LONGEST INCREASING SUBSEQUENCES

OF RANDOM COLORED PERMUTATIONS

Alexei Borodin

Department of Mathematics, The University of Pennsylvania Philadelphia, PA 19104-6395, U.S.A.

                       borodine@math.upenn.edu

Submitted: February 7, 1999; Accepted: February 15, 1999.

Abstract. We compute the limit distribution for the (centered and scaled) length of the longest increasing subsequence of random colored permutations. The limit distribution function is a power of that for usual random permutations computed recently by Baik, Deift, and Johansson (math.CO/9810105). In the two–colored case our method provides a different proof of a similar result by Tracy and Widom about the longest increasing subsequences of signed permutations (math.CO/9811154).

Our main idea is to reduce the ‘colored’ problem to the case of usual random permutations using certain combinatorial results and elementary probabilistic arguments.

1. Introduction

Baik, Deift, and Johansson recently solved a problem about the asymptotic behavior of the length l_{n} of the longest increasing subsequence for random permutations of order n as n → ∞ (with the uniform distribution on the symmetric group S_{n}). They proved, see [BDJ], that the sequence

\{\frac{l_{n} − 2\sqrt{n}}{n^{1/6}}\}

converges in distribution, as n → ∞, to a certain random variable whose distribution function we shall denote by F (x). This distribution function can be expressed via a solution of the Painlevé II equation, see [BDJ] for details. It was first obtained by Tracy and Widom [TW1] in the framework of Random Matrix Theory where it gives the limit distribution for the (centered and scaled) largest eigenvalue in the Gaussian Unitary Ensemble of Hermitian matrices.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 05A05, Secondary 60F99.

The problem of the asymptotics of l_{n} was first raised by Ulam [U]. Substantial contributions to the solution of the problem have been made by Hammersley [H], Logan and Shepp [LS], Vershik and Kerov [VK1, VK2].

A survey of the interesting history of this problem, further references, and a discussion of its intriguing connection with Random Matrix Theory can be found in [BDJ].

Soon after the appearance of [BDJ] Tracy and Widom computed the asymptotic behavior of the length l′_{n} of the longest increasing for the random ‘signed permutations’, see definitions in the next section. In [TW2] they showed that

\{\frac{l′_{n} − 2\sqrt{2n}}{2^{2/3}(2n)^{1/6}} \}

converges in distribution, as n → ∞, to a random variable with the distribution function F^{2}(x).

The present paper provides another proof of the result by Tracy and Widom. In our approach the distribution function F^{2}(x) arises as the distribution function of the maximum of two asymptotically independent variables each of which behaves as (l_{n} − 2\sqrt{n})/n^{1/6} (hence, by [BDJ], converges to the distribution given by F (x)).

The combinatorial techniques we use relies on recent works by Rains [R] and Fomin & Stanton [FS]. It also allows to handle a more general case of ‘colored permutations’ (the problem for ‘two–colored case’, essentially, coincides with that for signed permutations). We show that for the length l′′_{n} of the longest increasing subsequence of the random m–colored permutations of order n the sequence

\{\frac{l′′_{n} − 2\sqrt{mn}}{m^{2/3}(mn)^{1/6}} \}

converges in distribution, as n → ∞, to a random variable with distribution function F^{m}(x). The function F^{m}(x) naturally appears as the distribution function of the maximum of m asymptotically independent variables, each having F (x) as the limit distribution function.

Combinatorial quantities which we consider can be also interpreted as expectations of certain central functions on unitary groups, see Section 4.

I am very grateful to G. I. Olshanski for a number of valuable discussions.

2. Colored permutations and signed permutations

A colored permutation is a map from {1, …, n} to {1, …, n} × {1, …, m} such that its composition with the projection on the first component of the target set is a permutation (of order n). One can view such a map as a permutation with one of m colors attributed to each of n points which this permutation permutes. The set of all colored permutations of order n with m colors will be denoted by S_{n}^{(m)}.

An increasing subsequence of π ∈ S_{n}^{(m)} is a sequence 1 ≤ i_{1} < i_{2} < ⋯ < i_{k} ≤ n such that the first coordinates of π(i_{j}) increase in j and the second coordinates of π(i_{j}) are equal. Thus, elements of an increasing subsequence are of the same color, say, p. The length of such increasing subsequence is defined to be m(k − 1) + p.

