Random A-permutations and Brownian motion

ISSN 0081-5438, Proceedings of the Steklov Institute of Mathematics, 2013, Vol. 282, pp. 298–318. c© Pleiades Publishing, Ltd., 2013.Original Russian Text c© A.L. Yakymiv, 2013, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2013, Vol. 282, pp. 315–335.

Random A-Permutations and Brownian MotionA. L. Yakymiv a

Received March 2012

Abstract—We consider a random permutation τn uniformly distributed over the set of all de-gree n permutations whose cycle lengths belong to a fixed set A (the so-called A-permutations).Let Xn(t) be the number of cycles of the random permutation τn whose lengths are not greaterthan nt, t ∈ [0, 1], and l(t) =

∑i≤t, i∈A 1/i, t > 0. In this paper, we show that the finite-

dimensional distributions of the random process {Yn(t) = (Xn(t) − l(nt))/√

� ln n, t ∈ [0, 1]}converge weakly as n → ∞ to the finite-dimensional distributions of the standard Brownianmotion {W (t), t ∈ [0, 1]} in a certain class of sets A of positive asymptotic density �.

DOI: 10.1134/S0081543813060217

1. INTRODUCTION

Let A be an arbitrary nonempty subset of the set of positive integers N. Permutations whose cy-cle lengths belong to the set A are called A-permutations (see Sachkov’s book [24]). Let Tn = Tn(A)be the set of A-permutations of degree n and τn be a random permutation uniformly distributedover Tn. Denote by ζmn, m ∈ N, the number of length m cycles of the permutation τn. It is clear thatζmn = 0 for m /∈ A. Next, let ζn be the total number of cycles of the random permutation τn, i.e.,

ζn =∑

m∈N

ζmn.

Random A-permutations have been intensively studied by a number of authors over the last morethan thirty years. A survey of this field of research is contained in the author’s paper [31] andbook [32]. The first results in this direction were obtained by Bender [1] (1974), Bolotnikov et al. [8](1976), and Mineev and Pavlov [14, 15] (1976). Still earlier, Sachkov [21, 22] (1972, 1973) consideredmore general objects, namely, random mappings with limitations on the cycle lengths. For certainclasses of sets A, he studied the asymptotic behavior of the number of elements in the set Tn(A) asn → ∞ and the limiting behavior (in the weak sense) of the random variables ζmn and ζn, where mis a fixed element of A. Information on A-permutations can be found in eight monographs, whichwere written by Kolchin [11, 13] (1984, 2000), Sachkov [23–25] (1977, 1978, 2004), Timashev [28, 29](2011), and the author [32] (2005).

Throughout the paper, we will denote by |X| the number of elements of a finite set X. In thepresent paper, we will assume that

p(n) ≡ |Tn(A)|n!

= n�−1L(n), n ∈ N, (1.1)

where the function L(n) is slowly varying at infinity [27] and � ∈ (0, 1]. There is a wide range ofexamples of sets A that satisfy (1.1) (see the studies by Bender [1], Volynets [9], Kolchin [12, 7],Pavlov [16–19], and the author [32, Ch. 3]; a comprehensive survey of these studies is contained

a Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia.E-mail address: [email protected]

298

RANDOM A-PERMUTATIONS AND BROWNIAN MOTION 299

in [31]). Introduce random processes{

Xn(t) =∑

m≤nt

ζmn, t ∈ [0, 1]

}

and{

Yn(t) =Xn(t) − l(nt)√

� ln t, t ∈ [0, 1]

}

,

where

l(u) =

⎧⎪⎨

⎪⎩

∑

i∈A, i≤u

1i

if u ≥ min A,

0 if u < min A.

(1.2)

We will show that under assumption (1.1), the finite-dimensional distributions of the random process{Yn(t), t ∈ [0, 1]} converge weakly as n → ∞ to the finite-dimensional distributions of the standardWiener process {W (t), t ∈ [0, 1]}.

In the case of A = N, stronger results have been obtained. For example, DeLaurentis andPittel [3] showed that in this case the processes {Yn(t), t ∈ [0, 1]} converge in distribution in thespace D[0, 1] with the Skorokhod topology to the process {W (t), t ∈ [0, 1]} as n → ∞. Note thatsince the trajectories of the limit process {W (t), t ∈ [0, 1]} are continuous, the convergence to thisprocess in the Skorokhod topology is equivalent to convergence in the uniform topology. Hansen [6]proved an analogous functional limit theorem in the case when the set Sn of degree n permutationsis equipped with the Ewens measure [5], which assigns the appearance probability θk/θ(n) to eachpermutation in Sn with k cycles, where θ(n) = θ(θ + 1) . . . (θ + n − 1) and θ is a real positivenumber. Donnelly, Kurtz, and Tavare [4] reproved the same result by a different method. Ivchenkoand Medvedev [10] considered the random processes

{

Ln(t) =∑

i≤t

χ{ζin ≥ 1}, t ∈ [1, n]

}

and

{

Mn(t) =∑

i≤t

χ{ζin = 1}, t ∈ [1, n]

}

(here χ{B} is the indicator of the event B) and showed that for the equiprobable distribution on Sn,the corresponding “adjusted” random processes

{

L∗n(t) =

Ln(nt) − t ln n√ln n

, t ∈ [0, 1]}

and{

M∗n(t) =

Mn(nt) − t ln n√ln n

, t ∈ [0, 1]}

converge as n → ∞ in distribution to the same Wiener process {W (t), t ∈ [0, 1]}.However, the proof of the convergence of the processes {Yn(t), t ∈ [0, 1]} in the Skorokhod

topology to the process {W (t), t ∈ [0, 1]} in the case of an arbitrary set A encounters seriousanalytic difficulties. In particular, the methods used by the above-mentioned scientists fail to workhere. Therefore, the problem remains open for the present.

2. THE MAIN RESULT

Below we proveTheorem 1. Let relation (1.1) be satisfied. Then the finite-dimensional distributions of the

random process {Yn(t), t ∈ [0, 1]} converge weakly as n → ∞ to the finite-dimensional distributionsof the standard Wiener process {W (t), t ∈ [0, 1]}.

Remark 1. If (1.1) holds, then (see [31]) the limit

limn→∞

|k : k ∈ A, k ≤ n|n

= � (2.1)

PROCEEDINGS OF THE STEKLOV INSTITUTE OF MATHEMATICS Vol. 282 2013

300 A.L. YAKYMIV

exists and is called the asymptotic density of the set A in number theory (see Postnikov’s book [20,Sect. 3.1]). The converse is not true. For example, if A is the set of even numbers, then p(n) = 0for odd n.

Let us give some examples in which hypothesis (1.1) of Theorem 1 holds.Example 1. Suppose that the set A has unit asymptotic density, i.e., relation (2.1) holds with

� = 1. Then relation (1.1) (see [32, Sect. 3.1]) and, hence, the assertion of Theorem 1 are valid.Example 2. Suppose that starting from some number the elements of A form a periodic

sequence and there exists an r ∈ N such that the set A ∩ {s + 1, . . . , s + r} cannot be embedded inany lattice of integers with step greater than 1 for any nonnegative integer s. Then (see Kolchin’spapers [12, 7]) relation (1.1) is satisfied.

