The impatient collector
Anis Amri (IECL), Philippe Chassaing (IECL)

TL;DR
This paper studies the shape of the completion curve in the coupon collector problem under the condition of completing the collection unusually quickly, and applies findings to automata theory.
Contribution
It introduces the asymptotic shape of the completion curve conditioned on rapid collection, extending classical results and deriving a new formula for automata analysis.
Findings
Characterizes the asymptotic completion curve under fast collection conditions
Provides a new derivation of Koršunov's formula for automata
Enhances understanding of collection dynamics under atypical scenarios
Abstract
In the coupon collector problem with items, the collector needs a random number of tries to complete the collection. Also, after tries, the collector has secured approximately a fraction of the complete collection, so we call the (asymptotic) \emph{completion curve}. In this paper, for , we address the asymptotic shape of the completion curve under the condition , i.e. assuming that the collection is \emph{completed unlikely fast}. As an application to the asymptotic study of complete accessible automata, we provide a new derivation of a formula due to Kor\v{s}unov.
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Taxonomy
Topicssemigroups and automata theory · Advanced Combinatorial Mathematics · Algorithms and Data Compression
The impatient collector
Anis Amri Institut Élie Cartan, Université de Lorraine Email: [email protected]
Philippe Chassaing Institut Élie Cartan, Université de Lorraine Email: [email protected]
Abstract
In the coupon collector problem with items, the collector needs a random number of tries to complete the collection. Also, after tries, the collector has secured approximately a fraction of the complete collection, so we call the (asymptotic) completion curve. In this paper, for , we address the asymptotic shape of the completion curve under the condition , i.e. assuming that the collection is completed unlikely fast. As an application to the asymptotic study of complete accessible automata, we provide a new derivation of a formula due to Koršunov [7, 8].
Contents
1 Introduction
1.1 Main result
This section is intended as a concise introduction to the main results, at the price of, eventually, lacking details, for instance on the way we round real numbers where integers are expected. Details and context are given in the next section. In the standard coupon collector problem with items, our concern is the completion curve : after tries, according to [9, pp. 4-5], the collector has secured approximately a fraction
[TABLE]
of the complete collection111In [9, pp. 4-5], the coupons outside the collection are seen as the empty cells in a random allocation scheme.. Furthermore, the coupon collector needs a random number of tries to complete the collection, with expectation:
[TABLE]
In this paper, for any given , we address the asymptotic shape of the completion curve conditioned to the event :
[TABLE]
i.e. when the collection is completed much faster than in the classical model, hence the title. Since , one expects that the conditioning event has an exponentially small probability, see Section 5.2. Define through the relations:
[TABLE]
Formal definitions are given in the next section, but let us, for now, define the random variable as the fraction of the complete collection secured by the collector after tries. Let denote the principal branch of the Lambert W-function (i.e. the inverse of ), and set:
[TABLE]
Let denote the unique solution, on , of the Cauchy problem:
[TABLE]
The graph of stays in the set , and satisfies , see Section 5.3.
Let denote the conditional probability distribution of the coupon problem, given that . The asymptotic completion curve of the impatient collector is as follows :
Theorem 1**.**
For any , when with , i.e. with , we have
[TABLE]
Theorem 1 is an extension of Theorem 2 (see next section), to the conditional case: converges in probability to , uniformly in any interval . Thus is the -analog of . In the next section, we give a stronger result, in which convergence in probability is given with an explicit bound on the error, cf. Theorem 3. In Section 3, we discuss some applications of this result to random finite automata, including a new (to our knowledge) derivation of a formula by Koršunov [7]. Finally, in Sections 4 and 5 we precise (see Theorem 6) a classic asymptotic formula, due to Good, for Stirling numbers of the second kind, providing a bound that is key for our results, but could also be of independent interest.
