution for (V i , W i ) mentioned in Example 2.1 and a finite number of positive values i and i . As mentioned, the bivariate geometric distribution satisfies the conditions of the general assumptions of the joint distribution of (V i , W i ) . We assumed a strong dependence with p 00 = 0.85, p 01 = 0.03, p 10 = 0.02 and p 11 = 0.1.

We consider quite different models with different i and i to investigate the convergence rate. Let in the first case 1 = 0.8, 2 = 0.6, 3 = 0.4, 1 = 0.6, 2 = 0.45, 3 = 0.3 and i = 0 = i for i > 3, and in the second case 1 = 0.6, 2 = 0.35, 3 = 0.1, 1 = 0.5, 2 = 0.3, 3 = 0.1 and i = 0 = i for i > 3.

For each of these first two models we simulated 10'000 time series, selected n = 100 and 500 and derived the bivariate maxima (M (1) n , M (2) n ). Thus we compared the empirical (simulated) distribution functions (cdf) with the asymptotic cdf.

We plotted two cases with P(M (1) n −ũ n ≤ x, M (2) n −ṽ n ≤ x + ) where ũ n = u n − x and ṽ n = v n − y with u n , v n given in (24) and (25), respectively, using = 0 and 2 (see Figs. 1 and 2).

We notice from these simulations that the convergence rate is quite good, but it depends on the dependence, which is given by the thinning factors i and i . We find that the convergence rate is slower for the more dependent time series (the first case, Fig. 1 ) and that the factor has a negligible impact. This is even more clear in the second cases shown in Fig. 2 .

In some additional models we considered larger and more thinning factors different from 0. We show the simulations of the cases with i = (0.7) i , i = (0.6) i , for i ≤ 25 , and also with i = (0.9) i , i = (0.8) i , for i ≤ 40 . These cases are close to a infinite MA series, since i , i are very small for i > 26 or i > 41 , respectively. It means that such small values have an impact on the maxima. We figured out that the number of positive values is not so important. However, in these cases the second largest value of i or i is closer to the maximal value (=1), in particular in the second of these additional models. Considering the results of again 10'000 simulations (Fig. 3) , we show that the convergence rates are quite slower than in the first two models (Figs. 1 and 2) . We show the results of the two cases with n = 100 and 500 with = 0 only. We also figured out from the simulations of other models and distributions that if the correlation of the two components of the sequence is stronger, then the convergence to the limiting distribution (with asymptotic independence) is slower.

Proof of Proposition 2.1:

Observe first that By using the representation (4) for v and w large, and w > v > A (some large constant A) we deduce, for v ≤ w − 1, 

, the steps are similar, with * (v, w) such that

For v = w > A, A large, we get with similar steps as above The first partial sum is finite. The second one is bounded by

which is finite for s 2 < 1 + 2 due to (5). Analogously, the third one is bounded by

which is finite for s 1 < 1 + 1 . Finally, for the last partial sum we use Proposition 2.1. For simplicity we write g i (k) = s k i (1 + i ) −k k i+2 L i+2 (k) , for k ∈ ℕ , i = 1, 2 . Then, for large k, and some positive constants C 1 , C 2 , we get that this sum is bounded by

The sums ∑ ≥k g 2 ( ) and ∑ k≥ g 1 (k) are finite by applying the ratio criterium for g 2 ( ) and g 1 (k) . These sums are bounded by Cg 2 (k) and Cg 1 ( ) , respectively, with C a generic constant if s i < (1 + i ) . Then the convergence of the last sum is obtained for s 1 s 2 < (1+ 1 )(1+ 2 ) .

Proof of Proposition 2.3: Let q(k, ) = 1 − F (V,W) (k, ) and p(k, ) = P(V = k, W = ) . Then,

Write a 1 = s 1 + 1 − and a 2 = s 2 + 1 − . By Proposition 2.2, for s i ≠ 1, i = 1, 2, we have Proof of Proposition 2.6: Using Proposition 2.5, 1 − F X,Y (x, y) is given by the sum of three terms due to the assumption (4). Each term, defined by double sums, can be determined or bounded by (unique) sums associated to univariate tail functions satisfying Theorem 4 of Hall (2003) , see also Hall and Temido (2007) . The first sum can be approximated for x and y large, as

The second sum can be dealt with in the same way.

For the third term observe that due to the fact (v, w) is a bounded function, with bound , we get, for large integers x and y, Q X,Y (s 1 , s 2 ) = 1 − G X,Y (s 1 , s 2 )

(1 − s 1 )(1 − s 2 ) = 1 − G V,W ( s 1 + 1 − , s 2 + 1 − )

(1 − s 1 )(1 − s 2 ) = 1 − G V,W (a 1 , a 2 ) 1−a 1 1−a 2 = 1 − G V,W (a 1 , a 2 )

(1 − a 1 )(1 − a 2 ) = Q V,W (a 1 , a 2 ). Q X,Y (s 1 , s 2 ) = Q V,W ( s 1 + 1 − , s 2 + 1 − )

□ Proof of Lemma 3.1: a.) All moments of V exist, since the moment generating function of V exists for small positive values. Applying Taylor's expansion to the function f (1 + h) = (1 + h) k , h > 0 , we get, for k ≥ 2,

. Due to the fact that (1 + h) k > 1 + hk the proof of the first claim is complete.

b.) Similarly we have for some h * such that 0 < h < h * < 1 ∕ I . Then

where C 1 = E(V) ∑ i∈I i and C 2 = E(V 2 (1 + I h * ) V ) (1 − ) −y y+1 1 + 2 − max(1, ) 4 L 4 ( )

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