Example 3. Suppose that a function f defined on N satisfies the following conditions:

(1) f(m1m2) = f(m1)f(m2) for m1,m2 ∈ N, (m1,m2) = 1 (here and below, (m1,m2) is thegreatest common divisor of the numbers m1 and m2), and f(1) = 1;

(2) f takes only the values −1, 0, and 1;(3) for an arbitrary k ∈ N, k > 1, the function

ϕ(m) =∑

dk |m

f(d)

takes only the values 0 and 1.

Let A = A(f, k) be the set of m ∈ N such that ϕ(m) = 1. Then (see Pavlov’s paper [16]) (1.1)is valid. In [16], Pavlov presented known examples of functions f in number theory—such as theMobius function and the Liouville function—that satisfy the indicated requirements (1)–(3).

Example 4. Let M ∈ N, 1 ≤ i ≤ M , Ai = {m ∈ N : m = aik + bi, k = 0, 1, 2, . . .} withintegers ai > 1 and 1 ≤ bi ≤ ai − 1 such that (ai, bi) = 1, A =

⋃Mi=1 Ai, and the progressions Ai

and Aj be disjoint for i = j. Then (see Pavlov’s paper [19]) (1.1) is valid.Example 5. Let k1, . . . , ks ∈ N be such that

(1) ki ≥ 2 for i = 1, . . . , s;(2) (ki, kj) = 1 for i = j.

Set A = {m ∈ N : ki � m, i = 1, . . . , s}. Then (see Pavlov’s paper [19]) (1.1) is valid.Example 6. Suppose that (2.1) is satisfied and, for an arbitrary constant C > 1,

|k : k ≤ n, k ∈ A, m − k ∈ A|n

→ �2 (2.2)

uniformly in m ∈ [n,Cn] as n → ∞ (see [32, Ch. 3]). Then relation (1.1) holds.Note that Section 3.5 in the book [32] is devoted to the analysis of examples of sets A satisfy-

ing (2.2) and (2.1).Example 7. Let A be a random set, the random variables ξn = χ{n ∈ A}, n ∈ N, be jointly

independent, and pn = P{ξn = 1} → � > 0 as n → ∞, n ∈ B, for some set B ⊆ N of asymptoticdensity 1. Then relations (2.2) and (2.1) hold almost surely (a.s.) for A (see [32, Sect. 3.6]) and,hence,

P{Yn(t1) ≤ x1, . . . , Yn(tk) ≤ xk

∣∣ A

}→ P

{W (t1) ≤ x1, . . . ,W (tk) ≤ xk

}a.s.

as n → ∞ for arbitrary fixed k ∈ N, x1, . . . , xk ∈ R, and 0 ≤ t1 < . . . < tk ≤ 1.



Remark 2. In [18], using sets A satisfying (1.1), Pavlov constructed some other sets thatsatisfy this relation. For example, if A1, A2 ⊆ N are two disjoint sets and

pi(n) ≡ |Tn(Ai)|n!

= n�i−1Li(n), n ∈ N, i = 1, 2,

where �i > 0 and the functions Li(n) are slowly varying at infinity (i = 1, 2), then the set A =A1 ∪ A2 satisfies (1.1); moreover, � = �1 + �2 and

L(n) ∼ Γ(�1)Γ(�2)Γ(�)

L1(n)L2(n), n → ∞.

3. PROOFS

Just as in Sevastyanov’s book [26], here we will study the generating functions of the randomprocesses under consideration. To avoid cumbersome calculations, we prove Theorem 1 for two-dimensional distributions; the proof will it make clear how to proceed in the multidimensional case.For n, n1, n2 ∈ N, n1 ≤ n2, set

Z1(n1, n) =∑

i∈A(n1)

ζin, Z2(n1, n2, n) =∑

i∈A(n1,n2)

ζin,

where A(n1) = A ∩ [1, n1] and A(n1, n2) = A ∩ (n1, n2]. According to [24, formula (0.14)], forarbitrary t ∈ [0, 1) and si ∈ [0, 1], i ∈ N, we have

∑

n≥0

tnp(n)En∏

i=1

sζini =

∑

n≥0

tnp(n)E∞∏

i=1

sζini = exp

(

f(t) +∑

i∈A

(si − 1)ti

i

)

, (3.1)

where the function p(n) for n ∈ N is defined in (1.1), p(0) = 1, and

f(t) =∑

i∈A

ti

i. (3.2)

Therefore, for u, v ∈ [0, 1] and t ∈ [0, 1), we have

∑

n≥0

tnp(n)EuZ1(n1,n)vZ2(n1,n2,n) = exp

(

f(t) + (u − 1)∑

i∈A(n1)

ti

i+ (v − 1)

∑

i∈A(n1,n2)

ti

i

)

. (3.3)

In (3.3), we set

u = exp(

x1√l(j)

)

= h1(j), v = exp(

x2√l(j)

)

= h2(j) (3.4)

for fixed nonnegative numbers x1 and x2 and a variable j ∈ Z+ = {0, 1, 2, . . .}, with 1/√

l(j)assumed to be equal to 0 for j < minA. In addition, for n, n1, n2 ∈ N, n1 ≤ n2, let

a(j, n1, n2, n) = p(n)E exp(

x1Z1(n1, n)

√l(j)

)

exp(

x2Z2(n1, n2, n)

√l(j)

)

. (3.5)

Relations (3.3)–(3.5) imply

∑

n≥0

tna(j, n1, n2, n) = exp

(

f(t) + (h1(j) − 1)∑

i∈A(n1)

ti

i+ (h2(j) − 1)

∑

i∈A(n1,n2)

ti

i

)

(3.6)


302 A.L. YAKYMIV

for n1, n2 ∈ N, n1 ≤ n2, and j ∈ Z+. Fix β > 1. From (3.6), we obtain the following equality forthe generating function A(s, u, v, t) of the numbers jβ−1a(j, n1, n2, n):

A(s, u, v, t) ≡∑

j,n≥00≤n1≤n2

jβ−1a(j, n1, n2, n)sjun1vn2tn

=∑

j≥00≤n1≤n2

jβ−1 exp

(

f(t) + (h1(j) − 1)∑

i∈A(n1)

ti

i+ (h2(j) − 1)

∑

i∈A(n1,n2)

ti

i

)

, (3.7)

where s, u, v, t ∈ (0, 1). Let us describe the scheme of the proof of Theorem 1:

1. Analysis of the asymptotics of A(e−λ/n, e−μ/m, e−ν/k, e−θ/n) as n → ∞ for arbitrary fixedλ, μ, ν, θ > 0 and arbitrary positive sequences k = k(n) → ∞ and m = m(n) → ∞, k = o(m).

2. Analysis of the asymptotics of a(xn, ym, zk, un) for arbitrary fixed x, y, z, u > 0 as n → ∞based on the preceding step and the corresponding Tauberian lemma.

3. Proof of the weak convergence of the two-dimensional distributions of the process {Yn(t),t ∈ [0, 1]} to the two-dimensional distributions of the standard Wiener process as n → ∞ based onthe preceding step and the corresponding modification of the Curtiss theorem.