1.2 Context: coupon collector problem, Stirling numbers and random allocation
Let us define more precisely the classical model (resp. the conditioned model), that we shall call the patient model (resp. the impatient model). In the patient model, we consider a sequence
[TABLE]
of uniform i.i.d. integers in . Let denote the corresponding probability distribution on the set of infinite sequences. For , let
[TABLE]
denote the size of the collection after the th try, or the number of nonempty cells after the th allocation, so that can also be defined as follows: for ,
[TABLE]
Then, set:
[TABLE]
so that the completion curve is defined as:
[TABLE]
One finds easily:
[TABLE]
but also, more precisely, as a consequence of [9, Ch. 1.1-3],
Theorem 2**.**
In the patient model, in probability, for any ,
[TABLE]
As opposed to the patient model, in the impatient model, we consider the conditional distribution of given that : then only the prefix of matters. In the impatient model, is uniformly distributed on the sequences that are surjections on , a small subset of . Here, as usual, denotes the number of partitions of a set of elements in nonempty subsets, called Stirling number of the second kind. Thus , the conditional probability distribution of the coupon problem, given that , is the uniform distribution on . A stronger version of Theorem 2 is as follows:
Theorem 3**.**
For any , and for large enough, there exists such that, for ,
[TABLE]
The expression of is given at Section 5.6.6. If we assume that stays away from 0 and , then, according to the asymptotic analysis of Stirling numbers of the second kind, to be found in [6], the conditioning event has an exponentially small probability:
[TABLE]
in which is discussed in more detail in Section 5.2. Let us just mention, now, that is the unique positive solution of
[TABLE]
that , and that is decreasing and satisfies
[TABLE]
which entails that is positive. Together with
[TABLE]
the implicit function is known to play a special rôle in the asymptotic behavior of , see Section 4.
1.3 Asymptotics for the Stirling numbers of the second kind
First we need to set some notations. For some integers , the Stirling number of the second kind, denoted by , is the number of partitions of a set of elements into non-empty subsets. By convention , and for we have . Let denote the principal branch of the Lambert W-function (i.e. the inverse of ), and set:
[TABLE]
We set:
[TABLE]
The Stirling numbers of the second kind satisfy the following recurrence relation
[TABLE]
so that
[TABLE]
In Section 5.5, we prove that for , large, depends mostly on the ratio :
Theorem 4**.**
For any , there exist , both positive, such that, for any ,
[TABLE]
This bound proves to be crucial to our aims, for and can be seen as transition probabilities for a random walk closely related to the completion curve , cf. Proposition 1. At Section 5.6.6, we describe . To prove Theorem 4, we need a refinement of the asymptotic study of , originally made in [6]: set
[TABLE]
In Good [6], takes the alternative form
[TABLE]
As a first step toward Theorem 4, Good, followed by many others, established that is an estimate of the corresponding Stirling number:
Theorem 5** ([6]).**
When and both grow towards , with ,
[TABLE]
Though [6, (3)] hints at an asymptotic expansion for the relative error:
[TABLE]
it does not really provide a bound for , while such a bound is needed to prove Theorem 4. So Sections 4 and 5 are devoted to the proof of the following bound, of independent interest :
Theorem 6**.**
For any , there exist , both positive, such that for any ,and for ,
[TABLE]
2 The asymptotic behavior of the completion curve
2.1 A random walk related to Stirling numbers
In this section, with the help of Theorem 6, we prove Theorem 3, about the asymptotic behavior of the completion curve of an impatient coupon collector. For a suitable elementary (small) step , to be defined later in the section, we shall prove that
[TABLE]
in which , while, by definition,
[TABLE]
Then is the result of an Euler scheme with rounding errors . As such, provides a stochastic approximation for , in the spirit of [2, 4].
Actually, time-reversed versions of and , that start at time and end at time 0, are more convenient, for the approximations of Stirling numbers that we use are much worse for small arguments, making the convergence trickier when and are close to . The bound on is obtained through probabilistic and combinatorial tools applied to the discrete version of , before it is rescaled: for any surjection , consider a time-reversed version of the completion curve of , defined, for , by
[TABLE]
Actually the corresponding point of the curve has coordinates , and under , the probability distribution of has a slick description in terms of Stirling numbers of the second kind.
Proposition 1**.**
* is a Markov chain starting at , with transition probabilities described, for , by :*
[TABLE]
In other words, is an inhomogeneous Markov chain, with increments satisfying
[TABLE]
Proof.
