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Take a self-similar fragmentation process with dislocation measure $\nu$ and index of self-similarity $\alpha > 0$. Let $e^{-m_t}$ denote the size of the largest fragment in the system at time $t\geq 0$. We prove fine results for the asymptotics of the stochastic process $(m_t)_{t \geq 0}$ for a broad class of dislocation measures satisfying $$\nu(1-s_1 > \delta ) = \delta^{-\theta} \ell(1/\delta),$$ for some $\theta \in (0,1)$ and $\ell:(0,\infty) \to (0,\infty)$ slowly varying at infinity. Under this regularity condition, we find that if for $t \geq 0$ we have the almost-sure convergence $\lim_{t \to \infty} (m_t - g(t)) = 0$, where $$ g(t) :=\left( \log t - (1-\theta) \log \log t +f(t) \right)/\alpha$$ and $f(t) = o(\log \log t)$ can be given explicitly in terms of $\alpha,\theta$ and $\ell(\cdot)$. We prove a similar result in the finite activity case, which corresponds roughly speaking to the setting $\theta = 0$. Our results sharpen significantly the best prior result on general self-similar fragmentation processes, due to Bertoin, which states that $m_t = (1+o(1)) \log (t)/\alpha$., Comment: are welcome!

We study a branching random walk with independent and identically distributed, heavy tailed displacements. The offspring law is supercritical and satisfies the Kesten-Stigum condition. We treat the case when the law of the displacements does not lie in the max-domain of attraction of an extreme value distribution. Hence, the classical extreme value theory, which is often deployed in this kind of models, breaks down. We show that if the tails of the displacements are such that the absolute value of the logarithm of the tail is a slowly varying function, one can still effectively analyse the extremes of the process. More precisely, after a non-linear transformation the extremes of the branching random walk process converge to a cluster Cox process.

We develop a probabilistic approach to the celebrated Jacobian conjecture, which states that any Keller map (i.e. any polynomial mapping $F\colon \mathbb{C}^n \to \mathbb{C}^n$ whose Jacobian determinant is a nonzero constant) has a compositional inverse which is also a polynomial. The Jacobian conjecture may be formulated in terms of a problem involving labellings of rooted trees; we give a new probabilistic derivation of this formulation using multi-type branching processes. Thereafter, we develop a simple and novel approach to the Jacobian conjecture in terms of a problem involving shuffling subtrees of $d$-Catalan trees, i.e. planar $d$-ary trees. We also show that, if one can construct a certain Markov chain on large $d$-Catalan trees which updates its value by randomly shuffling certain nearby subtrees, and in such a way that the stationary distribution of this chain is uniform, then the Jacobian conjecture is true. Finally, we use the local limit theory of large random trees to show that the subtree shuffling conjecture is true in a certain asymptotic sense, and thereafter use our machinery to prove an approximate version of the Jacobian conjecture, stating that inverses of Keller maps have small power series coefficients for their high degree terms., Comment: 34 pages, 6 figures

We study the quenched behaviour of a perturbed version of the simple symmetric random walk on the set of integers. The random walker moves symmetrically with an exception of some randomly chosen sites where we impose a random drift. We show that if the gaps between the marked sites are i.i.d. and regularly varying with a sufficiently small index, then there is no strong quenched limit laws for the position of the random walker. As a consequence we study the quenched limit laws in the context of weak convergence of random measures.

We investigate a branching random walk where the displacements are independent from the branching mechanism and have a stretched exponential distribution. We describe the positions of the particles in the vicinity of the rightmost particle in terms of point process convergence. As a consequence we give a~new limit theorem for the position of the rightmost particle. Our methods rely on providing precise large deviations for sums of i.i.d. random variables with stretched exponential distributions outside the so-called one big jump regime., Comment: 30 pages

We study a class of kinetic-type differential equations $\partial \phi_t/\partial t+\phi_t=\widehat{\mathcal{Q}}\phi_t$, where $\widehat{\mathcal{Q}}$ is an inhomogeneous smoothing transform and, for every $t\geq 0$, $\phi_t$ is the Fourier--Stieltjes transform of a probability measure. We show that under mild assumptions on $\widehat{\mathcal{Q}}$ the above differential equation possesses a unique solution and represent this solution as the characteristic function of a certain stochastic process associated with the continuous time branching random walk pertaining to $\widehat{\mathcal{Q}}$. Establishing limit theorems for this process allows us to describe asymptotic properties of the solution, as $t\to\infty$., Comment: 24 pages, published in Stochastic Processes and Their Applications