Now we proceed to implement this scheme.Lemma 1. Suppose that relation (1.1) is valid. Then, for some β > 1, arbitrary fixed

λ, μ, ν, θ > 0, and arbitrary positive sequences k = k(n) → ∞ and m = m(n) → ∞, k = o(m), itholds that

A(e−λ/n, e−μ/m, e−ν/k, e−θ/n)kmn2r(n)

→ Γ(β)Γ(�)λβμνθ�

(3.8)

as n → ∞, where Γ(·) is the Euler gamma function and

r(n) = Cnβ−1 exp(

x1l(k)

√l(n)

+ x2l(k,m)√

l(n)

)

p(n), C = exp(

x21t12

+x2

2(t2 − t1)2

)

(3.9)

(the generating function A(s, u, v, t) is defined in (3.7)).Proof. Since

∞∑

l=0

p(l)tl = exp(f(t)) (3.10)

(see (3.3) for u = v = 1), by Karamata’s Abelian theorem and in view of (1.1) we have

exp(f(t)) ∼ np(n)θ−� Γ(�) (3.11)

for t = exp(−θ/n) as n → ∞. It follows from (3.11) and (3.10) that

A(s, u, v, t) ∼ np(n)θ−� Γ(�)Σ (3.12)

as n → ∞, where

Σ ≡∑

j≥00≤n1≤n2

jβ−1sjun1vn2 exp

(∑

i∈A(n1)

ti

i(h1(j) − 1) +

∑

i∈A(n1,n2)

ti

i(h2(j) − 1)

)

. (3.13)



It remains to find the asymptotics for Σ. We have

0 ≤∑

i∈A(n1)

1 − ti

i≤

∑

i∈A(n1)

iθ

in≤ n1θ

n≤ θ

for n1 ≤ n. Similarly,

0 ≤∑

i∈A(n1,n2)

1 − ti

i≤

∑

i∈A(n1,n2)

iθ

in≤ (n2 − n1)θ

n≤ θ

for n2 − n1 ≤ n. Therefore,

Σ′ ≡∑

j≥0, 0≤n1≤n2n1≤n, n2−n1≤n

jβ−1sjun1vn2 exp

(∑

i∈A(n1)

1 − ti

i(h1(j) − 1) +

∑

i∈A(n1,n2)

1 − ti

i(h2(j) − 1)

)

≤∑

j,n1,n2≥0

jβ−1sjun1vn2 exp(θ(h1(j) − 1) + θ(h2(j) − 1)

)

= O(1)∑

j,n1,n2≥0

jβ−1sjun1vn2 = O(1)nβ 11 − u

11 − v

= O(nβmk) (3.14)

as n → ∞ (recall that u and v depend on n). Next,

∑

i∈A(n1)

ti

i(h1(j) − 1) +

∑

i∈A(n1,n2)

ti

i(h2(j) − 1) ≤

∑

i∈A(n2)

ti

i(h(j) − 1),

where h(j) = exp(x/

√l(j)

)and x = max(x1, x2). By virtue of [32, Lemma 3.3.1],

∑

i∈A(n2)

ti

i≤ f(t) = l(n) + O(1)

as n → ∞. Hence, for sufficiently large n ≥ n′, we have

Σ′′ ≡∑

j≥0, 0≤n1≤n2max(n1,n2−n1)≥n

jβ−1sjun1vn2 exp

(∑

i∈A(n1)

ti

i(h1(j) − 1) +

∑

i∈A(n1,n2)

ti

i(h2(j) − 1)

)

≤ 2∑

j≥0, 0≤n1≤n2max(n1,n2−n1)≥n

jβ−1sjun1vn2 exp(l(n)(h(j) − 1)

)≤ 2

∑

n1,j≥0n2≥n

jβ−1sjun1vn2 exp(l(n)(h(j) − 1)

)

= 2∑

j≥0

jβ−1sj exp(l(n)(h(j) − 1)

) ∑

n1≥0, n2≥n

un1vn2 = O(1)nβ exp(x√

l(n)) 11 − u

vn

1 − v

= O(1)nβ exp(x√

l(n))km exp

(− νn

k

)= O(nβmk). (3.15)

Here we used Lemma 3.2.1 from [32] and the fact that k � nt1 . Next, we show that

Σ1 ≡∑

j≥00≤n1≤n2

jβ−1sjun1vn2 exp(l(n1)(h1(j) − 1) + l(n1, n2)(h2(j) − 1)

)

∼ C(μ, ν, λ, β, x1, x2)mknβL1(n),


304 A.L. YAKYMIV

where the slowly varying function L1(n) → ∞ as n → ∞. Therefore, it will follow from (3.15)and (3.14) that Σ ∼ Σ1 as n → ∞. First, consider the following part of the sum Σ1:

Σ2 ≡∑

j≥0jt1/ ln j≤n1≤jt1 ln jjt2/ ln j≤n2≤jt2 ln j

jβ−1sjun1vn2 exp(l(n1)(h1(j) − 1) + l(n1, n2)(h2(j) − 1)

)

∼∑


jβ−1sjun1vn2 exp(

x1l(n1)√

l(j)+ x2

l(n1, n2)√l(j)

)

exp(

x21t12

+x2

2(t2 − t1)2

)

= C∑


jβ−1sjun1vn2 exp(

x1l(n1)√

l(j)+ x2

l(n1, n2)√l(j)

)

∼ C1

1 − u

11 − v

∑

j≥0

jβ−1sj exp(

x1l(jt1)√

l(j)+ x2

l(jt1 , jt2)√

l(j)

)

∼ Cmk

μν

∑

j≥0

jβ−1sj exp(

x1l(jt1)√

l(j)+ x2

l(jt1 , jt2)√

l(j)

)

∼ Cmk

μνλβΓ(β)nβ exp

(

x1l(k)

√l(n)

+ x2l(k,m)√

l(n)

)

, (3.16)

where

C = exp(

x21t12

+x2

2(t2 − t1)2

)

.

We have used the fact that for i = 1, 2 and j → ∞

hi(j) − 1 =xi√l(j)

+x2

i

2l(j)+ O

((l(j))−3/2

)

and for x ≤ y

l(y) ≤ l(x) + lny

x.

In exactly the same way as in [34], we can show that the remaining parts of the sum Σ1 are “o”with respect to the asymptotics (3.16). Thus,

Σ ∼ Cmk

μνλβΓ(β)nβ exp

(

x1l(k)

√l(n)

+ x2l(k,m)√

l(n)

)

.

Finally, taking into account (3.9), (3.10), and (3.13), we conclude that (3.8) holds as n → ∞.Lemma 1 is proved.

Definition 1. Let positive sequences k = k(n) → ∞ and m = m(n) → ∞ be fixed. We saythat a sequence a(j, n1, n2, n) almost increases in the last variable along the sequence (n, k,m, n)if the following inequality holds as n → ∞ for j � n, n1 � k, n2 � m, n1 ≤ n2, w = w(n) > n,w − n = o(n), and fixed x > 0:

lim infn→∞

a(j, n1, n2, wx)a(j, n1, n2, nx)

≥ 1 (3.17)

(for noninteger x, y, z, u > 0, we set a(x, y, z, u) = a([x], [y], [z], [u])).