Let us compute the probability of a sample path
[TABLE]
for , in which and : the restriction to of any surjection resulting in has elements in its image, leading to choices for this restriction, then at each step we have either choices for if , or choices for if . The second case happens times exactly, and produces a factor . Thus
[TABLE]
while, if denotes the path —seen as a word—, we have, by the same formula, since :
[TABLE]
Thus the expression
[TABLE]
depends only on the final part of the sample path, on the couple . As a consequence, satisfies the Markov property, and is its transition probability, as expected.∎
2.2 Azuma inequality
Theorem 3 is a consequence of the following chain of approximations:
[TABLE]
and its proof results from bounds for the errors in this chain of approximation, as explained before. The first error is bounded with the help of the Azuma-Hoeffding inequality, as usual when the approximation stems from the law of large numbers, while the bound for the second error, given by Theorem 4, follows from the saddle-point method, as explained in Section 4. In order to use an Euler scheme, let us now divide the path into a sequence of, approximately, infinitesimal intervals, each of these intervals being a sequence of steps, . Consider then an integer of the form , , so that is the beginning of some interval, and is the end of the same interval. Then
[TABLE]
in which:
[TABLE]
the last equality due to (9). Rescaling time and space by a factor , we set and
[TABLE]
Finally, for , we set
[TABLE]
in such a way that, according to Theorem 6, is uniformly bounded for in , as long as is large enough, and the same holds true for Theorem 4. Now, for , by geometric considerations,
[TABLE]
Section 1.3 entails that
Lemma 1**.**
For large enough, and for , if and , we have
[TABLE]
Proof.
Recall that . If
[TABLE]
then
[TABLE]
but, if , we obtain, below, that
[TABLE]
entailing (11). Relation (12) follows from the Taylor inequality for , provided that both and belong to :
[TABLE]
and, since meets the conditions in Theorem 4,
[TABLE]
while, according to Section 5.4,
[TABLE]
For large enough,
[TABLE]
yielding successively (12), then (11). ∎
Also, for , let denote the -algebra , and
[TABLE]
The sequence is a martingale with respect to the filtration and for any , , thus Azuma’s inequality gives :
[TABLE]
in which . For , we obtain:
[TABLE]
Set
[TABLE]
The previous bounds lead to
Proposition 2**.**
For large enough, the set satisfies:
[TABLE]
2.3 Euler scheme
Thus, for , i.e. but for a probability at most , is obtained through an Euler scheme with step and rounding error such that
[TABLE]
For the choice , , and for large enough, depending on the choice of , we obtain that
[TABLE]
Then we can see , resp. , as the solution of the ODE at time (resp. the output of the Euler scheme after steps), and set
[TABLE]
Then, following [3] and according to Section 5.4, provided that the points and belong to , and that , we can write
[TABLE]
in which
[TABLE]
and
[TABLE]
The bounds for the supremums in (13) are obtained in Section 5.5, see Proposition 10. For , for small enough. Consider the bound (48) obtained for at Section 5.3. It entails that, for ,
[TABLE]
so that for . Thus if . Now, for ,
[TABLE]
Assume that, for , , so that we can write :
[TABLE]
Then
[TABLE]
for large enough, depending on , but not on , since :
[TABLE]
for . Relations (16) and (17) entail that so that (14) holds true and, in turn, , if necessary. It follows, recursively, that, for any ,
[TABLE]
that is, at the ends of any infinitesimal interval, the error is bounded accordingly. Between these ends the error can be larger by at most half the length of this infinitesimal interval, i.e. by , since both and are non increasing with slope smaller than 1. Finally, for large enough and , i.e. but for a probability at most , on the interval ,
[TABLE]
3 Coupon and automata
3.1 Koršunov’s formula
In , Koršunov [7, 8] proves a formula for the asymptotic enumeration of accessible complete and deterministic automata (ACDA) with states over a -letters alphabet. Later Nicaud [13] proves that ACDA are in bijection with a subset of , though he uses a different terminology : surjections are represented by boxed diagrams, and ACDA by Dyck boxed diagram. We recall briefly the definitions of these combinatorial objects in the next subsection. In this paper we assume that and we set . With these notations, we can rephrase Koršunov’s result as follows :
Theorem 7**.**
[TABLE]
In the notations of [10], . In Section 3.2, we describe following the lines of [13], then in Sections 3.3,3.4,3.5 we give a probabilistic proof of Theorem 7 : with the help of Theorem 3 and of the representation of ACDA, taken from [13], Theorem 7 reduces to the Pollaczeck-Khinchine formula for a simple random walk. In Section 3.6 we explain how Theorem 3 extends to ACDA.
