The analysis of strings of $n$ random variables with geometric distribution has recently attracted renewed interest: Archibald et al. consider the number of distinct adjacent pairs in geometrically distributed words. They obtain the asymptotic ($n\rightarrow\infty$) mean of this number in the cases of different and identical pairs. In this paper we are interested in all asymptotic moments in the identical case, in the asymptotic variance in the different case and in the asymptotic distribution in both cases. We use two approaches: the first one, the probabilistic approach, leads to variances in both cases and to some conjectures on all moments in the identical case and on the distribution in both cases. The second approach, the combinatorial one, relies on multivariate pattern matching techniques, yielding exact formulas for first and second moments. We use such tools as Mellin transforms, Analytic Combinatorics, Markov Chains.

In discrete time, ℓ-blocks of red lights are separated by ℓ-blocks of green lights. Cars arrive at random. We seek the distribution of maximum line length of idle cars, and justify conjectured probabilistic asymptotics algebraically for 2≤ℓ≤3 and numerically for ℓ≥4.

In this paper, we analyze the stochastic properties of some large size (area) polyominoe's perimeter such that the directed column-convex polyomino, the column-convex polyomino, the directed diagonally-convex polyomino, the staircase (or parallelogram) polyomino, the escalier polyomino, the wall (or bargraph) polyomino. All polyominoes considered here are made of contiguous, not-empty columns, without holes, such that each column must be adjacent to some cell of the previous column. We compute the asymptotic (for large size $n$) Gaussian distribution of the perimeter, including the corresponding Markov property of the chain of columns, and the convergence to classical Brownian motions of the perimeter seen as a trajectory according to the successive columns. All polyominoes of size $n$ are considered as equiprobable.

We obtain an asymptotic series $$\sum _{j=0}^\infty \frac{I_j}{n^j}$$ for the integral $$\int _0^1[x^n+(1-x)^n]^{\frac{1}{n}}\mathrm{{d}}x$$ as $$n\rightarrow \infty $$, and compute $$I_j$$ in terms of alternating (or “colored”) multiple zeta values. We also show that $$I_j$$ is a rational polynomial in the ordinary zeta values, and give explicit formulas for $$j\le 12$$. As a by-product, we obtain precise results about the convergence of norms of random variables and their moments. We study $$Z_n=\Vert (U,1-U)\Vert _n$$ as n tends to infinity and we also discuss $$W_n=\Vert (U_1,U_2,\dots ,U_r)\Vert _n$$ for standard uniformly distributed random variables.

The problem of estimating the number n of distinct keys of a large collection of N data is well known in computer science. A classical algorithm is the adaptive sampling (AS). n can be estimated by R2 J , where R is the final bucket size and J is the final depth at the end of the process. Several new interesting questions can be asked about AS (some of them were suggested by P.Flajolet and popularized by J.Lumbroso). The distribution of W = log(R2 J /n) is known, we rederive this distribution in a simpler way. We provide new results on the moments of J and W. We also analyze the final cache size R distribution. We consider colored keys: assume also that among the n distinct keys, m do have color K We show how to estimate p = m n. We study keys with some multiplicity : we provide a way to estimate the total number M of some color K keys among the total number N of keys. We consider the case where we know a priori the multiplicities but not the colors. There we want to estimate the total number of keys N. An appendix is devoted to the case where the hashing function provides bits with probability different from 1/2.

Let a word be a sequence of n i.i.d. integer random variables. The perimeter P of the word is the number of edges of the word, seen as a polyomino. In this paper, we present a probabilistic approach to the computation of the moments of P. This is applied to uniform and geometric random variables. We also show that, asymptotically, the distribution of P is Gaussian and, seen as a stochastic process, the perimeter converges in distribution to a Brownian motion.

In discrete time, $\ell$-blocks of red lights are separated by $\ell$-blocks of green lights. Cars arrive at random. We seek the distribution of maximum line length of idle cars, and justify conjectured probabilistic asymptotics for $2 \leq \ell \leq 3$.

In discrete time, ℓ-blocks of red lights are separated by ℓ-blocks of green lights. Cars arrive at random. The maximum line length of idle cars is fully understood for ℓ = 1, but only partially for 2 ≤ℓ≤ 3.