Definition 2. We say that a sequence a(j, n1, n2, n) almost decreases in the last variable alongthe sequence (n, k,m, n) if the following inequality holds as n → ∞ for j � n, n1 � k, n2 � m,n1 ≤ n2, w = w(n) > n, w − n = o(n), and fixed x > 0:

lim supn→∞

a(j, n1, n2, wx)a(j, n1, n2, nx)

≤ 1. (3.18)

Definition 3. A sequence a(j, n1, n2, n) is said to be almost monotone in the last variable alongthe sequence (n, k,m, n) if one of relations (3.17) or (3.18) is satisfied. Sequences a(j, n1, n2, n) thatare almost monotone in the first, second, and third variables along the sequence (n, k,m, n) aredefined in a similar way.

Lemma 2. The sequence a(j, n1, n2, n) defined by relation (3.5) is almost monotone in eachof the variables along the sequence (n, k,m, n) if (1.1) holds and k = o(m) as n → ∞.

Proof. According to (3.3)–(3.5), we have

∑

n≥0

tna(j, n1, n2, n) = exp

(

f(t) +∑

i∈A(n1)

ti

i(h1(j) − 1) +

∑

i∈A(n1,n2)

ti

i(h2(j) − 1)

)

. (3.19)

By virtue of (3.10), it follows from (3.19) that

a(j, n1, n2, n) = coef tn

∞∑

l=0

p(l)tl∞∑

k=0

(h1(j) − 1)k

k!

(∑

i∈A(n1)

ti

i

)k ∞∑

m=0

(h2(j) − 1)m

m!

(∑

s∈A(n1,n2)

ts

s

)m

= coef tn

∞∑

k,m=0

(h1(j) − 1)k

k!(h2(j) − 1)m

m!

∞∑

l=0

p(l)tl(

∑

i∈A(n1)

ti

i

)k(∑

s∈A(n1,n2)

ts

s

)m

=∞∑

k,m=0

(h1(j) − 1)k

k!(h2(j) − 1)m

m!Skm(n1, n2, n), (3.20)

where

Skm(n1, n2, n) =∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)

p(n − i1 − . . . − ik+m)i1 . . . ik+m

, k,m ∈ N (3.21)

(we assume that p(0) = 1 and p(s) = 0 for s < 0), S00(n1, n2, n) = p(n), and

Sk0(n1, n2, n) =∑

i1,...,ik∈A(n1)

p(n − i1 − . . . − ik)i1 . . . ik

, k ∈ N, (3.22)

S0m(n1, n2, n) =∑

i1,...,im∈A(n1,n2)

p(n − i1 − . . . − im)i1 . . . im

, m ∈ N. (3.23)

For an arbitrary sequence of positive integers w = w(n) > n with r = r(n) = w − n = o(n),there exists a sequence g = g(n) ∈ N such that g(n) → ∞ and r/g(n) → 0 as n → ∞. Forv(n) = n − g(n), set

I1(k,m, n1, n2, n) =∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)i1+...+ik+m∈[v(n),n]

p(n − i1 − . . . − ik+m)i1 . . . ik+m

. (3.24)


306 A.L. YAKYMIV

In the sum (3.24), there is an l = k + 1, . . . , k + m such that il ≥ v(n)/(k + m). Therefore,

I1(k,m, n1, n2, n) ≤ m∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)i1+...+ik+m∈[v(n),n]ik+m≥v(n)/(k+m)

p(n − i1 − . . . − ik+m)i1 . . . ik+m

. (3.25)

In the last sum, we change the variables:

y = n − i1 − . . . − ik+m (ik+m = n − i1 − . . . − ik+m−1 − y).

Since ik+m ≥ v(n)/(k + m), according to (3.25) we have

I1(k,m, n1, n2, n) ≤ m(k + m)v(n)

g(n)∑

y=0

p(y)∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)i1+...+ik+m−1≤n−1

1i1 . . . ik+m−1

≤ m(k + m)v(n)

g(n)∑

y=0

p(y)∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)

1i1 . . . ik+m−1

=m(k + m)

v(n)

g(n)∑

y=0

p(y)l(n1)kl(n1, n2)m−1 =m(k + m)

v(n)O(g(n)p(g(n)))l(n1)kl(n1, n2)m−1 (3.26)

in view of (1.1), where the function l(t) is defined in (1.2) and the function l(n1, n2) = l(n2) − l(n1).Since g(n) = o(n), it follows from (1.1) and (3.26) that

I1(k,m, n1, n2, n) ≤ δ(n)m(k + m)

v(n)np(n)l(n1)kl(n1, n2)m−1

= δ1(n)m(k + m)p(n)l(n1)kl(n1, n2)m−1, (3.27)

where δ(n) and δ1(n) tend to 0 as n → ∞. For c ∈ (0, 1), set

I2(k,m, n1, n2, n) =∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)

i1+...+ik+m∈(cn,n]

p(n − i1 − . . . − ik+m)i1 . . . ik+m

(3.28)

and

I3(k,m, n1, n2, n) =∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)

i1+...+ik+m≤cn

p(n − i1 − . . . − ik+m)i1 . . . ik+m

. (3.29)

Estimating I2(k,m, n1, n2, n) in exactly the same way as I1(k,m, n1, n2, n), we obtain

I2(k,m, n1, n2, n) = O(m(k + m)p(n)l(n1)kl(n1, n2)m−1

), (3.30)

where the constant in “O” is independent of k, m, n1, and n2. Next, due to (1.1), we have

I3(k,m, n1, n2, n) = O(p(n))∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)

i1+...+ik+m≤cn

1i1 . . . ik+m

,



where the constant in “O” is independent of k, m, n1, and n2. In turn,

∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)

i1+...+ik+m≤cn

1i1 . . . ik+m

≤∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)

1i1 . . . ik+m

= l(n1)kl(n1, n2)m.

Therefore,

I3(k,m, n1, n2, n) = O(p(n)l(n1)kl(n1, n2)m

). (3.31)

Now let us estimate from below the function Skm(n1, n2, n) for k ≤ cn/2 and mn1 ≤ cn/2. Accordingto (3.21) and (1.1),

Skm(n1, n2, n) =∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)

p(n − i1 − . . . − ik+m)i1 . . . ik+m

≥∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)

i1+...+ik+m≤cn

p(n − i1 − . . . − ik+m)i1 . . . ik+m

≥ c1p(n)∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)

i1+...+ik+m≤cn

1i1 . . . ik+m

.

Fix an arbitrary ε ∈ (0, 1 − t1). For m ≤ nε, the following inequalities hold:

cn

2m≥ c

2n1−ε ≥ n1.

Here the right inequality holds for sufficiently large n ≥ n0, because n1 � nt1 as n → ∞. Therefore,for n ≥ n0 we have

Skm(n1, n2, n) ≥ c1p(n)∑

i1,...,ik∈A(τ1)ik+1,...,ik+m∈A(n1,τ2)

1i1 . . . ik+m

= c1p(n)l(τ1)kl(n1, τ2)m, (3.32)

where

τ1 = min( cn

2k, n1

), τ2 = min

( cn

2m,n2

).