3.2 Basics on automata
In this section, we recall briefly some vocabulary on words and automata, taken from [11, Section 1.3], then we describe the representation of ACDA by boxed diagrams, following [13]. Let be a finite totally ordered set, called alphabet. The elements of are called letters or also symbols. A finite word on the alphabet is a finite sequence of elements of . The set of words is endowed with the operation of concatenation, also called product, in which two words and give the word . This operation is associative, and it has a neutral element, the empty word, denoted by . The length of a word , denoted , is the number of letters in the word (so that ). We denote by the set of finite words on the alphabet .
Definition 1**.**
A deterministic and complete automaton is a quintuplet consisting of:
- •
an alphabet , such that ,
- •
a set of states, such that ,
- •
an initial state ,
- •
a transition function, , that takes as argument a state and a symbol and returns a state, ,
- •
a set of final states .
The transition function has a straightforward extension to , that describes a path from a state to another state through a sequence of letters (=edges) in a directed graph related to , see the figure below. For instance, for ,
[TABLE]
Definition 2**.**
A deterministic finite automaton is accessible when for each state of , there exists a word such that .
Definition 3**.**
A word is recognized by an automaton when . The language recognized by an automaton is the set of words that it recognizes.
Two representations of an ACDA. Consider the ACDA given by the alphabet , , and . The transition table of is a first representation of , for instance :
[TABLE]
The symbol marks the initial state, here it is . The symbols mark the final state(s) (here there is only one final state, ).
Another representation is through a directed graph with edges labelled by and whose set of vertices is : for a directed edge with label is present in the graph of if and only if . Then each vertex of the graph has out-degree , and there is a path from to in the graph if and only if there exists a word such that , hence the term accessible. Only the initial state has an ingoing edge with no starting point, and the final states have an outgoing edge with no endpoint.
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The accessibility of an automaton depends only on its transition structure , not on its final states, thus one can discuss the accessibility of a complete deterministic transition structure (CDTS) : can be seen as a map from the set of edges , including thus plus the starting edge , such that and . The CDTS is accessible only if its transition function is a surjection, that is, has to belong to and this has to be the connection between the impatient collector and ACDA. However, two problems arise:
- •
though has elements, a total order would be handy to identify with , and with an element of ;
- •
the surjectivity of is not sufficient to insure the connexity of .
It turns out that the answer to the second point is also an answer to the first point : as usual for the connexity of graphs, a necessary and sufficient condition of connexity is a positivity condition for a path related to the breadth-first-search of the corresponding graph, and this breadth-first search also provides a total order on , which, with the alphabetic order on , induces a lexicographic order on , allowing to identify with . The path, then, is the completion curve for once is identified to .
More precisely, the search starts from the initial vertex, , of the first directed edge , end vertex relabelled for it is the first piece in the collection. Then one explores the edges starting from , in the alphabetic order, from to , and when this exploration is over, either there exists no new piece in the collection, meaning that is a connected component by itself, and meaning that is not accessible, or there exists some new piece. Thus the completion curve of an accessible CDTS must satisfy . The new vertices, at this stage, are of the form
[TABLE]
and they are sorted (and explored) according to the alphabetic order for the letters . They are also relabeled . Similarly, after the exploration of the neighbours of the second piece of the collection, we need , else would be a connected component. In general, the CDTS is accessible if and only if the completion curve of satisfies
[TABLE]
Now, according to [13], the boxed diagram of is just the completion curve of , decorated with one mark in each column, at height , meaning that .
Example: In Figure 1, we see how the breadth-first search of the graph produces a labeling of the vertices and edges, which, in turn, dictates the order of the search: the ends of the edges starting from a given vertex are searched in the alphabetic order, and the vertices are searched according to their order of apparition during the breadth-first search, beginning with the starting vertex. The 24 CDTS, obtained by permutation of the symbols as labels of the vertices of our example on the left, would produce the same labeling as is pictured on the left. Note that the correspondance edge-endpoint is a surjection from to , with a special property: the partition , here e.g.
[TABLE]
is necessarily sorted in increasing order of the smallest elements of the parts.