These definitions are due to Rains [R]. A slightly more general notion of hook permutation was introduced and intensively used earlier by Stanton and White [SW].

We shall consider S_{n}^{(m)} as a probability space with uniform distribution: probability of each colored permutation is |S_{n}^{(m)}|^{−1} = (m^{n}n!)^{−1}. Then the length of the longest increasing subsequence becomes a random variable on this space, it will be denoted as L_{n}^{col(m)}.

Let H_{n} be the hyperoctahedral group of order n defined as the wreath product ℤ_{2}^{n} ⋉ S_{n} (S_{n} is the symmetric group of order n). The elements of H_{n} are called signed permutations. This group can be naturally embedded in S_{2n} as the group of permutations σ of {−n, −n + 1, …, −1, 1, …, n − 1, n} subject to the condition σ(−x) = −σ(x). Indeed, each such permutation is parametrized by the permutation |σ| ∈ S_{n} and the set of signs of σ(1), …, σ(n). Using the natural ordering on the set {−n, −n + 1, …, −1, 1, …, n − 1, n} we can define the length of the longest increasing subsequence for each signed permutation. Assuming that every signed permutation has probability |H_{n}|^{−1} = (2^{n}n!)^{−1}, we get a random variable on H_{n} which will be denoted as L_{n}^{even}.

The group H_{n} can also be embedded into the symmetric group of order 2n+1: we add 0 to the set {−n, −n + 1, …, −1, 1, …, n − 1, n} and assume that the elements σ ∈ H_{n} satisfy the same condition σ(−x) = −σ(x). Clearly, this implies σ(0) = 0. The random variable on H_{n} equal to the length of the longest increasing subsequence with respect to this realization will be denoted by L_{n}^{odd}.

Note that for any element σ ∈ H_{n},

L_{n}^{odd}(σ) − L_{n}^{even}(σ) = 0 or 1.  (2.1)

3. Rim hook tableaux

We refer to the work [SW] for the definitions concerning rim hook tableaux. The next claim is a direct consequence of the Schensted algorithm, see [S].

Proposition 3.1. Permutations of order n with the length of longest increasing subsequence equal to l are in one–to–one correspondence with pairs of standard Young tableaux of the same shape with n boxes and width l.

Here is a generalization of this claim for colored permutations.

Proposition 3.2 [R], [SW]. Colored permutations with m colors of order n with the length of longest increasing subsequence equal to l are in one–to–one correspondence with pairs of m–rim hook tableaux of the same shape with mn boxes and width l.

In [SW] it was proved that ⌈\frac{l}{m}⌉ = ⌈\frac{w}{m}⌉ where w is the width of the rim hook tableau corresponding to a permutation with the length of longest increasing subsequence equal to l (⌈a⌉ stands for the smallest integer ≥ a). The refinement of this statement given above was published in [R].

Proposition 3.3 [R]. Signed permutations of order n embedded in S_{2n} with the length of longest increasing subsequence equal to l are in one–to–one correspondence with pairs of 2–rim hook tableaux of the same shape with 2n boxes and width l.

The length of the longest increasing subsequence for signed permutations embedded into the symmetric group of odd order can also be interpreted in terms of rim hook tableaux, see [R, proof of Theorem 2.3].

Note that Propositions 3.2 and 3.3 imply that the distributions of random variables L_{n}^{col(2)} and L_{n}^{even} coincide.

4. Expectations over unitary groups

Everywhere below the symbol 𝔼_{U∈U(k)}f(U) stands for the integral of f over U ∈ U(k) with respect to the Haar measure on the unitary group U(k) normalized so that 𝔼_{U∈U(k)}1 = 1 (i.e., 𝔼 denotes the expectation of f with respect to the uniform distribution on the unitary group).

Proposition 4.1 [R].

Prob{L_{n}^{col(m)} ≤ k} = (m^{n}n!)^{−1} · 𝔼_{U∈U(k)} (|T r(U^{m})^{n}|^{2}) .  (4.1)

Proposition 4.2 [R].

Prob{L_{n}^{even} ≤ k} = (2^{n}n!)^{−1} · 𝔼_{U∈U(k)} (|T r(U^{2})^{n}|^{2}) .  (4.2)

Prob{L_{n}^{odd} ≤ k} = (2^{n}n!)^{−1} · 𝔼_{U∈U(k)} (|T r(U^{2})^{n}T r(U)|^{2}) .  (4.3)

[DS] gives (4.1) for k ≥ mn, (4.2) for k ≥ 2n, and (4.3) for k ≥ 2n + 1. For such values of k the left–hand sides of (4.1), (4,2), (4.3) are all equal to 1.