Below, we assume that the following inequalities are satisfied for arbitrary fixed constants M,M1 > 0and 0 < u1 < u2 < ∞:

k ≤ M√

ln n, m ≤ M1

√ln n (3.33)

and

u1nt1 ≤ n1 ≤ u2n

t1 . (3.34)

Then, in view of (3.33) and (3.34), we have

n1 ≤ u2nt1 ≤ cn

2M√

lnn≤ cn

2k(3.35)

for n ≥ n3 with some n3 ≥ n0. Therefore, (3.32) implies that

Skm(n1, n2, n) ≥ c1p(n)l(n1)kl(n1, τ2)m (3.36)


308 A.L. YAKYMIV

for n ≥ n3. If n2 ≤ cn/(2m), then it follows from (3.36) that

Skm(n1, n2, n) ≥ c1p(n)l(n1)kl(n1, n2)m (3.37)

for n ≥ n3. Let n2 > cn/(2m). Since n2 � nt2 , we have t2 = 1 and

s1n ≤ n2 ≤ s2n (3.38)

for some constants 0 < s1 < s2 ≤ 1. Then

l(n2) − l( cn

2m

)=

n2∑

i=[cn/(2m)]+1, i∈A

1i≤

n2∑

i=[cn/(2m)]+1

1i≤ ln n − ln

[ cn

2m

]

≤ ln n − ln( cn

2m− 1

)≤ ln n − ln

cn

3m= ln

m

c/3= ln m + c2 (3.39)

for m ≤ cn/6, where c2 = ln(3/c). It follows from (3.39) that

l(n1, τ2)m =(l(n1, n2) −

(l(n2) − l

( cn

2m

)))m≥

(l(n1, n2) − ln m − c2

)m (3.40)

for m ≤ cn/6. Since

l(n1, n2) = l(n2) − l(n1) = (1 + o(1))�(t2 − t1) ln n = (1 + o(1))�(1 − t1) ln n

as n → ∞ (see (3.35) and (3.38)), there are constants �1 ∈ (0, �(1 − t1)) and n4 ≥ n3 such that

l(n1, n2) ≥ �1 ln n

for n ≥ n4, and (3.25) implies that for n ≥ n4

l(n1, τ2)m ≥ l(n1, n2)m(

1 − ln m + c2

l(n1, n2)

)m

≥ l(n1, n2)m(

1 − ln m + c2

�1 ln n

)m

= l(n1, n2)m(

1 + O

(m lnm

ln n

))

= l(n1, n2)m(1 + o(1))

as n → ∞, according to (3.33). Thus, there exists an n5 ≥ n4 such that

l(n1, τ2)m ≥ 12l(n1, n2)m

for n ≥ n5. Therefore, it follows from (3.36) that

Skm(n1, n2, n) ≥ c1

2p(n)l(n1)kl(n1, n2)m (3.41)

for n ≥ n5. From (3.30) and (3.31) we obtain

Skm(n1, n2, n) = I2(k,m, n1, n2, n) + I3(k,m, n1, n2, n)

= O(m(k + m)p(n)l(n1)kl(n1, n2)m−1

)+ O

(p(n)l(n1)kl(n1, n2)m

)

= O(m(k + m)p(n)l(n1)kl(n1, n2)m

). (3.42)



For i = 1, 2, according to (3.4), we have

hi(j) − 1 = exp(

x1√l(j)

)

− 1 = (1 + o(1))xi√� ln j

= (1 + o(1))xi√� ln n

for n → ∞ and j � n. Hence, for arbitrary constants s1 and s2, 0 < s1 < s2 < ∞, there exists ann6 ≥ n5 such that

hi(j) − 1 ≤ maxi=1,2 xi + 1√

�√

ln n≡ c3√

ln n(3.43)

for n ≥ n6 and j ∈ [s1n, s2n]. Next, in view of (3.43),

∞∑

k=0

m(k + m)p(n)l(n1)kl(n1, n2)m(h1(j) − 1)k

k!

≤ m2p(n)l(n1, n2)m∞∑

k=0

l(n1)k(ln n)−k/2

k!ck3 + mp(n)l(n1, n2)m

∞∑

k=0

kl(n1)k(ln n)−k/2

k!ck3

= m2p(n)l(n1, n2)m exp(

c3l(n1)√

ln n

)

+ mp(n)l(n1, n2)mc3l(n1)√

ln n

∞∑

i=0

l(n1)i(ln n)−i/2

i!ci3

= m2p(n)l(n1, n2)m exp(

c3l(n1)√

ln n

)

+ mp(n)l(n1, n2)mc3l(n1)√

ln nexp

(

c3l(n1)√

ln n

)

= mp(n)l(n1, n2)m exp(

c3l(n1)√

ln n

)(

m + c3l(n1)√

lnn

)

(3.44)

for n ≥ n6 and j ∈ [s1n, s2n]. Since n1 � nt1 , there exist positive constants v1 and v2 such that

v1nt1 ≤ n1 ≤ v2n

t1 (3.45)

for sufficiently large n. Thus,

l(n1) = (1 + o(1))� ln n1 = (1 + o(1))t1� ln n (3.46)

as n → ∞. Therefore, there exists an n7 ≥ n6 such that

c3l(n1)√

ln n≤ c3

(t1� + 1) ln n√ln n

≡ c4

√ln n (3.47)

for n ≥ n7. It follows from (3.44) and (3.47) that for n ≥ n7, s1n ≤ j ≤ s2n, and v1nt1 ≤ n1 ≤ v2n

t1 ,

∞∑

k=0

m(k + m)p(n)l(n1)kl(n1, n2)m(h1(j) − 1)k

k!≤ mp(n)l(n1, n2)m exp

(c4

√ln n

)(m + c4

√lnn

)

≤ 2m2p(n)l(n1, n2)m exp(c4

√ln n

)(3.48)

if m ≥ c4

√ln n. By virtue of (3.43), for an arbitrary constant M1 ≥ c4 we have

∑

m>M1

√ln n

m2l(n1, n2)m(h2(j) − 1)m

m!≤

∑

m>M1

√ln n

m2l(n1, n2)m(

c3√ln n

)m 1m!

(3.49)


310 A.L. YAKYMIV

for n ≥ n7 and s1n ≤ j ≤ s2n. Applying the Stirling formula for m!, from (3.49) we find that

∑

m>M1

√ln n

m2l(n1, n2)m(h2(j) − 1)m

m!≤ c5

∑

m>M1

√ln n

m3/2

(ec3l(n1, n2)√

ln n

)m

m−m (3.50)

for some positive constant c5, n ≥ n8 with n8 ≥ n7, and s1n ≤ j ≤ s2n. It follows from condi-tion (1.1) of Theorem 1 (see [33]) that l(n) ∼ � ln n as n → ∞; hence,

l(n1, n2) = l(n1) − l(n2) = (1 + o(1))t2� ln n − (1 + o(1))t1� ln n

= (1 + o(1))(t2 − t1)� ln n (3.51)

for n1 � nt1 and n2 � nt2 . Therefore, there exists an n8 ≥ n7 such that

l(n1, n2) ≤ t2� ln n ≡ c6 ln n (3.52)

for n ≥ n8, v1nt1 ≤ n1 ≤ v2n

t1 , and u1nt2 ≤ n2 ≤ u2n

t2 . Set c7 = c3c6. It follows from (3.50)and (3.51) that

∑

m>M1

√ln n

m2l(n1, n2)m(h2(j) − 1)m

m!≤ c5

∑

m>M1

√lnn

m3/2(ec7

√ln n

)mm−m

= c5

∑

m>M1

√lnn

m3/2

(ec7

√ln n

m

)m

(3.53)

for n ≥ n8, v1nt1 ≤ n1 ≤ v2n

t1 , u1nt2 ≤ n2 ≤ u2n

t2 , and s1n ≤ j ≤ s2n. Hence,

∑

m>M1

√ln n

m3/2

(ec7

√ln n

m

)m

≤∑

m>e1+cc7√

ln n

m3/2e−mc = O(ln5/4 n) exp(−e1+ccc7

√ln n

)(3.54)

for M1 = e1+cc7, where the constant c is chosen so that M1 ≥ c4. Relations (3.53) and (3.54) implythat for such a choice of M1,