Similarly, a sequence (a surjection) can be matched with a boxed diagram in in exactly ways, as follows : according to the coupon collector metaphor, the collection process produces an order among the elements of the collection:
[TABLE]
denotes the th element of to enrich the collection ; is a random uniform permutation of . Setting
[TABLE]
we obtain that
[TABLE]
Thus is a boxed diagram associated with the surjection , or with any surjection of the form , with . That is, if , produces the same boxed diagram as , and there exists exactly elements of with the same boxed diagram. Finally, a random uniform surjection produces a random uniform boxed diagram, while a random uniform CDTS produces a random uniform boxed diagram satisfying additionally the constraint (18) (such a boxed diagram is also called a -Dyck boxed diagram). Thus there is a correspondance in which each boxed diagram is related to different surjections of , and a similar (though different) correspondance in which a Dyck boxed diagram is related to different CDTS. Note that, according to (18),
[TABLE]
meaning that does not satisfy inequality (18).
3.3 Reduction to the NorthEast corner
Now denotes the subset of elements meeting the condition (18). Then Theorem 7 is equivalent to an assertion on the asymptotic behaviour of the Markov chain studied at Section 2.1 : the probability that the sample path of crosses the line outside its endpoints and converges to .
This probabilistic formulation of Koršunov’s formula hints at the relation with Theorem 3 : as Figure 3 shows, Theorem 3, with Proposition 9, prevents these crossings outside the close vicinity of the endpoints, but for a small probability. For large enough, a crossing inside the interval violates the convergence to the limit path at the rate given by Theorem 3, thus such crossings happen with a probability smaller than . As a consequence, the line of proof for Theorem 7 goes according to the following steps: set
[TABLE]
and let the event that a crossing happens inside the interval be denoted . Then
[TABLE]
in which, due to Theorem 3,
[TABLE]
We also have:
Proposition 3**.**
If is small enough,
[TABLE]
As a consequence, the asymptotic behaviour of the profile in the NorthEast corner should give simultaneously the limit of , and Koršunov’s formula. This is the topic of the next sections.
Proof.
An alternative formulation of the condition is as follows: *holds true for *. The proof of (21) has two steps :
[TABLE]
and
[TABLE]
Now:
[TABLE]
But
[TABLE]
for small enough, in which case we have :
[TABLE]
as expected.
For (23), note that :
[TABLE]
But, under , is a sequence of independent random variables (with geometric distributions and respective expectations ), so, according to [14]:
[TABLE]
or, equivalently,
[TABLE]
which is relation (23). ∎
3.4 Random walks and Pollaczek-Khintchine’s formula
In this section, and the next one, we shall prove that
[TABLE]
Relation (24) results from the Pollaczek-Khinchine formula, as we shall see now: relates to a crossing of the line by before time . But, before time , i.e. for close to , due to Proposition 1 and Theorem 4, the transition probabilities of are close to the constant , so that we expect to behave, early, like a random walk starting at , with step distribution
[TABLE]
Since , the trend is that does not cross the line, and if it does at all, the crossing has to take place early, hence we expect the crossing probability of Z to be the limit of . In the next section, we shall discuss the convergence (to ) of , and its speed. In this section, we compute the crossing probability of , in which:
[TABLE]
Proposition 4**.**
[TABLE]
Proof.
It is convenient to make some time and space changes to represent our crossing probability in more familar terms, i.e. in terms of a new random walk on the integers, with negative drift, starting from 0, and such that holds true (or such that and are closely related events, actually). If we set, for ,
[TABLE]
then is a random walk with step distribution
[TABLE]
with drift
[TABLE]
starting from 1 at time -1, and:
[TABLE]
Thus
[TABLE]
But we know, from the Pollaczek-Khinchine formula (cf. [1, Corollary 6.6]) that is the stationary distribution at 0 of the Lindsey process with step . For , let denote the average time needed by the Lindsey process (or, indifferently, by the random walk ) to hit 0 starting from position : Wald’s identity gives that
[TABLE]
On the other hand, if is the expected time of the first return to 0, starting from 0, then, by the Markov property,
[TABLE]
and finally:
[TABLE]
Thus, as expected, . ∎
3.5 Tail probabilities and Hoeffding’s inequality
For some process , let denote the section of the sample path . First, let us bound the distance between the random walk of Proposition 1, and :
Proposition 5**.**
Under , converges to in distribution. Moreover, for , there exists such that for large enough :
[TABLE]
Proof.