5. Rim hook lattices

Our main reference for this section is the work [FS] by Fomin and Stanton. For this section we fix an integer number m, all our rim hooks here will contain

exactly m boxes.

Let μ and λ be shapes (Young diagrams) such that μ ⊂ λ and λ − μ is a (m–)rim hook. Then we shall write μ ↗ λ.

We introduce a partial order on the set of Young diagrams as follows: λ ⪰ μ if and only if there exists a sequence ν_{1}, ν_{2}, …, ν_{k} of Young diagrams such that μ ↗ ν_{1} ↗ ν_{2} ↗ ⋯ ↗ ν_{k} ↗ λ. The empty Young diagram is denoted by ∅. We shall say that a Young diagram λ is m–decomposable if λ ⪰ ∅.

The poset of all m–decomposable shapes with ⪰ as the order is called rim hook lattice and is denoted by RH_{m}. (It can be shown that this poset is indeed a lattice).

For m = 1 we get the Young lattice: the poset of all Young diagrams ordered by inclusion. The Young lattice will be denoted by 𝕐.

Proposition 5.1 [FS]. The rim hook lattice RH_{m} is isomorphic to the Cartesian product of m copies of the Young lattice: RH_{m} ≅ 𝕐^{m}.

In other words, RH_{m} is isomorphic to the poset of m–tuples of Young diagrams with the following coordinate–wise ordering: one tuple is greater than or equal to another tuple if the kth coordinate of the first tuple includes (i.e., greater than or equal to) the kth coordinate of the second tuple for all k = 1, …, m.

Clearly, the number of m–rim hook tableaux of a given shape λ is equal to the number of paths ∅ ↗ ν_{1} ↗ ν_{2} ↗ ⋯ ↗ ν_{k} ↗ λ, k = |λ|/m − 1, from ∅ to λ (and is equal to 0 if λ is not m–decomposable), |λ| stands for the number of boxes in λ. We shall denote this number by dim_{m} λ and call it the m–dimension of the shape λ.

Take any λ ∈ RH_{m} and the corresponding m–tuple (λ_{1}, …, λ_{m}) ∈ 𝕐^{m}. Note that |λ| = m(|λ_{1}| + ⋯ + |λ_{m}|). We have

dim_{m} λ = \frac{(|λ_{1}| + ⋯ + |λ_{m}|)!}{|λ_{1}|! ⋯ |λ_{m}|!} · dim_{1} λ_{1} ⋯ dim_{1} λ_{m}.  (5.1)

Indeed, to specify the path from ∅ to (λ_{1}, …, λ_{m}) we need to specify m paths from ∅ to λ_{k} in the kth copy of 𝕐 for k = 1, …, m together with the order in which we make steps along those paths. The number of different orders is the combinatorial coefficient in the right–hand side of (5.1) while the number of different possibilities for the m paths in 𝕐 is the product of 1–dimensions dim_{1} λ_{1} ⋯ dim_{1} λ_{m}.

Note that the number of pairs of m–rim hook tableaux of the same shape λ is exactly

dim_{m}^{2} λ = (\frac{(|λ_{1}| + ⋯ + |λ_{m}|)!}{|λ_{1}|! ⋯ |λ_{m}|!})^{2} · dim_{1}^{2} λ_{1} ⋯ dim_{1}^{2} λ_{m}.  (5.2)

Let us denote by w(λ) the width of a Young diagram λ. We shall need the following

Observation 5.2. For any λ ∈ RH_{m} and corresponding m–tuple (λ_{1}, …, λ_{m}) ∈ 𝕐^{m} we have

m · max{w(λ_{1}), …, w(λ_{m})} − w(λ) ∈ {0, 1, …, m − 1}.  (5.3)

This follows immediately from the explicit construction of the isomorphism from Proposition 5.1, see [FS, §2].

Observation 5.2 will be crucial for our further considerations.