∑

m>M1

√lnn

m2l(n1, n2)m(h2(j) − 1)m

m!= O(ln5/4 n) exp

(−e1+ccc7

√ln n

). (3.55)

From (3.42), (3.48), and (3.55), we find that in view of the choice of the constant c,

Σ1 ≡∑

m>M1

√lnn

∞∑

k=0

Skm(n1, n2, n)(h1(j) − 1)k

k!(h2(j) − 1)m

m!

= O(p(n) exp

(c4

√ln n

))ln5/4 n exp

(−e1+ccc7

√ln n

)= o(p(n)) (3.56)

for n → ∞, n1 � nt1 , n2 � nt2 , and j � n. In exactly the same way we verify that there existsa constant M > 0 such that

Σ2 ≡∑

k>M√

ln n

∞∑

m=0

Skm(n1, n2, n)(h1(j) − 1)k

k!(h2(j) − 1)m

m!= o(p(n)) (3.57)



for n → ∞, n1 � nt1 , n2 � nt2 , and j � n. It follows from (3.20), (3.56), and (3.57) that

0 ≤ a(j, n1, n2, n) −[M

√ln n]∑

k=0

[M1

√ln n]∑

m=0

Skm(n1, n2, n)(h1(j) − 1)k

k!(h2(j) − 1)m

m!

≤ Σ1 + Σ2 = o(p(n)) (3.58)

for n → ∞, n1 � nt1 , n2 � nt2 , and j � n. To check that

lim infa(j, n1, n2, w)a(j, n1, n2, n)

≥ 1 (3.59)

for n → ∞, w > n, r = w − n = o(n), n1 � nt1 , n2 � nt2 , and j � n, in view of (3.58) it suffices toverify that

lim infb(j, n1, n2, w)b(j, n1, n2, n)

≥ 1 (3.60)

under the same conditions on j, n1, n2, n, and w, where

b(j, n1, n2, n) =[M

√ln n]∑

k=0

[M1

√ln n]∑

m=0

Skm(n1, n2, n)(h1(j) − 1)k

k!(h2(j) − 1)m

m!. (3.61)

Let us check (3.60). Set

I4(k,m, n1, n2, n) =∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)

i1+...+ik+m<v(n)

p(n − i1 − . . . − ik+m)i1 . . . ik+m

, (3.62)

where v(n) is the same as in (3.24). Set a = i1 + . . . + ik+m; then, according to (1.1), we have

p(n − a + r) − p(n − a)p(n − a)

=p(n − a + r)

p(n − a)− 1 =

(1 +

r

n − a

)�−1 L(n − a + r)L(n − a)

− 1

=(1 +

r

n − a

)�−1(1 + o(1)) − 1 = o(1) + (1 + o(1))

r(� − 1)n − a

= o(1) (3.63)

uniformly over n − a > g(n), because r(n) = o(g(n)) as n → ∞ by the hypothesis. It followsfrom (3.62) and (3.63) that

|I4(k,m, n1, n2, w) − I4(k,m, n1, n2, n)| ≤ ε(r, n)I4(k,m, n1, n2, n) ≤ ε(r, n)Skm(n1, n2, n), (3.64)

where ε(r, n) → 0 as n → ∞ for r(n) = o(n). Next,

I4(k,m, n1, n2, w) − I4(k,m, n1, n2, n) ≥ −I4(k,m, n1, n2, n). (3.65)

According to (3.27),

I1(k,m, n1, n2, n) ≤ δ1(n)m(k + m)p(n)l(n1)kl(n1, n2)m−1

≤ δ1(n)M1(M + M1)p(n)l(n1)kl(n1, n2)m−1 ln n

= δ1(n1, n2, n)p(n)l(n1)kl(n1, n2)m (3.66)


312 A.L. YAKYMIV

for k ≤ M√

ln n and m ≤ M1

√ln n, where δ1(n1, n2, n) → 0 as n → ∞ for n1 � nt1 and n2 � nt2 ,

because l(n1, n2) ∼ �(t2 − t1) ln n. It follows from (3.51) and (3.66) that for n ≥ n9 with somen9 ≥ n8 and for k ≤ M

√lnn, m ≤ M1

√ln n, n1 ∈ [u1n

t1 , u2nt1 ], and n2 ∈ [v1n

t2 , v2nt2],

I1(k,m, n1, n2, n) ≤ δ3(n1, n2, n)Skm(n1, n2, n). (3.67)

From (3.63), (3.64), and (3.67) we find that for n ≥ n9 and k ≤ M√

ln n, m ≤ M1

√lnn, n1 ∈

[u1nt1 , u2n

t1 ], and n2 ∈ [v1nt2 , v2n

t2 ],

Skm(n1, n2, w) − Skm(n1, n2, n)

=(I4(k,m, n1, n2, w) − I4(k,m, n1, n2, n)

)+

(I1(k,m, n1, n2, w) − I1(k,m, n1, n2, n)

)

≥ −(ε(r, n) − δ3(n1, n2, n)

)Skm(n1, n2, n) = −ε1(r, n1, n2, n)Skm(n1, n2, n), (3.68)

where ε1(r, n1, n2, n) → 0 as n → ∞ for n1 � nt1 , n2 � nt2 , and r = o(n). It follows from (3.61)and (3.68) that

b(j, n1, n2, w) − b(j, n1, n2, n)

≥[M

√lnn]∑

k=0

[M1

√lnn]∑

m=0

(Skm(n1, n2, w) − Skm(n1, n2, n)

) (h1(j) − 1)k

k!(h2(j) − 1)m

m!

≥ −ε1(r, n1, n2, n)[M

√ln n]∑

k=0

[M1

√ln n]∑

m=0

Skm(n1, n2, n)(h1(j) − 1)k

k!(h2(j) − 1)m

m!

= −ε1(r, n1, n2, n)b(j, n1, n2, n).

