1. Introduction and result
Stochastic processes appear in various constructions based upon permutations
$\sigma$
drawn at random from the symmetric group
$\mathbb{S}_n$
. Sometimes their distribution limit as
$n\to\infty$
is Brownian motion (see [Reference Babu and Manstavičius2, Reference DeLaurentis and Pittel13]); in other cases, it is some other process with independent or dependent increments (see, e.g., [Reference Babu and Manstavičius3, Reference Babu, Manstavičius, Zacharovas, Akiyama, Matsumoto, Murata and Sugita4]). The Poisson–Dirichlet process has received greater attention [Reference Arratia, Barbour and Tavaré1, Sections 5.5, 5.7, and 8.2]. In the present paper, we propose a new type of process construction based upon permutation divisors and prove a functional limit theorem in the Skorokhod space
$(\mathbb{D},\mathcal D)$
. Here,
$\mathbb{D}=\mathbb{D}[0, 1]$
denotes the space of right-continuous functions on [0, 1] that have left-hand limits, endowed with the Skorokhod topology, and
$\mathcal D$
represents the Borel sigma-algebra over
$\mathbb{D}$
[Reference Billingsley11, Chapter 3]. Let
$\mathbb{C}=\mathbb{C}[0, 1]$
be the subset of
$\mathbb{D}$
comprising continuous functions. The present paper has been highly influenced by progress in probabilistic number theory; in particular, the problem statement goes back to [Reference Manstavičius and Timofeev30]. Therefore, it is worth taking a wider look at the prehistory.
In 1979 [Reference Deshouillers, Dress and Tenenbaum14] established the following law for the natural divisors d of a uniformly drawn random natural number
$m\in [n]\,:\!=\,\{1,\dots, n\}$
. Namely, it was proved that
\begin{equation} \frac{1}{n}\sum_{m\leq n}\Bigg(\sum_{d\vert m} 1\Bigg)^{-1}\Bigg(\sum_{\substack{d\vert m\\ \log d\leq t\log n}}1\Bigg) = \frac2{\pi}\operatorname{arcsin}\sqrt t+O\big((\log n)^{-1/2}\big)\end{equation}
uniformly in
$0\leq t\leq 1$
and
$n\geq 2$
. Many generalisations of the last relation, nowadays called the DDT theorem, have appeared since then. In particular, [Reference Bareikis and Manstavičius9] dealt with the ratio
\begin{align*} T_n(m,t;\,g) \,:\!=\, \Bigg(\sum_{d\vert m} g(d)\Bigg)^{-1}\sum_{\substack{d\vert m\\ d\leq n^t}} g(d),\quad 0\leq t\leq 1,\end{align*}
where g is a nonnegative multiplicative arithmetic function taking a fixed value
$g(p)=\vartheta>0$
for all primes p and arbitrary values
$g(p^s)$
if
$s\geq 2$
. It was shown that
\begin{equation} \frac{1}{n}\sum_{m\leq n} T_n(m,t;\,g) = B(t;\,\theta,1-\theta) + O\big((\log n)^{-\min\{\theta,1-\theta\}}\big), \quad \theta \,:\!=\, \vartheta/(1+\vartheta).\end{equation}
Here,
\begin{align*} B(t;\,a,b) \,:\!=\, \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\int_0^t\frac{\mathrm{d}v}{v^a(1-v)^b}, \quad 0 < a,\,b < 1,\end{align*}
and
$\Gamma(\!\cdot\!)$
denotes Euler’s gamma function. For a concise bibliography of more recent results, we refer to [Reference Bareikis and Mačiulis8].
Long ago, in a personal discussion with the author, G. Tenenbaum asked if (1) relates to the so-called invariance principle where the arcsine law appears as the distribution of some functionals on the Wiener process. Actually, in the earlier article [Reference Dress15] announcing (1), an attempt to relate it to a random walk problem can be seen. However, in [Reference Manstavičius and Timofeev30], a new stochastic process lying behind the formulas (1) and (2) was revealed by proving a functional limit theorem. More specifically, the sequence
$T_n\,:\!=\,T_n(m,t;\,g)$
was treated as a random process in the Skorokhod space
$(\mathbb{D},\mathcal D)$
under the uniform probability measure on the set [n]. It was proved that
$T_n$
converges in distribution as
$n\to\infty$
to a limit process depending on
$\vartheta$
. In [Reference Tenenbaum34], Tenenbaum relaxed the condition
$g(p)=\vartheta$
for primes p and, in addition, showed that the limit trajectories belong to the space
$\mathbb{C}$
. In the following, we will return in more detail to the process that appears.
Recent decades have been marked by numerous works intertwining probabilistic number theory and random permutations (see, e.g., [Reference Arratia, Barbour and Tavaré1]). Reflecting this, and presenting new results, we will use the following notation and definitions. A permutation
$\sigma\in\mathbb{S}_n$
is a one-to-one (bijective) mapping
$\sigma\colon[n] \to[n]$
. It can be represented by the table
\begin{align*} \sigma=\binom{1\; 2 \; \cdots\; n\,}{i_1\, i_2\, \cdots\, i_n},\end{align*}
where
$\sigma(r)=i_r$
for
$1\leq r\leq n$
, or by the labelled directed graph
$G(\sigma)$
with vertex set
$V(\sigma)=[n]$
and edges
$r\to i_r$
, where
$1\leq r\leq n$
. Its components are oriented cycles. A typical cycle has a vertex set
$V(\varkappa)=\{k_1,\dots, k_j\}\subset[n]$
defined by
\begin{align*} k_1\xrightarrow{\sigma} k_2\xrightarrow{\sigma}\cdots\xrightarrow{\sigma} k_j\xrightarrow{\sigma} k_1.\end{align*}
Using mapping multiplication, we obtain the unique (up to the order of factors) decomposition of
$\sigma$
into cycles
$\varkappa_i$
on pairwise disjoint subsets
$V(\varkappa_i)$
, namely,
\begin{equation} \sigma=\varkappa_1\cdots\varkappa_w.\end{equation}
Here and in what follows,
$w=w(\sigma)$
denotes the number of cycles and
$[n]=V(\varkappa_1)\cup\cdots \cup V(\varkappa_w)$
. Let us also introduce the empty permutation
$\emptyset$
and
$\mathbb{S}_0=\{\emptyset\}$
.
A one-to-one mapping
$\delta\colon V(\delta)\to V(\delta)$
with
$V(\delta)\subset [n]$
is a divisor of
$\sigma$
if
$\sigma\vert_{V(\delta)}=\delta$
. The domain
$V(\delta)$
(the vertex set in the subgraph
$G(\delta)\subset G(\sigma)$
) is a fixed set of
$\sigma$
, that is, it satisfies the relation
$\sigma(V(\delta))=V(\delta)$
. The empty permutation
$\emptyset$
will be considered a divisor of each
$\sigma\in\mathbb{S}_n$
. A divisor is uniquely determined by an appropriate partial product of cycles in (3), thus each
$\sigma\in\mathbb{S}_n$
has
$2^{w(\sigma)}$
divisors. The cycles are analogues of primes in number theory. Following the latter, for a divisor
$\delta$
of
$\sigma$
, we will write
$\delta\vert \sigma$
and
$\sigma=\delta \tau$
, where
$\tau$
is another divisor acting on
$[n]\setminus V(\delta)$
and determined by the cycles from (3) not included in
$\delta$
. The notation
$\tau=\sigma/\delta$
will also be used where convenient.
Let us define some numerical parameters. If
$k_j(\delta)$
denotes the number of cycles in
$\delta$
of length j and
$1\leq j\leq n$
, then
$\bar k(\delta)\,:\!=\,(k_1(\delta),\dots, k_n(\delta))$
,
$\delta\vert\sigma$
, will be called the cycle vector of
$\delta$
. The zero vector corresponds to the empty permutation
$\delta=\emptyset$
. We introduce the notation
$\ell(\bar s)\,:\!=\,1s_1+\cdots+ ns_n$
for a vector
$\bar s=(s_1,\dots, s_n)\in \mathbb{Z}_+^n$
. Throughout the paper, the cardinality
$\#V(\delta)= \ell(\bar k(\delta))\,=\!:\, L(\delta)$
will be called the length of
$\delta$
. Let
$\nu_n$
denote the uniform (Haar) probability measure on
$\mathbb{S}_n$
. The facts about uniformly sampled permutations that we use can be found in [Reference Arratia, Barbour and Tavaré1].