6. Plancherel distributions

Using the correspondence from Proposition 3.2 (which is exactly the rim hook generalization of the Schensted algorithm, see [SW]), we can associate to each m– colored permutation of order n a Young diagram with mn boxes — the common shape of the corresponding pair of m–rim hook tableaux. The image of the uniform distribution on S_{n}^{(m)} under this map gives a probability distribution on m– decomposable Young diagrams with mn boxes; the weight of a Young diagram λ is, clearly, equal to (m^{n}n!)^{−1} · dim_{m}^{2} λ. As a consequence, we get (cf. [FS, Corollary 1.6])

∑_{|λ|=mn} dim_{m}^{2} λ = m^{n}n!.  (6.1)

Using the isomorphism of Proposition 5.1 we transfer our probability distribution to the set of m–tuples of Young disgrams with total number of boxes equal to n. Then by (5.2) we see that the probability of an m–tuple (λ_{1}, …, λ_{m}) ∈ 𝕐^{m} with |λ_{1}| + ⋯ + |λ_{m}| = n equals

Prob\{(λ_{1}, …, λ_{m})\} = \frac{1}{m^{n}n!} (\frac{n!}{|λ_{1}|! ⋯ |λ_{m}|!})^{2} · dim_{1}^{2} λ_{1} ⋯ dim_{1}^{2} λ_{m}.  (6.2)

This distribution will be called the Plancherel distribution.

Proposition 6.1. For any n = 1, 2, …

\begin{matrix}Prob\{|λ_{1}| = n_{1}, … |λ_{m}| = n_{m}; n_{1} + ⋯ + n_{m} = n\} \\ = \frac{1}{m^{n}} \frac{n!}{n_{1}! ⋯ n_{m}!}. \end{matrix}  (6.3)

Proof. Direct computation. Using (6.1) for m = 1 and (6.2) we get

Prob{|λ_{1}| = n_{1}, … |λ_{m}| = n_{m}; n_{1} + ⋯ + n_{m} = n}

= \frac{1}{m^{n}n!} ∑_{\begin{matrix}|λ_{k}|=n_{k} \\ k=1,...,m \end{matrix}} (\frac{n!}{|λ_{1}|! ⋯ |λ_{m}|!})^{2} · dim_{1}^{2} λ_{1} ⋯ dim_{1}^{2} λ_{m}

= \frac{1}{m^{n}n!} (\frac{n!}{n_{1}! ⋯ n_{m}!})^{2} · n_{1}! ⋯ n_{m}! = \frac{1}{m^{n}} \frac{n!}{n_{1}! ⋯ n_{m}!}. □

7. Two lemmas from Probability Theory

We shall denote by ⟶^{p} convergence of random variables in probability, and by ⟶^{D} convergence in distribution.

Lemma 7.1 [B, Theorem 4.1]. Let random variables ξ, {ξ_{n}}_{i=1}^{∞}, {η_{n}}_{i=1}^{∞} satisfy

ξ_{n} − η_{n} ⟶^{p} 0, ξ_{n} ⟶^{D} ξ.

Then

η_{n} ⟶^{D} ξ.

For m real random variables ξ_{1}, …, ξ_{m} we shall denote by ξ_{1} × ξ_{2} × ⋯ × ξ_{m} a ℝ^{m}–valued random variable with distribution function

F_{ξ_{1}×···×ξ_{m}}(x) = F_{ξ_{1}}(x_{1}) ⋯ F_{ξ_{m}}(x_{m}).

Lemma 7.2. Let {ξ_{n}^{(k)}}_{n=1}^{∞}, k = 1, …, m, be m > 1 sequences of real random variables convergent in distribution to random variables ξ^{(1)}, …, ξ^{(m)}, respectively. Denote by B_{n}^{(m)} the nth order m–dimensional fair Bernoulli distribution:

Prob\{B_{n}^{(m)} = (k_{1}, …, k_{m})\} = \{ \begin{matrix}\frac{1}{m^{n}} \frac{n!}{k_{1}!···k_{m}!}, & k_{1} + … + k_{m} = n, k_{i} ∈ \{0, …, n\} \\ 0, & otherwise \end{matrix}

Then the sequence

ζ_{n} = ξ_{B_{n,1}^{(m)}}^{(1)} × ξ_{B_{n,2}^{(m)}}^{(2)} × ⋯ × ξ_{B_{n,m}^{(m)}}^{(m)}

converges in distribution to ξ^{(1)} × ξ^{(2)} × ⋯ × ξ^{(m)}. In particular, ξ_{B_{n,1}^{(m)}}^{(1)} ⟶^{D} ξ^{(1)}.