When we first encountered these integer sequences, we assumed that their asymptotic developments had been well studied, but after extensively checking the literature, we believe that only special cases of the asymptotics have been analyzed. Due to the fundamental nature of these integer sequences, we decided to make a comprehensive characterization of the asymptotic growth of these integer sequences, as n→∞, for any (fixed) positive values m and k. These integer sequences are intimately connected with hypergeometric functions, as seen in equation (1). These integer sequences have also been of interest for a long time. The general family am,k(n) appears, for instance, in Barrucand [10]. The asymptotic growth of a2,k(n) has been known for (almost) a century [14], and perhaps longer. In the case k = 1, the values of am,k(n) are simply the powers of m, namely, am,1(n) = m n. The k = 2 case has myriad interpretations. They are used in Proposition 1 of Borwein et al. [3] and in the discussion and remarks after the proposition is proved. They use the notation “Wm(2n)” for our sequences am,2(n). Borwein and his co-authors point out that am,2(n) is the number of abelian squares of length 2n constructed from an alphabet that has m letters (i.e., strings of the form x1, . . . , xn, xσ(1), . . . , xσ(n) where σ is a permutation of 1, . . . , n); also see [15] for more details about such abelian squares. For another application of the family of sequences am,2(n), in number theory, see [2, p. 108]. The integers a4,2(n) are known as the Domb numbers; they enumerate the number of 2n-step polygons on a diamond lattice; see [6] and also OEIS #A002895. The sequences am,2(n) are also a key object of study in [17]. We close our discussion of the k = 2 case by noting ∗Université Libre de Bruxelles, Département d’Informatique, CP 212, Boulevard du Triomphe, B-1050 Bruxelles, Belgium, email: louchard@ulb.ac.be †Purdue University, Department of Statistics, 150 North University Street, West Lafayette, IN, USA, email: mdw@purdue.edu

Let $I(n):=\int_0^1 [x^n+(1-x)^n]^\frac1n dx.$ In this paper, we show that $I(n)= \sum_0^\infty \frac{I_i}{n^i},n\rightarrow \infty$ and we compute $I_i, i =0..5$, obtained by polylog functions and Euler sums. As a corollary, we obtain explicit expressions for some integrals involving functions $ u^i, exp(-u), (1 +exp(-u))^j , ln(1 + exp(-u))^k$ . As another asymptotic result, let $S_0(z):=\frac{Li_m(1)}{Li_m(1)-Li_m(z)}$, where $Li_m(z)$ is the polylog function. We provide the asymptotic behaviour of $S_n,n\rightarrow \infty$ where $S_n:=[z^n]S_0(z)$. This paper fits within the framework of analytic combinatorics.

Using the Saddle point method and multiseries expansions, we obtain from the exponential formula and Cauchy's integral formula, asymptotic results for the number $T(n,m,k)$ of partitions of $n$ labeled objects with $m$ blocks of fixed size $k$. We analyze the central and non-central region. In the region $m=n/k-n^\al,\quad 1>\al>1/2$, we analyze the dependence of $T(n,m,k)$ on $\al$. This paper fits within the framework of Analytic Combinatorics.

The classical secretary problem has been generalized over the years into several directions. In this paper we confine our interest to those generalizations which have to do with the more general problem of stopping on a last observation of a specific kind. We follow Dendievel, (where a bibliography can be found) who studies several types of such problems, mainly initiated by Bruss and Weber. Whether in discrete time or continuous time, whether all parameters are known or must be sequentially estimated, we shall call such problems simply Bruss-Weber problems. Our contribution in the present paper is a refined analysis of several problems in this class and a study of the asymptotic behaviour of solutions. The problems we consider center around the following model. Let $X_1,X_2,\ldots,X_n$ be a sequence of independent random variables which can take three values: $\{+1,-1,0\}.$ Let $p:=¶(X_i=1), p':=¶(X_i=-1), \qt:=¶(X_i=0), p\geq p'$, where $p+p'+\qt=1$. The goal is to maximize the probability of stopping on a value $+1$ or $-1$ appearing for the last time in the sequence. Following a suggestion by Bruss, we have also analyzed an x-strategy with incomplete information: the cases $p$ known, $n$ unknown, then $n$ known, $p$ unknown and finally $n,p$ unknown are considered. We also present simulations of the corresponding complete selection algorithm.