Construction of the processes we are interested in is based on the real-valued multiplicative functions q defined on
$\delta\vert\sigma$
where
$\sigma\in\mathbb{S}_n$
. Adopting the definition of a multiplicative function on partitions [Reference Vershik35, Section 2], we begin with a family of functions
$q_j\colon\mathbb{Z}_+\to\mathbb{R}$
,
$j\in\mathbb{N}$
, such that
$q_j(0)=1$
for every
$j\in \mathbb{N}$
, and put
\begin{equation} q(\delta) \,:\!=\, \prod_{j\leq n} q_j(k_j(\delta)), \quad \delta\vert\sigma,\, \sigma\in \mathbb{S}_n,\end{equation}
agreeing on
$q(\emptyset)=1$
. We say that
$q_j(s)$
,
$0\leq s\leq n$
, is the q function value prescribed to an s-subset of cycles of length j in a permutation divisor
$\delta$
. If
$q_j(k)=q_j(1)^{k}$
for each
$j\leq n$
and
$k\geq 0$
, where
$0^0\,:\!=\,1$
, the function q will be called completely multiplicative; its definition dates back to [Reference Manstavičius25]. Note that the multiplicative functions are structure dependent, that is, their values
$q(\delta)$
depend only on the vector
$\bar k(\delta)$
, irrespective of the numbers in
$V(\delta)$
. In the following, the multiplicative functions f, h, g, and q will have expressions as in (4) with
$f_j(\!\cdot\!)$
,
$h_j(\!\cdot\!)$
,
$g_j(\!\cdot\!)$
, and
$q_j(\!\cdot\!)$
, respectively. Let
${\mathcal M}$
and
${\mathcal M}_{\mathrm{c}}$
denote the classes of multiplicative and completely multiplicative functions. Given
$g,h\in{\mathcal M}$
, we introduce the convolution
\begin{align*} q(\sigma) \,:\!=\, \sum_{\sigma=\delta\tau}g(\delta)h(\tau) = \sum_{\delta\vert\sigma}g(\delta)h(\sigma/\delta).\end{align*}
Here, the first summation is over the ordered decompositions of
$\sigma$
into the product of divisors. Observe that the latter implies
\begin{equation} q_j(k)=\sum_{s=0}^{k} \binom{k}{s} g_j(s) h_j(k-s), \quad j\leq n.\end{equation}
Indeed, for each
$j\leq n$
, any
$\delta\vert\sigma$
has an s-subset,
$0\leq s\leq k_j(\sigma)\,=\!:\, k$
, of the cycles of length j from
$\sigma$
. The subset can be chosen in
$\binom{k}{s}$
ways and, moreover, the values of g on these subsets coincide, as do the values of h on the
$(k-s)$
subsets of the remaining cycles of the same length. Note also that the classes
${\mathcal M}$
and
${\mathcal M}_{\mathrm{c}}$
are closed under convolution. The recalled toolkit will be used in what follows.
Given a nonnegative
$g\in{\mathcal M}$
, we define the multiplicative function
\begin{equation} f(\sigma) \,:\!=\, \sum_{\delta\vert\sigma} g(\delta),\end{equation}
and the family of cumulative distribution functions supported by [0, 1],
\begin{equation} X_n(\sigma,t) \,:\!=\, X_n(\sigma,t;\,g) = \frac{1}{f(\sigma)}\sum_{\substack{\delta\vert\sigma\\L(\delta)\leq tn}}g(\delta), \quad \sigma\in\mathbb{S}_n,\, 0\leq t\leq 1.\end{equation}
In particular,
\begin{align*} X_n(\sigma,t;\,1) = 2^{-w(\sigma)}\#\{M\subset\mathbb{N}_n\colon \# M\leq tn,\, \sigma(M)=M\}, \quad \sigma\in\mathbb{S}_n,\, 0\leq t\leq 1,\end{align*}
is the relative density of the
$\sigma$
-invariant subsets
$M\subset\mathbb{N}_n$
with cardinality not exceeding tn. The recent result in [Reference Leung24] gives the following analogue of the DDT theorem:
\begin{align*} \frac1{n!}\sum_{\sigma\in\mathbb{S}_n}X_n(\sigma,t;\,1) = \frac2{\pi}\operatorname{arcsin}\sqrt t+O(n^{-1/2}), \quad 0\leq t\leq 1.\end{align*}
The latter concerns the asymptotic behaviour as
$n\to\infty$
of the averaged distribution of the random variable
$L(\delta)/n$
when
$\delta$
is drawn uniformly from the set of divisors of
$\sigma$
. A more general problem was explored in [Reference Bareikis and Manstavičius10] in which the divisor
$\delta\vert\sigma$
was taken according to the Ewens probability
\begin{equation*} \mathrm{Prob}_\vartheta(\{\delta\}) = \vartheta^{w(\delta)}\Bigg(\sum_{\delta\vert\sigma}\vartheta^{w(\delta)}\Bigg)^{-1} = \vartheta^{w(\delta)}(1+\vartheta)^{-w(\sigma)}.\end{equation*}
Here,
$\vartheta>0$
is a fixed constant and the last equality stems from (6) and (5). It was proved that
\begin{equation} \frac1{n!}\sum_{\sigma\in\mathbb{S}_n}\mathrm{Prob}_\vartheta(\delta\vert\sigma\colon L(\delta) \leq nt) = B(t;\,\theta,1-\theta) + O\big(n^{-\min\{\theta, 1-\theta\}}\big),\end{equation}
where
$\theta \,:\!=\, \vartheta/(1+\vartheta)$
and the constant in
$O(\!\cdot\!)$
can depend on
$\vartheta$
. Evidently, (8) corresponds to the number-theoretic result (2).
The exposed parallels between phenomena in number theory and combinatorics lead to the thought that an appropriate functional limit theorem is available for permutations. Let us denote by
$\nu_n$
the uniform (Haar) probability measure on
$\mathbb{S}_n$
and focus on the distribution
$\mathbb{P}_n \,:\!=\, \nu_n\cdot X_n^{-1}$
defined on
$\mathcal D$
by
\begin{equation*} \mathbb{P}_n(A) = \nu_n\cdot X_n^{-1}(A) = \nu_n(\sigma\in\mathbb{S}_n\colon X_n(\sigma,\cdot)\in A), \quad A\in\mathcal D.\end{equation*}
Theorem 1. Let
$\vartheta>0$
be a constant and
$g\in{\mathcal M}$
. Assume that
$g_j(1)=\vartheta>0$
for all
$j\geq 1$
, and
$g_j(k)\geq 0$
are arbitrary if
$j\geq 1$
and
$k\geq2$
. The sequence of distributions
$\mathbb{P}_n$
converges weakly as
$n\to\infty$
to a measure
$\mathbb{P}$
supported by
$\mathbb{C}$
.
In the proof of Proposition 2 below, we will see that the limit process X with distribution
$\mathbb{P}$
on the sigma-algebra
$\mathcal D$
coincides with that found in [Reference Manstavičius and Timofeev30]. Starting from [Reference Tenenbaum34], it has subsequently been explored in many aspects. Let
$\varphi$
be the variable in
$\mathbb{D}$
that is the space of trajectories, and let
\begin{equation} E(l,\bar t) \,:\!=\, \mathbb{E}\Bigg(\prod_{i\leq l}X(t_i)\Bigg) = \int_{\mathbb{D}}\prod_{i\leq l}\varphi(t_i)\,\mathrm{d}P(\varphi)\end{equation}
be the mixed moment of the process values
$X(t_i)$
, where
$\bar t\,:\!=\,(t_1,\dots,t_l)\in[0, 1]^l$
and
$l\in\mathbb{N}$
. Set
$\mathbb{E}_n$
to be the mean value with respect to the probability
$\nu_n$
.
Corollary 1. Assume that the conditions of Theorem 1 are satisfied, and
$l\in\mathbb{N}$
is fixed. Then
\begin{align*} \lim_{n\to\infty}\mathbb{E}_n\Bigg(\prod_{i\leq l}X_n(\sigma,t_i)\Bigg) = E(l,\bar t) \end{align*}
uniformly in
$\bar t\in [0, 1]^l$
.
Proof. Since
$0 < X_n(t)\leq 1$
, based on the dominated convergence theorem, from Theorem 1 we obtain convergence of the mixed moments to the expression defined in (9).