We study the asymptotics of the $k$-regular self-similar fragmentation process. For $\alpha > 0$ and an integer $k \geq 2$, this is the Markov process $(I_t)_{t \geq 0}$ in which each $I_t$ is a union of open subsets of $[0,1)$, and independently each subinterval of $I_t$ of size $u$ breaks into $k$ equally sized pieces at rate $u^\alpha$. Let $k^{ - m_t}$ and $k^{ - M_t}$ be the respective sizes of the largest and smallest fragments in $I_t$. By relating $(I_t)_{t \geq 0}$ to a branching random walk, we find that there exist explicit deterministic functions $g(t)$ and $h(t)$ such that $|m_t - g(t)| \leq 1$ and $|M_t - h(t)| \leq 1$ for all sufficiently large $t$. Furthermore, for each $n$, we study the final time at which fragments of size $k^{-n}$ exist. In particular, by relating our branching random walk to a certain point process, we show that, after suitable rescaling, the laws of these times converge to a Gumbel distribution as $n \to \infty$., Comment: 29 pages, 1 figure

We prove large deviation results for the position of the rightmost particle, denoted by $M_n$, in a one-dimensional branching random walk in a case when Cram\'er's condition is not satisfied. More precisely we consider step size distributions with stretched exponential upper and lower tails, i.e.~both tails decay as $e^{-|t|^r}$ for some $r\in( 0,1)$. It is known that in this case, $M_n$ grows as $n^{1/r}$ and in particular faster than linearly in $n$. Our main result is a large deviation principle for the laws of $n^{-1/r}M_n$ . In the proof we use a comparison with the maximum of (a random number of) independent random walks, denoted by $\tilde M_n$, and we show a large deviation principle for the laws of $n^{-1/r}\tilde M_n$ as well., Comment: arXiv admin note: text overlap with arXiv:1802.03960

We study the one-dimensional branching random walk in the case when the step size distribution has a stretched exponential tail, and, in particular, no finite exponential moments. The tail of the step size $X$ decays as $\mathbb{P}[X \geq t] \sim a \exp(-\lambda t^r)$ for some constants $a, \lambda > 0$ where $r \in (0,1)$. We give a detailed description of the asymptotic behaviour of the position of the rightmost particle, proving almost-sure limit theorems, convergence in law and some integral tests. The limit theorems reveal interesting differences betweens the two regimes $ r \in (0, 2/3)$ and $ r \in (2/3, 1)$, with yet different limits in the boundary case $r = 2/3$., Comment: 30 pages

It is well known that the product of two independent regularly varying random variables with the same tail index is again regularly varying with this index. In this paper, we provide sharp sufficient conditions for the regular variation property of product-type functions of regularly varying random vectors, generalizing and extending the univariate theory in various directions. The main result is then applied to characterize the regular variation property of products of iid regularly varying quadratic random matrices and of solutions to affine stochastic recurrence equations under non-standard conditions.

The integer points (sites) of the real line are marked by the positions of a standard random walk. We say that the set of marked sites is weakly, moderately or strongly sparse depending on whether the jumps of the standard random walk are supported by a bounded set, have finite or infinite mean, respectively. Focussing on the case of strong sparsity we consider a nearest neighbor random walk on the set of integers having jumps $\pm 1$ with probability $1/2$ at every nonmarked site, whereas a random drift is imposed at every marked site. We prove new distributional limit theorems for the so defined random walk in a strongly sparse random environment, thereby complementing results obtained recently in Buraczewski et al. (2018+) for the case of moderate sparsity and in Matzavinos et al. (2016) for the case of weak sparsity. While the random walk in a strongly sparse random environment exhibits either the diffusive scaling inherent to a simple symmetric random walk or a wide range of subdiffusive scalings, the corresponding limit distributions are non-stable., Comment: 35 pages, submitted

A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk $(X_n)_{n\in \mathbb{N}\cup\{0\}}$ in a sparse random environment $(S_k,\lambda_k)_{k\in\mathbb{Z}}$ is a nearest neighbor random walk on $\mathbb{Z}$ that jumps to the left or to the right with probability $1/2$ from every point of $\mathbb{Z}\setminus \{\ldots,S_{-1},S_0=0,S_1,\ldots\}$ and jumps to the right (left) with the random probability $\lambda_{k+1}$ ($1-\lambda_{k+1}$) from the point $S_k$, $k\in\mathbb{Z}$. Assuming that $(S_k-S_{k-1},\lambda_k)_{k\in\mathbb{Z}}$ are independent copies of a random vector $(\xi,\lambda)\in \mathbb{N}\times (0,1)$ and the mean $\mathbb{E}\xi$ is finite (moderate sparsity) we obtain stable limit laws for $X_n$, properly normalized and centered, as $n\to\infty$. While the case $\xi\leq M$ a.s.\ for some deterministic $M>0$ (weak sparsity) was analyzed by Matzavinos et al., the case $\mathbb{E} \xi=\infty$ (strong sparsity) will be analyzed in a forthcoming paper., Comment: submitted, 42 pages