Since ε1(r, n1, n2, n) → 0 as n → ∞ for n1 � nt1 , n2 � nt2 , and r = o(n), this proves (3.60). Next,for r, k ∈ [1, n2 − n1] and 1 ≤ n1 < n2 ≤ n,

Skm(n1 + r, n2, n) − Skm(n1, n2, n)

=∑

i1,...,ik∈A(n1+r)ik+1,...,ik+m∈A(n1+r,n2)

p(n − i1 − . . . − ik+m)i1 . . . ik+m

−∑

i1,...,ik∈A(n1)ik+1,...,ik+m∈A(n1,n2)

p(n − i1 − . . . − ik+m)i1 . . . ik+m

≤∑

i1,...,ik∈A(n1+r)ik+1,...,ik+m∈A(n1+r,n2)∃j∈[1,k] : ij∈A(n1,n1+r)

p(n − i1 − . . . − ik+m)i1 . . . ik+m

≤∑

i1,...,ik∈A(n1+r)ik+1,...,ik+m∈A(n1,n2)∃j∈[1,k] : ij∈A(n1,n1+r)

p(n − i1 − . . . − ik+m)i1 . . . ik+m

=k−1∑

l=0

(k

l

) ∑

i1,...,il∈A(n1)il+1,...,ik∈A(n1,n1+r)ik+1,...,ik+m∈A(n1,n2)

p(n − i1 − . . . − ik+m)i1 . . . ik+m

≤k−1∑

l=0

1nl

1

(k

l

) ∑

il+1,...,ik∈A(n1,n1+r)

Slm(n1, n2, n − il+1 − . . . − ik). (3.69)

Notice that under the summation conditions in (3.69),

il+1 + . . . + ik ≤ (n1 + r)k = o(n)



for n → ∞, n1 � nt1 , k = O(√

ln n), and r = o(n1). Therefore, in view of (3.68), for someconstant C we have

k−1∑

l=0

1nl

1

(k

l

) ∑

il+1,...,ik∈A(n1,n1+r)

Slm(n1, n2, n − il+1 − . . . − ik)

≤ Ck−1∑

l=0

(r

n1

)k−l(k

l

)

Slm(n1, n2, n) ≤ Cr

n1

k−1∑

l=0

(k

l

)

Slm(n1, n2, n) (3.70)

for n → ∞, n1 � nt1 , k = O(√

ln n), and r = o(n1). Note that

S0m(n1 + r, n2, n) − S0m(n1, n2, n) ≤ 0 (3.71)

for m ∈ Z+ and n1 < n1 + r ≤ n2 ≤ n. According to (3.69)–(3.71) and (3.53),

b(j, n1 + r, n2, n) − b(j, n1, n2, n)

=[M

√lnn]∑

k=0

[M1

√lnn]∑

m=0

(Skm(n1 + r, n2, n) − Skm(n1, n2, n)

) (h1(j) − 1)k

k!(h2(j) − 1)m

m!

≤ Cr

n1

[M√

ln n]∑

k=0

[M1

√lnn]∑

m=0

k−1∑

l=0

Slm(n1, n2, n)(

k

l

)(h1(j) − 1)k

k!(h2(j) − 1)m

m!

≤ Cr

n1

[M√

ln n]∑

k=0

[M1

√lnn]∑

m=0

k−1∑

l=0

Slm(n1, n2, n)(h1(j) − 1)k

l!(h2(j) − 1)m

m!

=r

n1O

(1√ln n

) [M√

ln n]∑

k=0

[M1

√ln n]∑

m=0

k−1∑

l=0

Slm(n1, n2, n)(h1(j) − 1)l

l!(h2(j) − 1)m

m!

=r

n1O

(1√ln n

) [M√

ln n]−1∑

l=0

[M1

√ln n]∑

m=0

Slm(n1, n2, n)(h1(j) − 1)l

l!(h2(j) − 1)m

m!

[M√

ln n]∑

k=0

1

=r

n1O(1)b(j, n1, n2, n) = o(1)b(j, n1, n2, n) (3.72)

for n → ∞, n1 � nt1 , n2 � nt2 , j � n, and r = o(n1). It follows from (3.72) that

lim supb(j, n1 + r, n2, n)

b(j, n1, n2, n)≤ 1 (3.73)

for n → ∞, n1 � nt1 , n2 � nt2 , j � n, and r = o(n1). Relations (3.58) and (3.73) imply that

lim supa(j, n1 + r, n2, n)

a(j, n1, n2, n)≤ 1 (3.74)

for n → ∞, n1 � nt1 , n2 � nt2 , j � n, and r = o(n1). Note that the sequence a(j, n1, n2, n) is mono-tone in the first and third variables. Thus, the almost monotonicity of the sequence a(j, n1, n2, n)is proved. The proof of Lemma 2 is complete.


314 A.L. YAKYMIV

Lemma 3. Suppose that the function

A(s, u, v, t) =∑

j,n≥00≤n1≤n2

jβ−1a(j, n1, n2, n)sjun1vn2tn

(β > 1, a(j, n1, n2, n) ≥ 0) is finite for all s, u, v, t ∈ (0, 1) and

A(e−λ/n, e−μ/m, e−ν/k, e−θ/n)kmn2r(n)

→ Γ(β)Γ(�)λβμνθ�

(3.75)

as n → ∞ for arbitrary fixed λ, μ, ν, θ > 0, some positive sequences k = k(n) → ∞ and m =m(n) → ∞, k = o(m), some � > 0, and a positive function r(n) of n (Γ(·) is the Euler gammafunction). If the sequence a(j, n1, n2, n) is almost monotone along (n, k,m, n) in each variable (seeDefinitions 1–3), then

a(xn, ym, zk, un) ∼ r(n)n1−βu�−1 (3.76)

as n → ∞ for arbitrary fixed x, y, z, u > 0.

Proof. Let B be the family of bounded Borel sets in R4+ = {(x1, x2, x3, x4) ∈ R

4 : xi ≥ 0∀i = 1, 2, 3, 4}. For an arbitrary B ∈ B, set

μn(B) =∑

j,n1,n2,i : (j/n,n1/k,n2/m,i/n)∈B

jβ−1a(j, n1, n2, i)kmn2r(n)

(3.77)

(we assume that a(j, n1, n2, i) = 0 for n1 > n2). It follows from (3.75) that

μn(y) ≡∫

R4+

e−(x,y)μn(dx) =A(e−y1/n, e−y2/m, e−y3/k, e−y4/n)

kmn2r(n)

→ ψ(y) = μ(y) ≡∫

R4+

e−(x,y)xβ−11 x�−1

4 dx1 dx2 dx3 dx4 (3.78)

for an arbitrary y ∈ int R4+ as n → ∞. Thus, the limit function ψ(y) in (3.78) is the Laplace

transform of an absolutely continuous measure μ on B:

μ(B) =∫

B

xβ−11 x�−1

4 dx1 dx2 dx3 dx4. (3.79)

In view of the Tauberian Theorem 1.3.2 in [32], it follows from (3.78) that

μn ⇒ μ

as n → ∞; i.e., for an arbitrary B ∈ B such that μ(∂B) = 0, we have

μn(B) → μ(B), n → ∞. (3.80)

Suppose for definiteness that the function a(j, n1, n2, n) satisfies (3.17) and similar relations arevalid for the first, second, and third arguments of this function. It will be clear from what followshow to proceed in the other cases. Fix an arbitrary x = (x1, x2, x3, x4) ∈ int R

4+. Notice that for



w = w(n) = (w1, w2, w3, w4) ≥ n = (n, k,m, n) (i.e., the corresponding componentwise inequalitieshold) and w1 − n = o(n), w2 − k = o(k), w3 − m = o(n), and w4 − n = o(n), we have

lim infn→∞

a(wx)a(nx)

≥ 1, (3.81)

where wx = (w1x1, w2x2, w3x3, w4x4). Indeed,

a(wx)a(nx)

=a(wx)

a(nx1, w2x2, w3x3, w4x4)a(nx1, w2x2, w3x3, w4x4)a(nx1, kx2, w3x3, w4x4)

× a(nx1, kx2, w3x3, w4x4)a(nx1, kx2,mx3, w4x4)

a(nx1, kx2,mx3, w4x4)a(nx)

.