In the proof of Proposition 1, we will observe that X depends only on
$g_j(1)=\vartheta$
, regardless of the values
$g_j(k)$
if
$j\geq 1$
and
$k\geq 2$
. In other words, the limit process is the same as if
$X_n$
were defined via
$g(\sigma)=\vartheta^{w(\sigma)}$
and
$f(\sigma)=(1+\vartheta)^{w(\sigma)}$
.
If
$l=1$
, Corollary 1 recovers (8) without the remainder term estimate. Let us exploit the known expressions for the mixed moments, starting with the simple case.
Corollary 2. Let
$E(l,\bar t)$
be as defined in (9). Assume that the conditions of Theorem 1 are satisfied,
$l=2$
,
$\bar t=(t,t)$
where
$0\leq t\leq 1$
, and
$\theta=\vartheta/(1+\vartheta)$
. Then Corollary 1 holds with
\begin{align*} E(2,\bar t) = \begin{cases} I(t,1-\theta) & \text{if}\ 0\leq t\leq \frac12, \\ I(1-t,\theta) + 2B(t;\,\theta,1-\theta)-1 & \text{if}\ \frac12 < t\leq 1. \end{cases} \end{align*}
Here,
\begin{align*} I(t,a) = \frac{2}{\Gamma(a^2)\Gamma(b^2)\Gamma^2(ab)}\int_0^t\frac{\mathrm{d}w}{w^{1-ab}} \int_0^{t-w}\frac{\mathrm{d}v}{v^{1-b^2}}\int_0^w\frac{u^{ab-1}\,\mathrm{d}u}{(1-w-v-u)^{1-a^2}} \end{align*}
if
$0\leq t\leq \frac12$
,
$0 < a<1$
, and
$b=1-a$
.
Proof. Apply Corollary 1 and [Reference Bareikis and Mačiulis7, Theorem 2.2]. That result states that the second moment of the number-theoretic process
$T_n(m,t;\,1)$
defined above equals
\begin{align*} \frac1{n}\sum_{m\leq n}T_n^2(m,t;\,1) = E(2,\bar t) + \text{remainder}, \quad \bar t=(t,t),\, 0\leq t\leq 1. \end{align*}
Since the limits of
$T_n(m,t;\,1)$
and
$X_n(\sigma,t;\,1)$
, as well as the limits of their moments, coincide, we can exploit this.
A few computer-drawn illustrations of
$I'_t(t,a)$
are presented in [Reference Bareikis and Mačiulis7]. Generalising [Reference Bareikis and Mačiulis5] and [Reference Bareikis and Mačiulis7], [Reference De la Bretèche and Tenenbaum12] succeeded in writing rather complex formulas for all
$E(l,\bar t)$
in (9). For the reader’s convenience, we include them here.
Let
$l\geq 1$
be fixed,
$r=2^l-1$
,
$\bar v=(v_1,\dots,v_r)\in[0, 1]^r$
, and
$\bar t=(t_1,\dots, t_l)\in[0, 1]^l$
. For a nonnegative integer m, introduce the base-2 digits
$d_j(m)\in\{0,1\}$
of m and the sum of digits
$d(m)=d_0(m)+d_1(m)+\cdots$
. Given
$\bar v\in \mathbb{R}^r$
, define the vector
$\bar u(\bar v)=(u_1(v), \dots,u_l(v))\in \mathbb{R}^l$
by
$u_j(v) \,:\!=\, \sum_{1\leq m\leq r} d_j(m) v_m$
,
$j\leq l$
. For an arbitrary vector
$\bar u=(u_1,\dots, u_l)\in\mathbb{R}^l$
, we will write
$\bar u\leq \bar t$
if
$u_j\leq t_j$
for each
$j\leq l$
. Set
$s(\bar v)=v_1+\cdots+ v_r$
and
$\mathrm{d}\bar v=\mathrm{d}v_1\cdots\mathrm{d}v_r$
. Define the region
$\Omega(\bar t) = \{\bar v\in[0, 1]^r\colon \bar u(\bar v) \leq \bar t,\, s(\bar v)\leq 1\}$
.
Corollary 3. Assume that the conditions of Theorem 1 are satisfied,
$l\geq 1$
,
$\bar t=(t_1,\dots, t_l)\in [0, 1]^l$
, and
$E(l,\bar t)$
is as defined in (9). Then Corollary 1 holds with
\begin{equation*} E(l,\bar t) = \prod_{0\leq m\leq r}\Gamma\big(\vartheta^{d(m)}(1+\vartheta)^{-l}\big)^{-1} \int_{\Omega(\bar t)}\prod_{1\leq m\leq r}v_m^{\vartheta^{d(m)}(1+\vartheta)^{-l}-1}(1-s(\bar v))^{(1+\vartheta)^{-l}-1}\, \mathrm{d}\bar v. \end{equation*}
Proof. Apply Corollary 1 and [Reference De la Bretèche and Tenenbaum12, (1.7), p. 4].
In a similar manner, to the result in [Reference Leung24] holding in the case
$\vartheta=1$
, we can add more facts concerning the limit process. Here are two examples.
Corollary 4. Assume that the process
$X_n$
is defined in (7) with
$\vartheta=1$
. Then:
-
(i) For all
$0 < s < t < 1$
, the increments
$X_n(t)-X_n(s)$
converge in distribution to a discrete random variable whose values are dyadic rational numbers. -
(ii) If
$\varphi(t)$
is a trajectory of the limit process, then, P-almost surely,
$\varphi'(t)=0$
for every
$0 < t<1$
.
Proof. Claim (i) is just [Reference Hall and Tenenbaum20, Theorem 20]. Claim (ii) is formulated in [Reference Tenenbaum34, p. 16].
Note that the latter corollary does not exhaust all the possible applications of item (i) of the theorem from [Reference Tenenbaum32].
The main lemmata required are collected in the next section. The proof of Theorem 1, split into three parts, is presented in Section 3. Namely, Proposition 1 shows that the influence of short cycles is negligible and provides the possibility of reducing the process to a simpler case. Proposition 2 establishes the convergence of marginal laws of finite order. Proposition 3 verifies the tightness criteria for the sequence of measures
$\{\mathbb{P}_n\}_{n=1}^{\infty}$
, in fact in a stronger form than needed for the application of [Reference Billingsley11, Theorem 15.5].
2. Lemmata
Let us recall the classical formulas due to Cauchy and Goncharov concerning the distribution of the cycle vector and its components.
Lemma 1 Let
$\bar k=(k_1,\dots, k_n)\in\mathbb{Z}_+^n$
and
$n\geq 1$
. Then
\begin{align*} \nu_n(\bar k(\sigma)=\bar k) = {\mathbf 1}\{\ell(\bar k)=n\}\prod_{j\leq n}\frac{1}{j^{k_j} k_j!}. \end{align*}
If
$j\leq n$
, then
\begin{align*} \nu_n(k_j(\sigma)=k) = \frac{1}{j^k k!}\sum_{0\leq s\leq (n/j)-k}\frac{(\!-\!1)^s}{j^s s!} \leq \frac1{j^k k!}. \end{align*}
Furthermore,
$\mathbb{E}_n k_j(\sigma)=1/j$
and
$\mathbb{E}_n k_j(\sigma)(k_j(\sigma)-1)={\mathbf 1}\{j\leq n/2\}/j^2$
for each
$j\leq n$
.
Proof. See [Reference Arratia, Barbour and Tavaré1, (1.3), (1.7), and (1.5), pp. 11–12].
The next lemma gives an analogue of the Turán–Kubilius inequality, well known for additive number-theoretic functions. The result below corresponds to that obtained in [Reference Kubilius22] and refined in [Reference Lee23].
Lemma 2. Let
$a_j$
,
$1\leq j\leq n$
, be arbitrary real numbers. Then
\begin{align*} \frac1{n!}\sum_{\sigma\in\mathbb{S}_n}\Bigg(\sum_{j\leq n}a_jk_j(\sigma)-\sum_{j\leq n}\frac{a_j}{j}\Bigg)^2 = \sum_{j\leq n}\frac{a_j^2}{j} - \sum_{\substack{i,j\leq n\\ i+j>n}}\frac{a_ia_j}{ij} \leq \frac32\sum_{j\leq n}\frac{a_j^2}{j}. \end{align*}
Proof. See [Reference Klimavičius and Manstavičius21].
Note that
$\frac32$
is the optimal constant if
$n\geq 2$
, and it can be substituted by 1 if
$a_j\geq 0$
.
The next lemma is an analogue of the number-theoretic result in [Reference Hall19].