We shall use that if
[TABLE]
then:
[TABLE]
Consider a sample path in which . Let denote its th increment. Under , as a consequence of Proposition 1, for any given ,
[TABLE]
while
[TABLE]
For and large enough, and for a suitable choice of , belongs to , so that, according to (12), for and ,
[TABLE]
Since the probability of a given sample path of , resp. , is a product of terms or of the following form
[TABLE]
in which is the increment for some step of the random walks of , resp. , and as a consequence of (26) and (25) (with ), the probabilities of these sample paths of length differ by at most
[TABLE]
That a set has at most admissible elements starting at position entails that Proposition 5 holds true with the choice . ∎
Let (resp. ) denote the set of finite (resp. finite or infinite) words on the alphabet , and for a finite word , set . For , let us define the crossing set as follows:
[TABLE]
so that, for instance,
[TABLE]
The next proposition completes the proof of Theorem 7 :
Proposition 6**.**
[TABLE]
Proof.
We shall prove successively that:
[TABLE]
for , in which . First, Proposition 5 entails (28) at once. Relations (27) and (29) both follow from Hoeffding’s inequality. For (27) it is rather straightforward : if we set , relation (44) entails that , so that
[TABLE]
Thus the probability of a crossing at some point after time satisfies
[TABLE]
Similarly
[TABLE]
but here we cannot use Hoeffding’s inequality directly, though is a sum of Bernoulli random variables, for these Bernoulli random variables are not independent. However, we can build, on the same probability space, a copy of and a random walk in such a way that, for , and ’s drift is smaller than , using a sequence of independent random variables, uniform on . Set and
[TABLE]
For and to have the same distribution, due to Proposition 1 , we need to set and
[TABLE]
For , and for , if we choose large enough, so that , and so that
[TABLE]
we can use (26) to obtain that
[TABLE]
Hence
[TABLE]
the second inequality due to Hoeffding’s inequality. Relation (29) follows. ∎
3.6 The profile of an accessible automaton
Jointly with Theorem 7, Theorem 3 has a straightforward consequence : the completion curve of a uniform ADCA has the same limit curve. More precisely, if denotes the uniform distribution on ADCA with vertices and letters, or, equivalently is the conditional probability given :
[TABLE]
then
Lemma 2**.**
For a sequence of events ,
[TABLE]
Thus, Theorem 3 translates to large automata at once, and we obtain
Theorem 8**.**
For any , there exists such that, for ,
[TABLE]
Recall that is defined at the end of Section 1.2.
4 Saddle-point method and Stirling numbers
This section is devoted to the proof of Theorem 6.
4.1 Generating function and Cauchy formula
Recall the notations:
[TABLE]
According to [6, (6)] or [5, Example III.11, p.179], we have :
[TABLE]
in which
[TABLE]
By the Cauchy formula,
[TABLE]
in which
[TABLE]
to be compared to the asymptotic equivalent to given by [6, (3)], , that satisfies:
[TABLE]
We expect that for any , or , since , as a power series in , has positive coefficients. We also expect that, around 0,
[TABLE]
and more precisely, since is a saddle-point, we expect that
[TABLE]
According to (46),
[TABLE]
which entails that
[TABLE]
Set
[TABLE]
in which a suitable choice of is made later, so that:
[TABLE]
In the next sections, in order to prove Theorem 6, we obtain that
[TABLE]
4.2 Central term
In this section we obtain a saddlepoint bound for , following [5]. We write
[TABLE]
in which :
[TABLE]
is the characteristic function of any Poisson random variable with expectation . For our aims, we need a precise estimation of , obtained through the Taylor-Laplace inequality, see Section 5.6. There, we prove that, for suitable constants ,
[TABLE]
in which, according to Section 5.6.2,
[TABLE]
Note that, according to (46),
[TABLE]
We can write
[TABLE]
Now
[TABLE]
Thus, has to be large for to be . On the other hand,
[TABLE]
in which, for ,
[TABLE]
With the help of (53), since is bounded for , we obtain that
[TABLE]
in which is discussed at Section 5.6.2. For to be small, cannot be too large:
[TABLE]
yields that
[TABLE]
and that, for , and ,
[TABLE]
Now
[TABLE]
in which the dependence of on is studied at Section 5.6, in order to complete the proof of Theorem 6. Finally
[TABLE]
For inequality (38), note that, due to inequalities (54), if , then for large enough. For the first term, since
[TABLE]
and , we have
[TABLE]
for large enough. Also:
[TABLE]
Thus
[TABLE]
4.3 Tail
As for , relation (34) yields that:
[TABLE]
Set:
[TABLE]
Following [12, Lemma 1& 2], we prove that
Lemma 3**.**
For ,
[TABLE]
or equivalently
[TABLE]
Proof.