Since the lower limit of the product of positive factors is not less than the product of their lowerlimits and the lower limit of each of the four factors in the above relation is not less than 1,inequality (3.81) holds.

According to (3.79) and (3.80), for an arbitrary fixed δ = (δ1, δ2, δ3, δ4) ∈ (0, 1)4 we have

Σ ≡∑

i∈Z4, i/n∈B

iβ−11 a(i)

kmn2r(n)→ xβ

1 ((1 + δ1)β − 1)δ2x2δ3x3x�4((1 + δ4)� − 1)

β�, (3.82)

where i/n = (i1/n, i2/k, i3/m, i4/n) and B = {y ∈ R4+ : xi ≤ yi ≤ xi(1 + δi), i = 1, 2, 3, 4}. In view

of (3.81), for an arbitrary ε ∈ (0, 1) there exist δ = (δ1, δ2, δ3, δ4) ∈ (0, 1)4 and n0 ∈ N such that

a(i) ≥ a(nx)(1 − ε) (3.83)

for n ≥ n0 and i ∈ B ∩ Z4, where nx = (nx1, kx2,mx3, nx4). It follows from (3.82) and (3.83) that

Σ ≥ (1 − ε)a(nx)∑

i∈Z4, i/n∈B

iβ−11

kmn2r(n)≥ (1 − ε)

a(nx)(x1n)β−1

kmn2r(n)

∑

i∈Z4, i/n∈B

1.

Hence we obtaina(nx)

n1−βr(n)≤ xβ−1

1

1 − ε

Σkmn2

∑i∈Z4, i/n∈B 1

;

therefore,

lim supn→∞

a(nx)n1−βr(n)

≤ xβ−11

1 − ε

xβ1 ((1 + δ1)β − 1)δ2x2δ3x3x

�4((1 + δ4)� − 1)

β�δ1x1δ2x2δ3x3δ4x4

=1

1 − ε

((1 + δ1)β − 1)x�−14 ((1 + δ4)� − 1)

β�δ1δ4. (3.84)

Since the left-hand side of (3.84) is independent of ε and δ, letting ε and δ on the right-hand sidego to zero yields

lim supn→∞

a(nx)n1−βr(n)

≤ x�−14 .

Replacing B by the set B1 = {y ∈ R4+ : xi(1 − δi) ≤ yi ≤ xi, i = 1, 2, 3, 4} and using similar

inequalities in the opposite direction, we verify that

lim infn→∞

a(nx)n1−βr(n)

≥ x�−14 .

The last two inequalities imply (3.75). The proof of Lemma 3 is complete.


316 A.L. YAKYMIV

All the measures considered below are assumed to be nonnegative, σ-additive, σ-finite, anddefined on the δ-ring A of all bounded Borel sets in R

n.We say that a sequence of measures (Uk, k ∈ N) converges weakly to a measure U and write

Uk ⇒ U (k → ∞) if Uk(A) → U(A) (k → ∞) for every A ∈ A such that U(∂A) = 0 (here andbelow, ∂A denotes the boundary of the set A).

The following generalization of the well-known Curtiss theorem [2] for moment-generating func-tions is valid (see [36]).

Lemma 4. Suppose that for a sequence of measures (Uk, k ∈ N) in some domain D ⊆ Rn

there exist moment-generating functions

Uk(z) =∫

Rn

e(z,x) Uk(dx) < ∞ ∀z ∈ D.

1. If

Uk(z) → ω(z) < ∞ ∀z ∈ D (3.85)

as k → ∞, then ω(z) is a moment-generating function of some measure U on Rn and Uk ⇒ U

as k → ∞.2. Conversely, if Uk ⇒ U as k → ∞ and Uk(a) are bounded for an arbitrary a ∈ ∂D (here

the domain D is assumed to be bounded), then relation (3.85) is satisfied, the measure U has amoment-generating function U(z) for z ∈ D, and U(z) = ω(z) for z ∈ D.

In contrast to the Curtiss theorem, the domain D ⊆ Rn in Lemma 4 does not necessarily

contain the zero point, which turns out to be essential for applications. If the measures Uk areprobabilistic, then for the limit measure U to be probabilistic it suffices that the function ω(z) havean analytic continuation to some neighborhood of zero of the n-dimensional complex space C

n andω(0) = 1. Note also that the existence of a moment-generating function of the measure U in theform defined in Lemma 3 in some domain D not containing zero in no way requires the existenceof any moments of this measure. However, in the name of this transform we follow the traditions(see, for example, [24, 25, 30]). A one-dimensional variant of Lemma 4 is presented in the authors’spaper [35, Lemma 5] and is used there in the proof of a limit theorem for the logarithm of the orderof a random A-permutation.

Proof of Theorem 1. Applying Lemma 4 and relation (3.5), we find that

a(n, k,m, n) = p(n)E exp(

x1Z1(k, n)√

l(n)

)

exp(

x2Z2(k,m, n)

√l(n)

)

∼ Cp(n) exp(

x1l(k)

√l(n)

+ x2l(k,m)√

l(n)

)

as n → ∞; thus

E exp(

x1Z1(nt1 , n)

√l(n)

)

exp(

x2Z2(nt1 , nt2 , n)

√l(n)

)

∼ C exp(

x1l(nt1)√

l(n)+ x2

l(nt1 , nt2)√

l(n)

)

,

or

E exp(

x1Z1(nt1 , n) − l(nt1)

√l(n)

)

exp(

x2Z2(nt1 , nt2 , n) − l(nt1 , nt2)

√l(n)

)

→ C = exp(

x21t12

+x2

2(t2 − t1)2

)

. (3.86)



Since (3.86) holds for all nonnegative x1 and x2, from Lemma 4 we find that the joint distributionsof the random variables Yn(t1) and Yn(t2) − Yn(t1) converge weakly to the joint distribution ofthe random variables W (t1) and W (t2) − W (t1), because it follows from (1.1) that l(n) ∼ �nas n → ∞. Therefore, the two-dimensional distributions of the sequence of random processes{Yn(t), t ∈ [0, 1]} converge weakly as n → ∞ to the two-dimensional distributions of the Wienerrandom process {W (t), t ∈ [0, 1]}. The proof of the convergence of finite-dimensional distributions ofhigher dimension reproduces the previous arguments almost word for word. The proof of Theorem 1is complete.

ACKNOWLEDGMENTS

In conclusion, I express my deep gratitude to the referee for useful remarks that allowed me tosignificantly improve the exposition of the paper and to make it more precise and accurate.

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Translated by I. Nikitin


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Random A-permutations and Brownian motion