Lemma 3. If
$q\in{\mathcal M}$
satisfies
$0\leq q_j(k)\leq 1$
for
$jk\leq n$
, then
\begin{align*} \mathbb{E}_n q(\sigma) \ll \exp\Bigg\{\sum_{j\leq n}\frac{q_j(1)-1}{j}\Bigg\}. \end{align*}
Proof. If
$q\in{\mathcal M}_{\mathrm{c}}$
, the required estimate is known [Reference Manstavičius28, Proposition 5] even with the exact asymptotic constant.
According to the definition of a multiplicative function, q takes the same value for all permutations having the cycle structure vector
$\bar k$
; therefore, by Lemma 1,
\begin{align*} \mathbb{E}_m q(\sigma) = \sum_{\ell(\bar k)=m}\prod_{j\leq m}\frac{q_j(k_j)}{j^{k_j}k_j!}, \quad 1\leq m\leq n, \end{align*}
and
$\mathbb{E}_0 q(\sigma)=1$
. This leads to the formal product expression for the generating series
\begin{align*} Z(x;\,q) \,:\!=\, \sum_{m=0}^\infty\mathbb{E}_m q(\sigma)x^m = \prod_{j=1}^\infty\Bigg(1+\sum_{k\geq 1}\frac{q_j(k)x^{jk}}{j^k k!}\Bigg). \end{align*}
If
$jk>n$
, the values
$q_j(k)$
are not present in the expressions for
$\mathbb{E}_m q(\sigma)$
for
$m\leq n$
. We may assume them to be equal to zero. Thus, Z(x,q) may be considered as a finite product of polynomials. Consequently,
\begin{align} \sum_{0\leq m\leq n}\mathbb{E}_m q(\sigma) \leq \prod_{j\leq n}\Bigg(1+\sum_{k\leq n/j}\frac{q_j(k)}{j^k k!}\Bigg) & = \exp\Bigg\{\sum_{j\leq n}\frac{q_j(1)}j\Bigg\} \prod_{j\leq n}\mathrm{e}^{-q_j(1)/j}\Bigg(1+\sum_{k\leq n/j}\frac{q_j(k)}{j^k k!}\Bigg) \nonumber \\ & \ll \exp\Bigg\{\sum_{j\leq n}\frac{q_j(1)}j\Bigg\}. \end{align}
The proof that the product over
$j\leq n$
is bounded by an absolute constant is straightforward; we omit it here.
Next, differentiating the generating series, we obtain the following equality:
\begin{align*} xZ'(x, q) = \sum_{j\geq 1}\sum_{k\geq 1}\frac{q_j(k)x^{\,jk}}{j^{k-1}(k-1)!}Z(x,q^{(\,j)}) = \sum_{j\geq 1}\sum_{k\geq 1}\frac{q_j(k)x^{jk}}{j^{k-1}(k-1)!}\sum_{m=0}^\infty\mathbb{E}_m q^{(j)}(\sigma) x^m. \end{align*}
Here,
$q^{(j)}=q^{(j)}(\sigma)\in{\mathcal M}$
are obtained from q by setting
$q_j(k)=0$
for all
$k\geq 1$
. Equalising the coefficients of
$x^n$
and applying the condition of the lemma, we have
\begin{align*} n\mathbb{E}_n q(\sigma) = \sum_{jk\leq n}\frac{q_j(k)}{j^{k-1}(k-1)!}\mathbb{E}_{n-jk}q^{(j)}(\sigma) \leq \sum_{m\leq n}\mathbb{E}_{n-m}q(\sigma)\sum_{jk=m}\frac{1}{j^{k-1}(k-1)!}. \end{align*}
Since the inner sum is bounded by an absolute constant, (10) yields
\begin{align*} \mathbb{E}_n q(\sigma) \ll \frac1n\sum_{0\leq m < n}\mathbb{E}_{m}q(\sigma) \ll \exp\Bigg\{\sum_{j\leq n}\frac{q_j(1)-1}j\Bigg\}. \end{align*}
The lemma is proved.
If
$1\leq r\leq m$
and
$m\geq 2$
are arbitrary integers, then we can uniquely decompose
$\sigma\in\mathbb{S}_m$
into the product
$\sigma=\sigma'\sigma''$
, where
$\sigma'$
is the so-called r-friable (smooth) divisor and
$\sigma''$
the r-free divisor. Namely,
$\sigma'$
comprises all cycles in
$\sigma$
whose lengths do not exceed r, while
$\sigma''$
contains the remaining ones. Counting such divisors, we will reduce the task to enumerating the r-friable and the r-free permutations in
$\mathbb{S}_m$
where
$r\leq m\leq n$
. Now we present a few known results.
As usual, let
$\rho\colon[0,\infty[\!\to\!]0,1]$
denote the Dickman–de Bruijn function defined as
$\rho(u)=1$
for [0, 1] and, for the rest of its range, by the delay differential equation
$u\rho'(u)+\rho(u-1)=0$
. Recall (see, e.g., [Reference Tenenbaum33, Section 5.4]) that
$\rho(u)=u^{-u+o(u)}$
as
$u\to\infty$
.
Lemma 4. If
$ 1\leq r\leq m$
and
$u\,:\!=\,m/r$
, then
\begin{align*} \nu_m(\sigma\in\mathbb{S}_m\colon \sigma\ \text{is}\ r\text{-friable}) = \rho(u)\bigg(1 + O\bigg(\frac{u\log(u+1)}{r}\bigg)\bigg). \end{align*}
Proof. This is [Reference Gorodetsky18, Proposition 1.8]. Actually, the lemma is a quick consequence of [Reference Ford17, Theorem 1.17]. If
$\sqrt{m\log(m+1)} \leq r \leq m$
, the result dates back to [Reference Manstavičius and Petuchovas29].
Let
$\omega\colon[0,\infty[\!\to\!]0,1]$
denote Buchstab’s function defined as
$\omega(u)=1/u$
for
$u\in [1,2]$
and by the delay differential equation
$(u\omega(u))'=\omega(u-1)$
for
$u\geq 2$
. [Reference Tenenbaum33, Theorem 4, p. 402] gives us that
$\omega(u)-\mathrm{e}^{-\gamma} \ll \rho(u)\log^{-1}(u+1)$
if
$u\geq 1$
.
Lemma 5. If
$ 1\leq r\leq m$
and
$u\,:\!=\,m/r\geq 1$
, then
\begin{align*} \nu_m(\sigma\in\mathbb{S}_m\colon\sigma\ \text{is}\ r\text{-free}) = \exp\Bigg\{{-}\sum_{j\leq r}\frac1j\Bigg\}\big(\mathrm{e}^\gamma\omega(u) + O(r^{-1})\big). \end{align*}
Proof. This is [Reference Manstavičius26, Theorem 3]. See [Reference Ford17, Reference Petuchovas31] for state-of-the-art surveys on enumeration of the r-free permutations.
3. Proof of Theorem 1
We split the proof into three parts.
3.1. Long cycles are essential
We will discover that the divisors having only long cycles
$\varkappa$
, that is, large
$L(\varkappa)$
, determine the asymptotic behaviour of the process
$X_n(\sigma,t)$
defined in (7) via the function
$g\in{\mathcal M}$
. If
$0 < \varepsilon<1$
, we let
\begin{align*} \sigma'(\varepsilon) = \prod_{\substack{\varkappa\vert\sigma\\ L(\varkappa)\leq\varepsilon n}}\varkappa, \qquad \sigma(\varepsilon) = \prod_{\substack{\varkappa\vert\sigma\\ \varepsilon n < L(\varkappa)\leq n}}\varkappa\end{align*}
denote the
$(\varepsilon n)$
-friable and
$(\varepsilon n)$
-free divisors, respectively. We introduce the process
\begin{equation} X_n(\sigma(\varepsilon),t) = \frac1{f(\sigma(\varepsilon))}\sum_{\substack{\delta\vert\sigma(\varepsilon)\\ L(\delta)\leq tn}}g(\delta), \quad 0\leq t\leq 1.\end{equation}
For brevity, set
$\alpha=(\log(1/\varepsilon))^{1/2}$
.
Proposition 1. Let
$X_n(\sigma,t)$
and
$X_n(\sigma(\varepsilon),t)$
be the processes defined in (7) and (11). There exist absolute positive constants
$c_0$
,
$\varepsilon_0$
, and C such that
$\nu_n(X_n(\sigma,t)\not=X_n(\sigma(\varepsilon),t)) \leq C\alpha^{-1}$
uniformly in
$\varepsilon^c\leq t\leq 1-\varepsilon^c$
, provided that
$0 < c\leq c_0$
,
$0 < \varepsilon\leq\varepsilon_0$
, and
$n\geq n_0(\varepsilon)$
.