Proof. The conditions ξ_{n}^{(k)} ⟶^{D} ξ^{(k)}, k = 1, …, m, are equivalent to the pointwise convergence of distribution functions

F_{ξ_{n}^{(k)}}(x) → F_{ξ^{(k)}}(x).

We have

F_{ζ_{n}}(x) = ∑_{\begin{matrix}k_{1}+···+k_{m}=n \\ k_{i}=1,...,n \end{matrix}} \frac{1}{m^{n}} \frac{n!}{k_{1}! ⋯ k_{m}!} · F_{ξ_{k_{1}}^{(1)}}(x_{1}) ⋯ F_{ξ_{km}^{(m)}}(x_{m}).

For m convergent number sequences {a_{n}^{(k)}} with limits a^{(k)}, k = 1, …, m, the sequence

c_{n} = ∑_{\begin{matrix}k_{1}+···+k_{m}=n \\ k_{i}=1,...,n \end{matrix}} \frac{1}{m^{n}} \frac{n!}{k_{1}! ⋯ k_{m}!} a_{k_{1}}^{(1)} ⋯ a_{k_{m}}^{(m)}

converges to the product a^{(1)} ⋯ a^{(m)}, so we conclude, that {F_{ζ_{n}}(x)} converges to F_{ξ}^{(1)}(x_{1}) ⋯ F_{ξ}^{(m)}(x_{m}) pointwise. □

Lemma 7.2 implies that random variables ξ_{B_{n,1}^{(m)}}^{(1)}, ξ_{B_{n,2}^{(m)}}^{(2)}, …, ξ_{B_{n,m}^{(m)}}^{(m)} are asymptot-

ically independent as n → ∞.

8. Asymptotics

Baik, Deift, and Johansson recently proved, see [BDJ], that the sequence

\frac{L_{n}^{col(1)} − 2\sqrt{n}}{n^{1/6}}

converges in distribution to a certain random variable, whose distribution function will be denoted by F (x). This function can be expressed through a particular solution of the Painlevé II equation, see [BDJ] for details.

In this section we shall study the asymptotic behavior of L_{n}^{col(m)} for m ≥ 2 when n → ∞. Our main result is the following statement.

Theorem 8.1. For any m = 2, 3, … the sequence

\frac{L_{n}^{col(m)} − 2\sqrt{mn}}{(mn)^{1/6}}

converges in distribution, as n → ∞, to a random variable with distribution function F^{m}(m^{− \frac{2}{3}} x).

This result for m = 2 was proved by Tracy and Widom in [TW2]. Since the distributions of L_{n}^{col(2)} and L_{n}^{even} coincide (see Section 3), we immediately get two other asymptotic formulas.

Corollary 8.2 [TW2]. The sequence

\frac{L_{n}^{even} − 2\sqrt{2n}}{(2n)^{1/6}}

converges in distribution, as n → ∞, to a random variable with distribution function F^{2}(2^{− \frac{2}{3}} x).

Corollary 8.3 [TW2]. The sequence

\frac{L_{n}^{odd} − 2\sqrt{2n}}{(2n)^{1/6}}

converges in distribution, as n → ∞, to a random variable with distribution function F^{2}(2^{− \frac{2}{3}} x).

Proof of Corollary 8.3. Relation (2.1) implies that with probability 1,

(L_{n}^{odd} − L_{n}^{even}) (2n)^{−1/6} → 0.

Since the convergence with probability 1 implies the convergence in probability, the claim follows from Lemma 7.1 and Corollary 8.2. □

Proof of Theorem 8.1. The proof will consist of 4 steps. Step 1. Following Sections 3, 5, 6, we shall interpret L_{n}^{col(m)} as a random variable on the space of all m–tuples (λ_{1}, …, λ_{m}) of Young diagram with total number of boxes equal to n supplied with the Plancherel distribution. Let us denote by l_{1}^{(n)}, …, l_{m}^{(n)} the widths of the Young diagrams λ_{1}, …, λ_{m}:

l_{k}^{(n)} = w(λ_{k}), k = 1, …, m.

Observation 5.2 implies that with probability 1

\frac{L_{n}^{col(m)} − m · max\{l_{1}^{(n)}, …, l_{m}^{(n)}\}}{n^{1/6}} → 0.