For , set:
[TABLE]
so that:
[TABLE]
and:
[TABLE]
But
[TABLE]
Thus
[TABLE]
For ,
[TABLE]
leading to:
[TABLE]
∎
Thus, according to Section 5.6.5, for large enough,
[TABLE]
Finally we are ready to prove Theorem 6.
Proof.
Inequality (40) holds true for large enough, and, on , its coefficients are positive continuous functions of , thus, for , they are bounded. One can deal similarly with inequality (41) (see Section 5.6.5). ∎
5 Appendix: Some special functions
5.1 as an implicit function of
We have seen that is an essential parameter in the asymptotic behaviour of the Stirling number
[TABLE]
in which is defined by , and is an implicit function of , defined by
[TABLE]
For instance, the completion curve of Theorem 3 is defined in terms of and in terms of . Thus we need to list some of the properties of that are of interest in our proofs, not all of them being straightforward, for instance in order to prove Theorem 4 in Section 5.5.
Proposition 7**.**
The function is increasing, nonnegative and concave, and is increasing, nonnegative and concave as well. Also, we have:
[TABLE]
Proof.
Proof of (42). Relation (2) entails
[TABLE]
at once. Since ,
[TABLE]
so, from
[TABLE]
we deduce that
[TABLE]
In order to prove that , we need to prove that
[TABLE]
but the last inequality holds true for any positive number , as a consequence of
[TABLE]
Proof of monotony and concavity of and . Note that
[TABLE]
entails . For we have no additional trouble: when ,
[TABLE]
Now, from the implicit function theorem, we obtain:
[TABLE]
entailing that , and as well, are increasing. Then
[TABLE]
so that
[TABLE]
It follows that and are concave. Finally, (43) is an easy consequence of (2), and (44) follows from:
[TABLE]
and from (42).∎
We also need that :
Lemma 4**.**
[TABLE]
Proof.
The relation (46) can be written successively:
[TABLE]
the last one being clearly true. ∎
5.2 Large deviation for the coupon collector
Since, for , we have:
[TABLE]
Theorem 5 entails that
[TABLE]
in which
[TABLE]
One can write :
[TABLE]
and finally
[TABLE]
Also:
[TABLE]
Thus, is decreasing and
[TABLE]
which entails that is positive.
5.3 Properties of the limit path.
The properties of the sample path solution of
[TABLE]
between and matter to our saddle-point estimates for the Stirling numbers, since these estimates are valid only when the sample path is far away from , or, equivalently, when is large enough and is far from . We are specially interested by the solution obtained on the interval when , for it is the asymptotic completion curve mentionned in Theorem 3. In this section, we prove that satisfies
[TABLE]
and stays away from , if is large enough, as shown in Figure 5. This follows from the variations of along the curve , where we have:
[TABLE]
Proposition 8**.**
The solution to (47) going through satisfies, for ,
[TABLE]
Proof.
From (47), we obtain the differential equation for :
[TABLE]
**Lower bounds. ** Relations (50) and (46) yields that
[TABLE]
thus, for ,
[TABLE]
leading to the lower bounds of (49).
**Upper bounds. ** With (42) and (50) together, we obtain:
[TABLE]
leading to the upper bounds in Proposition 8. ∎
These estimates are also useful in the proof of Koršunov’s formula, in which we need that a strip close to the limit path intersects the forbidden zone only close to its endpoints and . We have :
Proposition 9**.**
For , and ,
[TABLE]
Proof.