Furthermore, the estimate continues to hold if, defining
$X_n(\sigma(\varepsilon), t)$
by (11), we replace
$g(\delta)$
by
$\vartheta^{w(\delta)}$
and
$f(\sigma(\varepsilon))$
by
$(1+\vartheta)^{w(\sigma(\varepsilon))}$
.
Proof. We adopt the original arguments used in [Reference Manstavičius and Timofeev30]. We may start with
$\varepsilon_0\leq\mathrm{e}^{-1}$
and
$n_0(\varepsilon)>\varepsilon^{-1}\geq\mathrm{e}$
and refine the choice in the proof process.
Each
$\delta\vert\sigma'(\varepsilon)\sigma(\varepsilon)$
splits into a product of two divisors,
$\delta=\delta_1\delta_2$
, such that
$\delta_1\vert\sigma'(\varepsilon)$
and
$\delta_2\vert\sigma(\varepsilon)$
. Hence,
\begin{align*} X_n(\sigma,t) & = \frac{1}{f(\sigma'(\varepsilon))}\sum_{\delta_1\vert\sigma'(\varepsilon)}g(\delta_1)\cdot \frac{1}{f(\sigma(\varepsilon))}\sum_{\substack{\delta_2\vert\sigma(\varepsilon)\\ L(\delta_2)\leq nt}}g(\delta_2) \\ & \quad - \frac{1}{f(\sigma)}\sum_{\delta_1\vert\sigma'(\varepsilon)}g(\delta_1) \sum_{\substack{\delta_2\vert\sigma(\varepsilon)\\ tn-L(\delta_1) < L(\delta_2)\leq nt}}g(\delta_2) \\ & \,=\!:\, X_n(\sigma(\varepsilon),t) - Y_n(\sigma,t). \end{align*}
The largest
$(\varepsilon n)$
-friable divisor of
$\sigma$
is
$\sigma'(\varepsilon)$
. Observe that the subset of
$\sigma$
having comparatively large
$L(\sigma'(\varepsilon))$
is sparse. Indeed,
\begin{align*} \nu_n(\sigma\in\mathbb{S}_n\colon L(\sigma'(\varepsilon))>\varepsilon\alpha n) & = \nu_n\Bigg(\sum_{j\leq\varepsilon n}jk_j(\sigma) > \varepsilon\alpha n\Bigg) \\ & \leq \frac{1}{\varepsilon\alpha n}\sum_{j\leq\varepsilon n}j\mathbb{E}_nk_j(\sigma) = \frac{1}{\varepsilon\alpha n}\lfloor{\varepsilon}n\rfloor \leq \frac1{\alpha} \end{align*}
by Lemma 1. Hence,
\begin{align} & \nu_n(Y_n(\sigma,t)\not=0) \nonumber \\ & \qquad \leq \alpha^{-1} + \nu_n\big(\sigma\in\mathbb{S}_n\colon L(\sigma'(\varepsilon)) \leq \varepsilon\alpha n, \\ & \qquad\qquad\qquad\qquad\qquad\quad \text{there exists}\ \delta_2\vert\sigma(\varepsilon),\,tn-L(\sigma'(\varepsilon)) < L(\delta_2)\leq tn\big) \nonumber \\ & \qquad \,=\!:\, \alpha^{-1} + \mu_n(t). \nonumber \end{align}
Let us focus on the
$\sigma$
s counted in
$\mu_n(t)$
. If
$\sigma(\varepsilon)=\delta_2\tilde\delta_2$
, then
$\sigma=\delta_2\tilde\delta_2\sigma'(\varepsilon)$
. Solving the inequalities between the parentheses in
$\mu_n(t)$
, by virtue of
$L(\delta_2)+L(\tilde\delta_2)=n-L(\sigma'(\varepsilon))$
we have
\begin{equation} (t-\varepsilon\alpha)n \leq L(\delta_2) \leq tn, \qquad (1-t-\varepsilon\alpha)n \leq L(\tilde\delta_2) \leq (1-t)n \end{equation}
provided that
$\varepsilon\alpha\leq t\leq 1-\varepsilon\alpha$
. Now, it is essential to verify that at least one of
$\delta_2$
and
$\tilde\delta_2$
has a comparatively small number of cycles. By the definition of the number-of-cycles function,
\begin{align*} w(\sigma(\varepsilon))=\sum_{\varepsilon n < j\leq n} k_j(\sigma). \end{align*}
According to Lemma 2, we have
\begin{equation} \frac1{n!}\sum_{\sigma\in\mathbb{S}_n}(w(\sigma(\varepsilon))-h(\varepsilon n,n))^2 \leq h(\varepsilon n,n), \quad n\geq 2, \end{equation}
where
$h(y,x)=\sum_{y<j\leq x}1/j$
, which satisfies
$|h(y,x)-\log(x/y)| \leq 1/y$
. This and the Chebyshev inequality yield
\begin{equation} \nu_n(w(\sigma(\varepsilon))>(3/2) \alpha^2) \leq \nu_n(w(\sigma(\varepsilon))-h(\varepsilon n, n) > \alpha^2/4) \leq 32\alpha^{-2} \end{equation}
if
$n \geq 4\varepsilon^{-1}\alpha^{-2}$
. Thus, for all but
$O(n!\alpha^{-2})$
permutations
$\sigma\in\mathbb{S}_n$
, one of the above
$\delta_2$
and
$\tilde\delta_2$
has no more than
$\frac34\alpha^2$
cycles. In either case, apart from
$O(n!\alpha^{-2})$
permutations
$\sigma$
counted in the frequency
$\mu_n(t)$
, we obtain a decomposition
$\sigma=\delta\tau(\varepsilon)$
, where
$\tau(\varepsilon)$
is
$(\varepsilon n)$
-free and
$\delta$
belongs to the set
$\Delta_t\,:\!=\,\big\{\delta\vert\sigma\colon w(\delta(\varepsilon)) \leq \frac34\alpha^2,\, (t-\varepsilon\alpha)n \leq L(\delta) \leq (t+\varepsilon\alpha)n\big\}$
for t (or
$(1-t)$
) from
$[\varepsilon\alpha,1-\varepsilon\alpha]$
, as indicated in (13).