Hence, by Lemma 7.1, it is enough to prove that

Prob \{\frac{m · max\{l_{1}^{(n)}, …, l_{m}^{(n)}\} − 2\sqrt{mn}}{(mn)^{1/6}} ≤ x\} → F^{m}(m^{− \frac{2}{3}} x)

Our strategy is to prove that l_{1}^{(n)}, …, l_{m}^{(n)} are asymptotically independent, and that each of them asymptotically behaves as L_{[n/m]}^{col(1)}. Then Theorem 8.1 will follow from the result of [BDJ] stated in the beginning of this section. Step 2.

Prob \{\frac{m · max\{l_{1}^{(n)}, …, l_{m}^{(n)}\} − 2\sqrt{mn}}{(mn)^{1/6}} ≤ x\}

= Prob \{\frac{m · l_{1}^{(n)} − 2\sqrt{mn}}{(mn)^{1/6}} ≤ x; …; \frac{m · l_{m}^{(n)} − 2\sqrt{mn}}{(mn)^{1/6}} ≤ x\}

= Prob \{\frac{l_{1}^{(n)} − 2\sqrt{n/m}}{(n/m)^{1/6}} ≤ m^{\frac{2}{3}} x; …; \frac{l_{m}^{(n)} − 2\sqrt{n/m}}{(n/m)^{1/6}} ≤ m^{\frac{2}{3}} x\} .

Thus, it suffices to prove that

Prob \{\frac{l_{1}^{(n)} − 2\sqrt{n/m}}{(n/m)^{1/6}} ≤ x_{1}; …; \frac{l_{m}^{(n)} − 2\sqrt{n/m}}{(n/m)^{1/6}} ≤ x_{m}\} → F (x_{1}) ⋯ F (x_{m}). (8.1)

Step 3. Denote by n_{i}^{(n)} the number of boxes in λ_{i}, i = 1, …, m. Then n_{1}^{(n)} + ⋯ + n_{m}^{(n)} = n. We claim that for all i = 1, …, m

\frac{l_{i}^{(n)} − 2\sqrt{n/m}}{(n/m)^{1/6}} − \frac{l_{i}^{(n)} − 2\sqrt{n_{i}^{(n)}}}{(n_{i}^{(n)})^{1/6}} ⟶^{p} 0.  (8.2)

Proposition 6.1 implies that

Prob\{n_{i}^{(n)} = k\} = \frac{(m − 1)^{n−k}}{m^{n}} \binom{n}{k}.

This means that n_{i}^{(n)} can be interpreted as the sum ξ_{1} + ⋯ + ξ_{n} of n independent identically distributed Bernoulli variables with

Prob\{ξ_{j} = 0\} = \frac{m − 1}{m}, Prob\{ξ_{j} = 1\} = \frac{1}{m}.

The central limit theorem implies that with probability converging to 1,

\frac{n}{m} − (\frac{n}{m})^{\frac{1}{2} +ε} < n_{i}^{(n)} < \frac{n}{m} + (\frac{n}{m})^{\frac{1}{2} +ε}

for any ε > 0. Hence,

\begin{matrix}|(\frac{n}{m})^{\frac{1}{3}} − (n_{i}^{(n)})^{\frac{1}{3}} | < |(\frac{n}{m})^{\frac{1}{3}} − (\frac{n}{m} ± (\frac{n}{m})^{\frac{1}{2} +ε})^{\frac{1}{3}} | \\ = (\frac{n}{m})^{\frac{1}{3}} |1 − (1 ± (\frac{n}{m})^{− \frac{1}{2} +ε})^{\frac{1}{3}} | < const · (\frac{n}{m})^{− \frac{1}{6} +ε} \end{matrix}  (8.3)

for sufficiently large n.

Similarly, we have

\begin{matrix}|(\frac{n}{m})^{− \frac{1}{6}} − (n_{i}^{(n)})^{− \frac{1}{6}} | < |(\frac{n}{m})^{− \frac{1}{6}} − ((\frac{n}{m}) ± (\frac{n}{m})^{\frac{1}{2} +ε})^{− \frac{1}{6}} | \\ = (\frac{n}{m})^{− \frac{1}{6}} |1 − (1 ± (\frac{n}{m})^{− \frac{1}{2} +ε})^{− \frac{1}{6}} | < const · (\frac{n}{m})^{− \frac{1}{2} − \frac{1}{6} +ε} < const · (\frac{n}{m})^{− \frac{1}{2} −ε} \end{matrix}

(8.4)

if we choose ε < 1/12.