We study one-dimensional nearest neighbour random walk in site-random environment. We establish precise (sharp) large deviations in the so-called ballistic regime, when the random walk drifts to the right with linear speed. In the sub-ballistic regime, when the speed is sublinear, we describe the precise probability of slowdown., Comment: 26 pages

We consider solutions to so-called stochastic fixed point equation $R \stackrel{d}{=} \Psi(R)$, where $\Psi $ is a random Lipschitz function and $R$ is a random variable independent of $\Psi$. Under the assumption that $\Psi$ can be approximated by the function $x \mapsto Ax+B$ we show that the tail of $R$ is comparable with the one of $A$, provided that the distribution of $\log (A\vee 1) $ is tail equivalent. In particular we obtain new results for the random difference equation., Comment: 19 pages

We consider the branching process in random environment $\{Z_n\}_{n\geq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We describe precise asymptotics of upper large deviations, i.e. $\mathbb{P}[Z_n > e^{\rho n}]$. Moreover in the subcritical case, under the Cram\'er condition on the mean of the reproduction law, we investigate large deviations-type estimates for the first passage time of the branching process in question and its total population size., Comment: 28 pages

An infinite convergent sum of independent and identically distributed random variables discounted by a multiplicative random walk is called perpetuity, because of a possible actuarial application. We give three disjoint groups of sufficient conditions which ensure that the distribution right tail of a perpetuity $\mathbb{P}\{X>x\}$ is asymptotic to $ax^ce^{-bx}$ as $x\to\infty$ for some $a,b>0$ and $c\in\mathbb{R}$. Our results complement those of Denisov and Zwart [J. Appl. Probab. 44 (2007), 1031--1046]. As an auxiliary tool we provide criteria for the finiteness of the one-sided exponential moments of perpetuities. Several examples are given in which the distributions of perpetuities are explicitly identified., Comment: To appear in Journal of Applied Probability, 55, no. 2, 2018

We investigate so-called generalized Mandelbrot cascades at the freezing (critical) temperature. It is known that, after a proper rescaling, a~sequence of multiplicative cascades converges weakly to some continuous random measure. Our main question is how the limiting measure $\mu$ fluctuates. For any given point $x$, denoting by $B_n(x)$ the ball of radius $2^{-n}$ centered around $x$, we present optimal lower and upper estimates of $\mu(B_n(x))$ as $n \to \infty$., Comment: 27 pages

For a given random sequence $(C,T_{1},T_{2},\ldots)$ with nonzero $C$ and a.s. finite number of nonzero $T_{k}$, the nonhomogeneous smoothing transform $\mathcal{S}$ maps the law of a real random variable $X$ to the law of $\sum_{k\ge 1}T_{k}X_{k}+C$, where $X_{1},X_{2},\ldots$ are independent copies of $X$ and also independent of $(C,T_{1},T_{2},\ldots)$. This law is a fixed point of $\mathcal{S}$ if the stochastic fixed-point equation (SFPE) $X\stackrel{d}{=}\sum_{k\ge 1}T_{k}X_{k}+C$ holds true, where $\stackrel{d}{=}$ denotes equality in law. Under suitable conditions including $\mathbb{E} C=0$, $\mathcal{S}$ possesses a unique fixed point within the class of centered distributions, called the canonical solution to the above SFPE because it can be obtained as a certain martingale limit in an associated weighted branching model. The present work provides conditions on $(C,T_{1},T_{2},\ldots)$ such that the canonical solution exhibits right and/or left Poisson tails and the abscissa of convergence of its moment generating function can be determined. As a particular application, the right tail behavior of the Quicksort distribution is found., Comment: 26 pages

Consider a sequence of i.i.d. random Lipschitz functions $\{\Psi_n\}_{n \geq 0}$. Using this sequence we can define a Markov chain via the recursive formula $R_{n+1} = \Psi_{n+1}(R_n)$. It is a well known fact that under some mild moment assumptions this Markov chain has a unique stationary distribution. We are interested in the tail behaviour of this distribution in the case when $\Psi_0(t) \approx A_0t+B_0$. We will show that under subexponential assumptions on the random variable $\log^+(A_0\vee B_0)$ the tail asymptotic in question can be described using the integrated tail function of $\log^+(A_0\vee B_0)$. In particular we will obtain new results for the random difference equation $R_{n+1} = A_{n+1}R_n+B_{n+1}$..