Indeed, if
$w(\delta_2)\leq\frac34\alpha^2$
, then we set
$\tau(\varepsilon)=\tilde\delta_2$
and
$\delta=\delta_2\sigma'(\varepsilon)$
, where
$\delta_2=\delta(\varepsilon)$
satisfies the left-hand-side inequalities in (13). In the second case, if
$w(\tilde\delta_2)\leq\frac34\alpha^2$
, then we take
$\tau(\varepsilon)=\delta_2$
and
$\delta=\tilde\delta_2\sigma'(\varepsilon)$
, where
$\tilde\delta_2=\delta(\varepsilon)$
satisfies the right-hand-side inequalities in (13). Consequently, excluding the permutations counted in (15), for the shorter interval
$[\varepsilon^c,1-\varepsilon^c]\subset[\varepsilon\alpha,1-\varepsilon\alpha]$
, where
$0 < c\leq \frac14$
, we have
\begin{align} \mu_n & \,:\!=\, \max_{\varepsilon^c\leq t\leq 1-\varepsilon^c}\mu_n(t) \nonumber \\ & \ll \alpha^{-2} + \nu_n(\sigma=\delta\tau(\varepsilon)\in\mathbb{S}_n\colon\text{there exists}\ t\in[\varepsilon^c,1-\varepsilon^c]\ \text{such that}\ \delta\in\Delta_t) \nonumber \\ & \ll \alpha^{-2} + \max_{\varepsilon^c\leq t\leq 1-\varepsilon^c}\frac1{n!}\sum_{\substack{\sigma=\delta\tau(\varepsilon)\in\mathbb{S}_n\\ \delta\in\Delta_t}}1. \end{align}
As described in [Reference Bareikis and Manstavičius10] or [Reference Flajolet and Sedgewick16, Chapter II], the sum over decompositions in (16) can be reduced to a summation over permutations belonging to respective symmetric groups of lower order. For that, the vertex labels in
$V(\tau(\varepsilon))$
and, simultaneously, the labels in
$V(\delta)$
belonging to [n], can be substituted by the numbers from
$[L(\tau(\varepsilon))]$
and
$[L(\delta)]$
so that the former orders of the labels for either of the divisors are preserved. If
$L(\delta)=k$
, then exactly
$\binom{n}{k}$
of the pairs
$(\delta,\tau(\varepsilon))$
are reduced to one pair
$(\delta,\tau(\varepsilon))$
, with
$\delta\in\mathbb{S}_k$
and
$\tau(\varepsilon)\in\mathbb{S}_{n-k}$
. Here, we are leaving the same notation after the relabelling since the reduction does not change the cycle structure of the divisors; in particular, neither the lengths nor
$w(\delta(\varepsilon))$
. Consequently, (16) attains the form
\begin{equation} \mu_n \ll \alpha^{-2} + \max_{\varepsilon^c\leq t\leq 1-\varepsilon^c}\frac1{n!}\sum_{(t-\varepsilon\alpha)n\leq k\leq (t+\varepsilon\alpha)n}\binom{n}{k} \sum_{\substack{\delta\in\mathbb{S}_k\\ w(\delta(\varepsilon))\leq(3/4)\alpha^2}}\sum_{\tau(\varepsilon)\in\mathbb{S}_{n-k}}1. \end{equation}
The innermost sum counts the
$(\varepsilon n)$
-free permutations in the symmetric group
$\mathbb{S}_{n-k}$
. Since
\begin{align*} \frac{(n-k)}{(\varepsilon n)} \geq \frac{(1-t-\varepsilon\alpha)}{\varepsilon} \geq \frac{\varepsilon^{c-1}}{2} \geq 1 \end{align*}
for
$0 < c\leq \frac14$
, ensuring
$\varepsilon\alpha\leq\frac12\varepsilon^{c}$
if
$\varepsilon\leq\mathrm{e}^{-2}$
, we can apply Lemma 5 to get
\begin{align*} \frac1{(n-k)!}\sum_{\tau(\varepsilon)\in\mathbb{S}_{n-k}}1 \ll \frac1{\varepsilon n}. \end{align*}
Inserting this into the estimate (17), we obtain
\begin{equation} \mu_n \ll \alpha^{-2} + \max_{\varepsilon^c\leq t\leq 1-\varepsilon^c}\frac1{\varepsilon n} \sum_{(t-\varepsilon\alpha)n\leq k\leq(t+\varepsilon\alpha)n}\nu_k\bigg(\delta\in\mathbb{S}_k\colon w(\delta(\varepsilon)) \leq \frac34\alpha^2\bigg). \end{equation}
The summation is over large k, namely
$k\geq(t-\varepsilon\alpha)n\geq\frac12\varepsilon^cn$
. Therefore, we can again use the inequality (14) with k instead of n, centralising
$w(\delta(\varepsilon))$
by
$h(\varepsilon n, k)$
. The frequency under the sum does not exceed
$\nu_k \,:\!=\, \nu_k\big(\delta\in\mathbb{S}_k\colon w(\delta(\varepsilon))-h(\varepsilon n, k) \leq -\frac14\alpha^2 + R\big)$
, where
\begin{align*} R = \alpha^2 - \log\bigg(\frac{k}{\varepsilon n}\bigg) + \frac1{\varepsilon n} = \log\bigg(\frac{n}{k}\bigg) + \frac1{\varepsilon n} \leq c\alpha^2 + \log 2 + \frac1{\varepsilon n}. \end{align*}
If
$c\leq 1/24 \,=\!:\, c_0$
,
$\varepsilon<2^{-24}\,=\!:\,\varepsilon_0$
, and
$n\geq n_0(\varepsilon)\,:\!=\, 24\varepsilon^{-1}\alpha^{-2}$
, then
$R\leq\frac18\alpha^2$
. Hence, by Chebyshev’s inequality and Lemma 2,
\begin{align*} \nu_k \leq \nu_k\bigg(\delta\in\mathbb{S}_k\colon w(\delta(\varepsilon))-h(\varepsilon n, k) \leq -\frac18\alpha^2\bigg) \ll \alpha^{-4}h(\varepsilon n,k) \ll \alpha^{-2}. \end{align*}
The last estimate and (18) yield
$\mu_n \ll \alpha^{-2} + (\varepsilon n)^{-1} \cdot \alpha^{-2} \cdot \varepsilon \alpha n \ll \alpha^{-1}$
. Combining the inequalities (16) and (12), we complete the proof of the main part of Proposition 1.
Next, recall that
$g_j(1)=\vartheta$
for every j, and that all the cycle lengths of
$\sigma(\varepsilon)$
belong to the interval
$(\varepsilon n, n]$
. Consequently,
$X_n(\sigma(\varepsilon), t)$
with
$g(\delta)$
as defined in (11) differs from the process
\begin{align*} (1+\vartheta)^{-w(\sigma(\varepsilon))}\sum_{\substack{\delta\vert\sigma(\varepsilon)\\ L(\delta)\leq tn}} \vartheta^{w(\delta)} \end{align*}
at most for
$\sigma\in\mathbb{S}_n$
having a cycle
$\varkappa$
of length
$j\in (\varepsilon n,n]$
such that
$k_j(\sigma)\geq 2$
. Using Lemma 1, we can estimate the frequency of such permutations:
\begin{align*} & \nu_n(\sigma\in\mathbb{S}_n\colon\,\text{there exists}\ j\in(\varepsilon n,n]\ \text{such that}\ k_j(\sigma)\geq 2) \\ & \qquad \leq \sum_{\varepsilon n < j\leq n}\sum_{2\leq m\leq n/j}\nu_n(k_j(\sigma)=m) \leq \sum_{\varepsilon n < j\leq n}\sum_{2\leq m\leq n/j}\frac1{j^m m!} \leq \sum_{\varepsilon n < j\leq n}\frac1{j^2} \ll \frac1{\varepsilon n} \ll \alpha^2. \end{align*}
This furnishes the proof of Proposition 1.
3.2. Convergence of finite-dimensional distributions
This step is devoted to the finite-dimensional distributions of the process
$X_n(t)$
. Let
$l\in\mathbb{N}$
and
$0\leq t_1 < \cdots < t_l\leq 1$
be arbitrary fixed numbers. Define the vectors
$T=(t_1,\dots,t_l)$
,
$U=(u_1,\dots,u_l)\in[0, 1]^l$
, and the distribution function
\begin{align*} F_n(U, T)=\nu_n\big( X_n(t_1)\leq u_1, \dots, X_n(t_l)\leq u_l\big).\end{align*}
Proposition 2. For all vectors T and U, the distribution function
$F_n(U,T)$
converges as
$n\to\infty$
to an l-dimensional distribution function.
Proof. Recall that the function
$f\in {\mathcal M}$
is defined by
$f_j(k)$
, where, according to (5),
$f_j(0)=1$
,
$f_j(1)=1+\vartheta$
, and
\begin{align*} f_j(k)=\sum_{s=0}^k\binom{k}{s}g_j(k)\geq 1+\vartheta, \quad k\geq2. \end{align*}
Since
$0 < 1/f(\sigma)=X_n(\sigma,0)\leq X_n(\sigma,t)\leq X_n(\sigma,1)=1$
,
$\sigma\in\mathbb{S}_n$
and
$0\leq t\leq 1$
, without loss of generality we can assume that
$t_l<1$
and
$u_1>0$
.
To settle the case
$t_1=0$
, we observe that, by virtue of
$1/f\in {\mathcal M}$
and
$0 < 1/f(\sigma)\leq 1$
, Lemma 3 yields
\begin{align*} \mathbb{E}_n\bigg(\frac1f\bigg) \ll \exp\Bigg\{\sum_{j\leq n}\frac{(1+\vartheta)^{-1}-1}{j}\Bigg\} \ll \exp\{-\vartheta/(1+\vartheta)\log n\} \ll n^{-\vartheta/(1+\vartheta)}. \end{align*}
Using this estimate, we can evaluate the difference
\begin{align*} 0 & \leq \nu_n(X_n(t_2)\leq u_2, \dots, X_n(t_l)\leq u_l) - \nu_n(X_n(0)\leq u_1, X_n(t_2)\leq u_2, \dots, X_n(t_l)\leq u_l) \\ & \leq \nu_n(X_n(0)>u_1) \leq u_1^{-1}\mathbb{E}_n(1/f) \ll n^{-\vartheta/(1+\vartheta)}. \end{align*}
Thus, if
$t_1=0$
, our task reduces to the
$(l-1)$
-dimensional problem.
