Limit laws are proven by the contraction method for random vectors of a recursive nature as they arise as parameters of combinatorial structures such as random trees or recursive algorithms, where we use the Zolotarev metric. In comparison to previous applications of this method, a general transfer theorem is derived which allows us to establish a limit law on the basis of the recursive structure and the asymptotics of the first and second moments of the sequence. In particular, a general asymptotic normality result is obtained by this theorem which typically cannot be handled by the more common 2 metrics. As applications we derive quite automatically many asymptotic limit results ranging from the size of tries or m-ary search trees and path lengths in digital structures to mergesort and parameters of random recursive trees, which were previously shown by different methods one by one. We also obtain a related local density approximation result as well as a global approximation result. For the proofs of these results we establish that a smoothed density distance as well as a smoothed total variation distance can be estimated from above by the Zolotarev metric, which is the main tool in this article.

We prove convergence in distribution for the profile (the number of nodes at each level), normalized by its mean, of random recursive trees when the limit ratio˛of the level and the logarithm of tree size lies in OE0; e/. Convergence of all moments is shown to hold only for˛2 OE0; 1 (with only convergence of finite moments when˛2 .1; e/). When the limit ratio is 0 or 1 for which the limit laws are both constant, we prove asymptotic normality for˛D 0 and a "quicksort type" limit law for˛D 1, the latter case having additionally a small range where there is no fixed limit law. Our tools are based on contraction method and method of moments. Similar phenomena also hold for other classes of trees; we apply our tools to binary search trees and give a complete characterization of the profile. The profiles of these random trees represent concrete examples for which the range of convergence in distribution differs from that of convergence of all moments.

Nonstationarity of the event rate is a persistent problem in modeling time series of events, such as neuronal spike trains. Motivated by a variety of patterns in neurophysiological spike train recordings, we define a general class of renewal processes. This class is used to test the null hypothesis of stationary rate versus a wide alternative of renewal processes with finitely many rate changes (change points). Our test extends ideas from the filtered derivative approach by using multiple moving windows simultaneously. To adjust the rejection threshold of the test, we use a Gaussian process, which emerges as the limit of the filtered derivative process. We also develop a multiple filter algorithm, which can be used when the null hypothesis is rejected in order to estimate the number and location of change points. We analyze the benefits of multiple filtering and its increased detection probability as compared to a single window approach. Application to spike trains recorded from dopamine midbrain neurons in anesthetized mice illustrates the relevance of the proposed techniques as preprocessing steps for methods that assume rate stationarity. In over 70% of all analyzed spike trains classified as rate nonstationary, different change points were detected by different window sizes.

We study the profile $X_{n,k}$ of random search trees including binary search trees and $m$-ary search trees. Our main result is a functional limit theorem of the normalized profile $X_{n,k}/\mathbb{E}X_{n,k}$ for $k=\lfloor\alpha\log n\rfloor$ in a certain range of $\alpha$. A central feature of the proof is the use of the contraction method to prove convergence in distribution of certain random analytic functions in a complex domain. This is based on a general theorem concerning the contraction method for random variables in an infinite-dimensional Hilbert space. As part of the proof, we show that the Zolotarev metric is complete for a Hilbert space.Comment: Published in at http://dx.doi.org/10.1214/07-AAP457 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

We characterize all limit laws of the quicksort type random variables defined recursively by X n d = X In + X * n−1−In + T n when the "toll function" T n varies and satisfies general conditions, where (X n), (X * n), (I n , T n) are independent, X n d = X * n , and I n is uniformly distributed over {0,. .. , n − 1}. When the "toll function" T n (cost needed to partition the original problem into smaller subproblems) is small (roughly lim sup n→∞ log E(T n)/ log n ≤ 1/2), X n is asymptotically normally distributed; non-normal limit laws emerge when T n becomes larger. We give many new examples ranging from the number of exchanges in quicksort to sorting on broadcast communication model, from an in-situ permutation algorithm to tree traversal algorithms, etc.