The result of [BDJ] implies that L_{n}^{col(1)}n^{− \frac{1}{2} −δ} ⟶^{D} 0 for any δ > 0.

From Proposition 6.2 it follows that in the notation of Lemma 7.2

{l_{i}^{(n)}}_{n=1}^{∞} = {L_{(B_{n,1}^{(m)})}^{col(1)}}_{n=1}^{∞} .

Applying Lemma 7.2 we see that

\frac{l_{i}^{(n)}}{(n_{i}^{(n)})^{\frac{1}{2} +δ}} ⟶^{D} 0,

which means that l_{i}^{(n)} < (n_{i}^{(n)})^{\frac{1}{2} +δ} with probability converging to 1.

Since n_{i}^{(n)} < n/m + (n/m)^{\frac{1+ε}{2}} with probability converging to 1, we get that

\begin{matrix}l_{i}^{(n)} < (\frac{n}{m} + (\frac{n}{m})^{\frac{1+ε}{2}} )^{\frac{1}{2} +δ} = (\frac{n}{m})^{\frac{1}{2} +δ} (1 + (\frac{n}{m})^{\frac{−1+ε}{2}} )^{\frac{1}{2} +δ} \\ < (\frac{n}{m})^{\frac{1}{2} +δ} + const · (\frac{n}{m})^{δ+ \frac{ε}{2}} < const · (\frac{n}{m})^{\frac{1}{2} +δ} \end{matrix}  (8.5)

with probability converging to 1.

Making use of the relations (8.3), (8.4), (8.5), we now see that

\begin{matrix}|\frac{l_{i}^{(n)} − 2\sqrt{n/m}}{(n/m)^{1/6}} − \frac{l_{i}^{(n)} − 2\sqrt{n_{i}^{(n)}}}{(n_{i}^{(n)})^{1/6}} | ≤ l_{i}^{(n)} |(\frac{n}{m})^{− \frac{1}{6}} − (n_{i}^{(n)})^{− \frac{1}{6}} | \\ +2 |(\frac{n}{m})^{\frac{1}{3}} − (n_{i}^{(n)})^{\frac{1}{3}} | < const · (\frac{n}{m})^{\frac{1}{2} +δ} (\frac{n}{m})^{− \frac{1}{2} −ε} + const · (\frac{n}{m})^{− \frac{1}{6} +ε} \end{matrix}

with probability converging to 1. Since the last expression converges to zero as n → ∞ if δ < ε < 1/6, the proof of (8.2) is complete.

The relation (8.2) implies that we have asymptotic equivalence of two m–dimensional vectors

\begin{matrix}(\frac{l_{1}^{(n)} − 2\sqrt{n/m}}{(n/m)^{1/6}}; …; \frac{l_{m}^{(n)} − 2\sqrt{n/m}}{(n/m)^{1/6}}) \\ − (\frac{l_{1}^{(n)} − 2\sqrt{n_{1}^{(n)}}}{(n_{1}^{(n)})^{1/6}} ; …; \frac{l_{m}^{(n)} − 2\sqrt{n_{m}^{(n)}}}{(n_{m}^{(n)})^{1/6}} ) ⟶^{p} 0. \end{matrix}  (8.6)

Step 4. The random m–dimensional vector

ζ_{n} = (\frac{l_{1}^{(n)} − 2\sqrt{n_{1}^{(n)}}}{(n_{1}^{(n)})^{1/6}} ; …; \frac{l_{m}^{(n)} − 2\sqrt{n_{m}^{(n)}}}{(n_{m}^{(n)})^{1/6}} )  (8.7)

is obtained from m identical 1–dimensional random variables

ξ_{n}^{(i)} = \frac{L_{n}^{col(1)} − 2\sqrt{n}}{n^{1/6}}, i = 1, …, m

by the procedure described in Lemma 7.2:

ζ_{n} = ξ_{B_{n,1}^{(m)}}^{(1)} × ξ_{B_{n,2}^{(m)}}^{(2)} × ⋯ × ξ_{B_{n,m}^{(m)}}^{(m)}

(this follows from Proposition 6.1). Since sequences {ξ_{n}^{(i)}} converge in distribution, as n → ∞, to a random variable with distribution function F (x) (this is the result of [BDJ]), the sequence (8.7) also converges in distribution to a random variable with the distribution function F (x_{1}) ⋯ F (x_{m}). Then Lemma 7.1 and (8.6) conclude the proof of (8.1). □

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