Heavy tailed solutions of multivariate smoothing transforms

Let $N > 1$ be a fixed integer and $(C_1,..., C_N,Q)$ a random element of $GL(d, \R)^N x \R^d$. We consider solutions of multivariate smoothing transforms, i.e. random variables $R$ satisfying R \eqdist \sum_{i=1}^N C_i R_i +Q where \eqdist denotes equality in distribution, and $R, R_1,..., R_N$ are independent identically distributed $\R^d$-valued random variables, and independent of $(C_1,..., C_N, Q)$. We briefly review conditions for the existence of solutions, and then study their asymptotic behaviour. We show that under natural conditions, these solutions exhibit heavy tails. Our results also cover the case of complex valued weights $(C_1,..., C_N)$.Comment: 35 page

Runs in superpositions of renewal processes with applications to discrimination

AbstractWald and Wolfowitz [Ann. Math. Statist. 11 (1940) 147–162] introduced the run test for testing whether two samples of i.i.d. random variables follow the same distribution. Here a run means a consecutive subsequence of maximal length from only one of the two samples. In this paper we contribute to the problem of runs and resulting test procedures for the superposition of independent renewal processes which may be interpreted as arrival processes of customers from two different input channels at the same service station. To be more precise, let (Sn)n⩾1 and (Tn)n⩾1 be the arrival processes for channel 1 and channel 2, respectively, and (Wn)n⩾1 their be superposition with counting process N(t)=defsup{n⩾1:Wn⩽t}. Let further Rn* be the number of runs in W1,…,Wn and Rt=RN(t)* the number of runs observed up to time t. We study the asymptotic behavior of Rn* and Rt, first for the case where (Sn)n⩾1 and (Tn)n⩾1 have exponentially distributed increments with parameters λ1 and λ2, and then for the more difficult situation when these increments have an absolutely continuous distribution. These results are used to design asymptotic level α tests for testing λ1=λ2 against λ1≠λ2 in the first case, and for testing for equal scale parameters in the second

Weak convergence to the t-distribution

We present a new limit theorem for random means: if the sample size is not deterministic but has a negative binomial or geometric distribution, the limit distribution of the normalised random mean is a t-distribution with degrees of freedom depending on the shape parameter of the negative binomial distribution. Thus the limit distribution exhibits exhibits heavy tails, whereas limit laws for random sums do not achieve this unless the summands have innite variance. The limit law may help explain several empirical regularities. We consider two such examples: rst, a simple model is used to explain why city size growth rates are approximately t-distributed. Second, a random averaging argument can account for the heavy tails of high-frequency returns. Our empirical investigations demonstrate that these predictions are borne out by the data.convergence, t-distribution, limit theorem

How long is the convex minorant of a one-dimensional random walk?

We prove distributional limit theorems for the length of the largest convex minorant of a one-dimensional random walk with independent identically distributed increments. Depending on the increment law, there are several regimes with different limit distributions for this length. Among other tools, a representation of the convex minorant of a random walk in terms of uniform random permutations is utilized

Polyharmonic Functions And Random Processes in Cones

We investigate polyharmonic functions associated to Brownian motions and random walks in cones. These are functions which cancel some power of the usual Laplacian in the continuous setting and of the discrete Laplacian in the discrete setting. We show that polyharmonic functions naturally appear while considering asymptotic expansions of the heat kernel in the Brownian case and in lattice walk enumeration problems. We provide a method to construct general polyharmonic functions through Laplace transforms and generating functions in the continuous and discrete cases, respectively. This is done by using a functional equation approach

On multivariate stochastic fixed point equations:the smoothing transform and random difference equations

In dieser Arbeit werden Verteilungen studiert, welche multivariate stochastische Fixpunktgleichungen lösen. Im Falle der multivariaten Smoothing Transform (homogen und inhomogen) wird die Menge der alpha-elementaren Fixpunkte charakterisiert, und ein Markov-Erneuerungssatz wird bewiesen. Das Tailverhalten des eindeutigen Fixpunktes einer affinen stochastischen Rekursion wird mithilfe der Theorie Harris-rekurrenter Markov-Ketten untersucht. This thesis is concerned with the study of probability measures on R^d, being fixed points of multivariate versions of the smoothing transform (ST) as well as random difference equations (RDE). Considering the ST, a full description of the set of alpha-elementary fixed points is obtained, both for the homogeneous and inhomogeneous case and a simple Markov renewal theorem is proven. Considering RDEs, heavy tail properties of the unique fixed point are studied using regeneration techniques from the theory of Harris recurrent Markov chains

Ein Prinzip moderater Abweichungen für die Größe der größten Komponente in einem Erdös-Rényi-Zufallsgraphen im superkritischen Fall

Diese Arbeit untersucht die größte Komponente eines Erdös-Rényi-Zufallsgraphen G(n,p) mit p=an. Für a>1 ist bekannt, dass die größte Komponente eines solchen Graphen asymptotisch von der Größe cn ist, wenn c die Überlebenswahrscheinlichkeit eines Galton-Watson-Prozesses mit Reproduktionsverteilung Poi(a) ist. Für diesen Fall wir hier ein Prinzip moderater Abweichungen um diesen Wert bewiesen, welches die Lücke zwischen den bekannten Resultaten eines zentralen Grenzwertsatzes und eines Prinzips großer Abweichungen schließt. Dieses Ergebniss wird schließlich auf das Modell G(n,M) übertragen

Large deviations for random matrices in the orthogonal group and Stiefel manifold with applications to random projections of product distributions

We prove large deviation principles (LDPs) for random matrices in the orthogonal group and Stiefel manifold, determining both the speed and good convex rate functions that are explicitly given in terms of certain log-determinants of trace-class operators and are finite on the set of Hilbert-Schmidt operators $M$ satisfying $\|MM^*\|<1$. As an application of those LDPs, we determine the precise large deviation behavior of $k$-dimensional random projections of high-dimensional product distributions using an appropriate interpretation in terms of point processes, also characterizing the space of all possible deviations. The case of uniform distributions on $\ell_p$-balls, $1\leq p \leq \infty$, is then considered and reduced to appropriate product measures. Those applications generalize considerably the recent work [Johnston, Kabluchko, Prochno: Projections of the uniform distribution on the cube - a large deviation perspective, Studia Mathematica 264 (2022), 103-119].Comment: 41 page