Henceforth, let
$0 < \varepsilon < \min\big\{t_1^{1/c_0},(1- t_l)^{1/c_0}\big\}$
, where
$c_0$
was found in Proposition 1. We introduce the distribution functions
\begin{equation*} G_n(U, T) \,:\!=\, \nu_n(\sigma\colon X_n(\sigma(\varepsilon),t_1) \leq u_1,\dots, X_n(\sigma(\varepsilon),t_l) \leq u_l) \,=\!:\, \nu_n\big(\sigma\colon \overline{X}_n(\sigma(\varepsilon),T) \in A\big), \end{equation*}
where the process
$X_n(\sigma(\varepsilon),t)$
was defined in the second assertion of Proposition 1 via
$g(\delta)=\vartheta^{w(\delta)}$
, and
\begin{align*} \overline{X}_n(\sigma(\varepsilon),T) \,:\!=\, (X_n(\sigma(\varepsilon),t_1),\dots,X_n(\sigma(\varepsilon),t_l)), \qquad A\,:\!=\,\prod_{i\leq l}\mathopen{]}0, u_i\mathclose{[} \subset \mathopen{]}0, 1\mathclose{[}^l. \end{align*}
By Proposition 1,
\begin{equation*} F_n(U,T) = G_n(U,T) + O(\alpha^{-1}) = \frac1{n!}\sum_{\substack{\sigma=\sigma(\varepsilon)\delta\in\mathbb{S}_n,\\ \delta\,\text{is}\,(\varepsilon n)\text{-friable}}} {\mathbf 1}\big\{\overline{X}_n(\sigma(\varepsilon),T) \in A\big\} + O(\alpha^{-1}). \end{equation*}
As in the derivation of (17), we can apply the reduction of labels (then
$\binom{n}{k}$
of the
$(\varepsilon n)$
-free permutations
$\sigma(\varepsilon)$
reduce to one
$(\varepsilon n)$
-free permutation
$\tau\in\mathbb{S}_k$
) and rewrite
\begin{align*} F_n(U,T) & = \sum_{\varepsilon n\leq k\leq n}\frac1{k!}\sum_{\substack{\tau\in\mathbb{S}_k\\ \tau\,\text{is}\,(\varepsilon n)-\text{free}}} {\mathbf 1}\big\{\overline{X}_n(\tau, T)\in A\big\} \\ & \quad \times \nu_{n-k}(\delta\in\mathbb{S}_{n-k}\colon\delta\ \text{is}\ (\varepsilon n)\text{-friable}) + O(\alpha^{-1}) \\ & = \sum_{\varepsilon n\leq k\leq n}\frac1{k!}\sum_{\substack{\tau\in\mathbb{S}_k\\ \tau\,\text{is}\,(\varepsilon n)-\text{free}}} {\mathbf 1}\big\{\overline{X}_n(\tau,T)\in A\big\}\rho\bigg(\frac{n-k}{\varepsilon n}\bigg) + O(\alpha^{-1}) \end{align*}
by Lemma 4 with
$m=n-k\geq n(1-\varepsilon)$
and
$r=\lfloor\varepsilon n\rfloor$
, taking into account that
$\rho(u)\ll u^{-u/2}$
for
$u=(n-k)/(\varepsilon n)\geq \varepsilon^{-1}/2$
. Similarly, here we can get rid of
$\tau$
having cycles with repeated lengths. For that, we can apply the next estimate obtained using Lemma 1. Namely,
\begin{multline*} \sum_{\varepsilon n\leq k\leq n}\nu_k(\tau\in\mathbb{S}_k\colon\tau\ \text{is}\ (\varepsilon n)\text{-free},\, \text{there exists}\ j\ \text{such that}\ k_j(\tau)\geq 2) \\ \leq \frac12\sum_{\varepsilon n\leq k\leq n}\sum_{\varepsilon n\leq j\leq k}\mathbb{E}_k(k_j(\tau)(k_j(\tau)-1)) \leq \frac12\sum_{\varepsilon n\leq k\leq n}\sum_{\varepsilon n\leq j\leq k}\frac1{j^2} \ll \frac1{\varepsilon}. \end{multline*}
Multiplied by
$\rho(\varepsilon^{-1}/2)$
, this quantity gives also the remainder
$O(\alpha^{-1})$
. So. after simplifications, we arrive at
\begin{align} & F_n(U,T) \nonumber \\ & = \sum_{\varepsilon n\leq k\leq n}\rho\bigg(\frac{n-k}{\varepsilon n}\bigg) \nu_k\big(\tau\in\mathbb{S}_k\colon\tau\ \text{is}\ (\varepsilon n)\text{-free},\, k_j(\tau)\leq 1,\,\varepsilon n\leq j\leq k,\,\overline{X}_n(\tau, T)\in A\big) \nonumber \\ & \quad + O(\alpha^{-1}) \nonumber \\ & = \sum_{m\leq \varepsilon^{-1}}\sum_{\varepsilon n\leq k\leq n}\rho\bigg(\frac{n-k}{\varepsilon n}\bigg) \nu_k\big(\tau\in\mathbb{S}_k(m)\colon\overline{X}_n(\tau, T)\in A\big) + O(\alpha^{-1}). \end{align}
Here,
$\mathbb{S}_k(m)$
denotes the subset of
$(\varepsilon n)$
-free permutations
$\tau$
in
$\mathbb{S}_k$
that have exactly m,
$1\leq m\leq\varepsilon^{-1}$
, cycles of different lengths. We further intend to change the summation over k by a summation over m-tuples of the cycle lengths appearing in all accounted
$\tau$
.
Let
$\tau\in\mathbb{S}_k(m)$
and its cycle lengths in the fixed order be
$(j_1,\dots, j_m)\in\mathopen{]}\varepsilon n,n\mathclose{]}^m$
provided that
$j_1+\cdots+j_m=k$
. The number of such
$\tau$
s equals
$k!/ (j_1\cdots j_m)$
. What is the meaning of the condition
$\overline{X}_n(\tau, T)\in A$
in terms of the cycle lengths
$ (j_1,\dots, j_m)$
for the given
$\tau$
?
The following observations hold for any such
$\tau$
. Firstly,
$f(\tau)=(1+\vartheta)^m$
. Secondly, the divisors
$\delta\vert\tau$
are enumerated by the vector of indicators
$(i_1,\dots,i_m)$
, where
$i_r=1$
if the cycle of length
$j_r$
appears in
$\delta$
. Hence,
$L(\delta)=i_1j_1+\cdots+i_mj_m$
. Thirdly, the vector
$(j_1/n,\dots,j_m/n)$
necessarily belongs to the intersection, denoted by
$D_m(\varepsilon)$
, of the following two sets of the vectors
$\bar x=(x_1,\dots, x_m)$
:
$\{\bar x\in\mathopen{]}\varepsilon,1\mathclose{]}^m\colon x_1+\cdots+x_m\leq 1\}$
and
\begin{align*} \bigcap_{j\leq l}\Bigg\{\bar x\colon \sum_{(i_1,\dots,i_m)\in\{0,1\}^m}\vartheta^{i_1+\dots + i_m} {\mathbf 1}\{i_1x_1+\cdots+i_m x_m\leq t_j\} \leq u_j(1+\vartheta)^m\Bigg\}. \end{align*}
Having all this in mind, we obtain, from (19),
\begin{align*} F_n(U,T) & = \sum_{m\leq\varepsilon^{-1}}\sum_{(j_1/n,\dots,j_m/n)\in D_m(\varepsilon)}\rho\Bigg(\frac{1}{\varepsilon} \Bigg(1-\sum_{r\leq m}\frac{j_r}{n}\Bigg)\Bigg)\frac1{j_1\cdots j_m} + O(\alpha^{-1}) \\ & = \sum_{m\leq\varepsilon^{-1}}\int_{D_m(\varepsilon)}\rho\Bigg(\frac{1}{\varepsilon} \Bigg(1-\sum_{r\leq m}x_r\Bigg)\Bigg)\frac{\textrm{d}x_1\cdots\textrm{d}x_m}{x_1\cdots x_m} + O(\alpha^{-1}) + o_\varepsilon(1) \end{align*}
as
$n\to\infty$
, by the definition of the m-dimensional Riemann integral. We have arrived at the situation described in [Reference Manstavičius and Timofeev30, p. 6]. Successively letting
$n\to\infty$
and
$\varepsilon\to 0$
, by virtue of the notation
$\alpha=(\log(1/\varepsilon))^{-1/2}$
, we verify that
\begin{align*} \limsup_{\varepsilon\to0}I(\varepsilon) \leq \liminf_{n\to\infty}F_n(U,T) \leq \limsup_{n\to\infty}F_n(U,T) \leq \liminf_{\varepsilon\to0}I(\varepsilon), \end{align*}
where
$I(\varepsilon)$
denotes the sum of integrals in the above relation. This shows that, as claimed in Proposition 2, the limits
$\lim_{\varepsilon\to0}I(\varepsilon) = \lim_{n\to\infty}F_n(U,T)$
exist for all vectors T and U.