Due to (49), we only need to prove that
[TABLE]
when is in the interval, and it follows easily from the fact that
[TABLE]
holds true at the endpoints. ∎
5.4 Small variations of
In this section, we bound the variations of in order to obtain the accuracy of the Euler scheme used in Theorem 3, cf. (13), and also to obtain the precision of the approximation of the completion curve by a random walk in the proof of Koršunov’s formula. Since , according to (45)
[TABLE]
Thus, according to (50), a sample path solution of (47) satisfies
[TABLE]
Similarly, anywhere in the domain,
[TABLE]
since holds true, for it reduces to . Thus we have:
Proposition 10**.**
If ,
[TABLE]
5.5 Proof of Theorem 4
5.5.1 Small variations of
For , and , , , we set, for any real function ,
[TABLE]
Then we have:
Proposition 11**.**
For , , we have
[TABLE]
Proof.
We need a bound for
[TABLE]
Thus,
[TABLE]
In order to bound , after some computations starting from:
[TABLE]
we obtain:
[TABLE]
since
[TABLE]
yielding a bound, for the second derivative, that entails the desired result. ∎
5.5.2 Final argument
Now we can use Theorem 6 to bound the error in the approximation of the transition probability by :
[TABLE]
Set :
[TABLE]
so that :
[TABLE]
First :
[TABLE]
Since
[TABLE]
we have
[TABLE]
According to Theorem 6,
[TABLE]
thus, for large enough, and
[TABLE]
Now, for some ,
[TABLE]
in which , that turns out to be , is defined as follows :
[TABLE]
We write
[TABLE]
with
[TABLE]
The factor in is the reason why we need the second order approximations of Section 5.5.1. Now we see, from Proposition 11, that :
[TABLE]
so that
[TABLE]
But, more precisely, Proposition 11 entails:
[TABLE]
that is, . Now, for , since , , , are increasing and concave, then is increasing and concave too, being composed with an increasing and concave function, so, due to Taylor-Lagrange formula, all these functions satisfy:
[TABLE]
and that yields:
[TABLE]
Also, for ,
[TABLE]
so that, using (46),
[TABLE]
and, using (46) again, for ,
[TABLE]
so that, for and , we have and
[TABLE]
For instance, this holds true for .
5.6 Explicit bounds for the second order asymptotics of
In this section, we provide detailed computations in order to bound , thus completing the proof of Theorem 6.
5.6.1 Taylor coefficients
As usual, the derivatives of a characteristic function such as are bounded as follows:
[TABLE]
thus, due to (36), we need the first moments of the Poisson distribution, given by the Touchard polynomials:
[TABLE]
in order to compute the coefficients in the Taylor-Laplace formula for , for the derivatives of are obtained through the Leibniz rule, as follows:
[TABLE]
This gives the coefficients in the Taylor-Laplace inequality:
[TABLE]
that is:
[TABLE]
computations needed in order to bound the coefficients in (40).
5.6.2 Upper bound for
Finally, for , the fifth derivative is bounded as follows:
[TABLE]
in which we use again and again and , cf. Figure 6.
Thus we just proved that:
[TABLE]
in which . This leads to contributing to the bound for through
[TABLE]
so that entails
[TABLE]
In the next sections, we shall also use the following inequalities:
[TABLE]
5.6.3 Upper bound for
The choice insures that
[TABLE]
so that
[TABLE]
Then , and, as a function of , is bounded for , for any choice of , just like , , and the other coefficients in relation (40). More precisely, for , for instance, we have
[TABLE]
For , and large enough, we have , thus entails that
[TABLE]
and , as a consequence,
[TABLE]
Then
[TABLE]
but, since ,
[TABLE]
and
[TABLE]
Finally
[TABLE]
and does the trick. Finally, entails that the corresponding contribution to is bounded as follows :
[TABLE]
5.6.4 Upper bound for and for
First, inequality (39) holds true, for instance, when
[TABLE]
while inequality (38) holds true if , for instance if
[TABLE]
Then
[TABLE]
holds true if one chooses:
[TABLE]
Finally, entails that the corresponding contribution to is bounded as follows :
[TABLE]
Similarly
[TABLE]
5.6.5 Upper bound for
According to (41),
[TABLE]
we have
[TABLE]
as desired, provided that:
[TABLE]
Due to the variations of , , this amounts to:
[TABLE]
5.6.6 Conclusion
Finally, for and large enough (be more precise),
[TABLE]
more precisely,
[TABLE]
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