In this paper, we analyze the asymptotic number I ( m, n ) of involutions of large size n with m singletons. We consider a central region and a non-central region. In the range m = n − n α , 0 < α < 1, we analyze the dependence of I ( m, n ) on α . This paper fits within the framework of Analytic Combinatorics.

In this paper, we analyze the asymptotic number \( I(m,n) \) of involutions of large size \( n \) with \( m \) singletons. We consider a central region and a non-central region. In the range \( m = n - n^\alpha \), \( 0 < \alpha < 1 \), we analyze the dependence of \( I(m,n) \) on \( \alpha \). This paper fits within the framework of Analytic Combinatorics.

The objective of this paper is to find in a setting of n sequential observations of objects a good online policy to select the k bestof these n uniquely rankable objects. This focus is motivated by the fact that it is hard to find closed form solutions of optimalstrategies for general k and n. Selection is without recall, and the idea is to investigate threshold functions which maintain allpresent information, that is thresholds which are functions of all selections made so far. Our main interest lies in the asymptoticbehaviour of these thresholds as n -> infinity and in the corresponding asymptotic performance of the threshold algorithm.

The following type of dice games has been mentioned and/or studied in the literature. Players take turns in rolling a fair die successively, each player accumulating his or her scores as long as the outcome 1 does not occur. If the result 1 turns up, the accumulated score is wiped out, and the turn ends, that is the player gives the die to the next player. At any stage after a roll, the player (she, say) can choose to end her turn and bank her accumulated score. The winner is the first player to reach some xed target n 2 N. We present some new results on optimal strategies and winning probability in a one or two players game. For just one player there is no competition of course, and in this case we suppose that the player simply wants to minimize her total expected number of tosses over all possible banking strategies.

The following classical asymmetric leader election algorithm has obtained quite a bit of attention lately. Starting with n players, each one throws a coin, and the k of them which have each thrown a head (with probability q) go on, and the leader will be found amongst them, using the same strategy. Should nobody advance, the party will repeat the procedure. One of the most interesting parameter here is the number J(n) of rounds until a leader has been identified. In this paper we investigate, in the classical leader election algorithm, what happens near the end of the game, namely we fix an integer n and we study the behaviour of the number of survivors G at level J(n) n. In our asymptotic analysis (for n co) we are focusing on the limiting distribution functions. We also investigate what happens, if the parameter p = 1 q gets small (p 0) or large (p 1). We use three efficient tools: an urn model, a Mellin-Laplace technique for harmonic sums and some asymptotic distributions related to one of the extreme -value distributions: the Gumbel law. This study was motivated by a recent paper by Kalpathy, Mahmoud and Rosenkrantz, where they consider the number of survivors S,t, after t election rounds, in a broad class of fair leader election algorithms starting with n candidates.

Using the Saddle point method and multiseries expansions, we obtain from the generating function of the Eulerian numbers An;k and Cauchy’s integral formula, asymptotic results in noncentral region. In the region k = n n ; 1 > > 1=2, we analyze the dependence of An;k on . This paper ts within the framework of Analytic Combinatorics.

The objective of this paper is to find in a setting of n sequential observations of objects a good online policy to select the κ best of these n uniquely rankable objects. This focus is motivated by the fact that it is hard to find closed form solutions of optimal strategies for general κ and n. Selection is without recall, and the idea is to investigate threshold functions which maintain all present information, that is thresholds which are functions of all selections made so far. Our main interest lies in the asymptotic behaviour of these thresholds as n → ∞ and in the corresponding asymptotic performance of the threshold algorithm.

The present paper makes three distinct improvements over an earlier investigation of Kalpathy and Ward. We analyze the length of the entire election process (not just one participant’s duration), for a randomized election algorithm, with a truncated geometric number of survivors in each round. We not only analyze the mean and variance; we analyze the asymptotic distribution of the entire election process. We also introduce a new variant of the election that guarantees a unique winner will be chosen; this methodology should be more useful in practice than the previous methodology. The method of analysis includes a precise analytic (complex-valued) approach, relying on singularity analysis of probability generating functions.