We stress once more that the limiting finite-dimensional distribution agrees with that found in [Reference Manstavičius and Timofeev30] for the corresponding number-theoretic process
$T_n(m,t)$
.
3.3. Tightness
This subsection is devoted to showing the tightness of the sequence of distributions
$\{\mathbb{P}_n\}_{n=1}^\infty$
. We take advantage of the idea [Reference Tenenbaum34] of verifying a stronger tightness criterion than that used in the Skorokhod space
$\mathbb{D}[0, 1]$
. According to [Reference Billingsley11, Theorem 15.5], the following assertion also ensures that the weak limit
$\mathbb{P}$
of a subsequence
$\{\mathbb{P}_{n'}\}$
as
$n'\to\infty$
is supported by a subset of the space
$\mathbb{C}[0, 1]$
.
Proposition 3. For every
$ 0 < a,\epsilon < 1$
,
$\nu_n\big(\sup_{|s-t|\leq a}|X_n(t)-X_n(s)| \geq \epsilon\big) \ll \epsilon^{-1}a^{\vartheta/(1+\vartheta)^2}$
.
Proof. As noted in Proposition 1, in the definition in (7) of
$X_n(t)=X_n(\sigma,t)$
, we may take
$g(\delta)=\vartheta^{w(\delta)}$
and
$f(\sigma)=(1+\vartheta)^{w(\sigma)}$
. By virtue of the monotonicity of
$X_n(t)$
,
$0\leq t\leq 1$
, the modulus of continuity is
\begin{align*} \sup_{|s-t|\leq a}|X_n(t)-X_n(s)| = \sup_{0\leq t\leq 1-a}(1+\vartheta)^{-w(\sigma)} \sum_{\substack{\delta\vert\sigma\\ tn < L(\delta)\leq(t+a)n}}\vartheta^{w(\delta)} \,=\!:\, Q_n(\sigma, a), \end{align*}
where
$0\leq s\leq t\leq 1$
and
$0 < a < 1$
. Thus, the estimate of Proposition 3 will follow from
\begin{equation} \mathbb{E}_n Q_n(\sigma, a)\ll a^{\vartheta/(1+\vartheta)^2}. \end{equation}
For a given
$\sigma\in\mathbb{S}_n$
, the introduced
$Q_n(\sigma, a)$
is just the concentration function of the random variable taking values
$L(\delta)/n$
with probability
$\vartheta^{w(\delta)}(1+\vartheta)^{-w(\sigma)}$
for each
$\delta\vert\sigma$
. By [Reference Tenenbaum33, Chapter III, Lemma 2.6.1],
$Q_n(\sigma,a) \leq 3an\int_0^{1/(an)}|H(\sigma,v)|\,\textrm{d}v$
, where, according to the toolkit presented in the introduction,
\begin{align*} H(\sigma,v) & = (1+\vartheta)^{-w(\sigma)}\sum_{\delta\vert\sigma}\vartheta^{w(\delta)}\textrm{e}^{ivL(\delta)} \\ & = (1+\vartheta)^{-w(\sigma)}\sum_{\delta\vert\sigma}\prod_{j\leq n}(\vartheta\textrm{e}^{ivj})^{k_j(\delta)} = \prod_{j\leq n}\bigg(\frac{1+\vartheta\textrm{e}^{ivj}}{1+\vartheta}\bigg)^{k_j(\sigma)}\in{\mathcal M}_{\mathrm{c}}. \end{align*}
Consequently,
\begin{equation} \mathbb{E}_n Q_n(\sigma, a) \ll an\int_0^{1/(an)}\mathbb{E}_n|H(\sigma,v)|\,\textrm{d}v. \end{equation}
Since
$|H(\sigma, v)|\leq 1$
, by Lemma 3 we obtain
\begin{equation} \mathbb{E}_n|H(\sigma,v)| \ll \exp\Bigg\{\sum_{j\leq n} \bigg(\frac{|1+\vartheta\textrm{e}^{i v j}|}{1+\vartheta}-1\bigg)\frac1j\Bigg\} \leq \exp\Bigg\{{-}\frac{\vartheta}{(1+\vartheta)^2}\sum_{j\leq n}\frac{1-\cos v j}{j}\Bigg\}. \end{equation}
Here we have applied the inequality
\begin{align*} |1+\vartheta\textrm{e}^{ix}| \leq 1+\vartheta-\frac{\vartheta}{1+\vartheta}(1-\cos x), \quad x\in\mathbb{R}, \end{align*}
whose proof is straightforward. To estimate the sum in (22), we apply the relation
\begin{multline*} \sum_{j\leq n}\frac{1-\cos v j}{j}-\log\frac{|1-\textrm{e}^{-1/n+iv}|}{ 1-\textrm{e}^{-1/n}} \\ = \sum_{j\leq n}\frac{1-\cos v j}{j}(1-\textrm{e}^{-j/n}) - \sum_{j>n}\frac{1-\cos v j}{j}\textrm{e}^{-j/n} \leq 2+2\int_1^\infty\textrm{e}^{-v}\frac{\textrm{d}v}v \ll 1. \end{multline*}
Thus, (22) yields
\begin{align*} \mathbb{E}_n|H(\sigma,v)| \ll \bigg(\frac{1-\textrm{e}^{-1/n}}{|1-\textrm{e}^{-1/n+iv}|}\bigg)^{\vartheta/(1+\vartheta)^2} \leq \bigg(\frac{ 1}{n|1-\textrm{e}^{-1/n+iv}|}\bigg)^{\vartheta/(1+\vartheta)^2} \end{align*}
with an absolute constant in the symbol
$\ll$
. Returning to (21), we proceed as follows:
\begin{align*} \mathbb{E}_n Q_n(\sigma,a) & \ll an^{1-\vartheta/(1+\vartheta)^2}\int_0^{1/(an)}\frac{\textrm{d}v}{|1-\textrm{e}^{-1/n+iv}|^{\vartheta/(1+\vartheta)^2}} \\ & \ll a + an^{1-\vartheta/(1+\vartheta)^2}\int_{1/n}^{1/(an)}v^{-\vartheta/(1+\vartheta)^2}\,\textrm{d}v \ll a^{\vartheta/(1+\vartheta)^2}. \end{align*}
This is the desired estimate (20).
4. Concluding remarks
Doing more cumbersome work, we can extend Theorem 1 in two ways. First, the function
$g\in{\mathcal M}$
might be more general. Adopting the argument from [Reference Tenenbaum34], we can replace the condition on
$g_j(1)=\vartheta>0$
so that their weighted sums over
$\varepsilon n\leq j\leq n$
behave in the required manner. The analytic technique, analogous to that developed in [Reference Bareikis and Mačiulis7], would work here as well. Second, as seen from [Reference Bareikis and Manstavičius10, Theorem 3], similar results could be established for permutations drawn according to the Ewens probability on
$\mathbb{S}_n$
with another parameter
$\vartheta_1>0$
. Then, a stochastic process depending on
$\vartheta$
and
$\vartheta_1$
should appear in the limit. Use of the generalised Ewens probabilities defined by multiplicative weights (see, e.g., [Reference Manstavičius27]) should also be possible.
Permutations comprise just one class of decomposable combinatorial structures. The distribution of divisors of a random element in an arithmetical semigroup has already attracted attention. Announcing [Reference Leung24, Theorem 1.2] for polynomials over a finite field, Leung rediscovered the result in [Reference Bareikis and Mačiulis6] from 2013. We are convinced that the methodology in proving the functional limit theorem elaborated in the present paper will have further combinatorial extensions.
Acknowledgements
We express sincere thanks to both reviewers for their benevolent suggestions and insightful critical remarks.
Funding information
This research has been carried out as part of a project implemented at the Department of Probability Theory and Number Theory.
Competing interests
There were no competing interests to declare during the preparation or publication process of this article.
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