arXiv subjects

Vitali Wachtel

Publications and source records attributed to Vitali Wachtel.

At least 19 recordsLinked to original sources

arXiv2026

A model of discrete interacting updates

We consider $N$ counters taking integer values which are subject to the following dynamics. At every time, a pair of distinct counters is chosen uniformly at random and their states are updated according to the following rule. If the states are different, then the smaller one is increased by $1$, while if the states are the same, both of them are increased by $1$. We show that, for a fixed $N$, the distances between consecutive ordered counters form a positive recurrent Markov chain and there exists the speed $V(N)$ defined as the average number of counters updated per time step in the stationary regime. We provide non-trivial upper and lower bounds for $V(N)$ as $N\to \infty$. Despite the simple formulation of the problem, its analysis seems to be highly complicated. We also provide a list of open problems and discuss various methods one may want to use, and obstacles one encounters.

math.PR↗

arXiv2026

Broadcasting on Random Trees: Martingale Methods and Error Bounds

The broadcasting problem on trees concerns the transmission of binary information along the edges of a tree. In the formulation studied here, each vertex carries one of two colours, and along every edge the colour is transmitted with error probability $q$. The central question is whether the root colour can be reconstructed from the observed colours of all vertices in the tree, but crucially, without knowing which vertex is the root. We study the problem of reconstructing the root colour in a broadcasting process on random trees, focusing on the majority rule estimator. We introduce a novel and adaptable martingale approach for analysing the colour imbalance. This method provides a simpler and more powerful alternative to existing techniques. Our martingale approach unifies the analysis of the different tree models and leads to linear upper bounds for the error probability of the majority rule estimator in each case. This shows that, for a broad class of random tree models, the error probability of the majority rule admits a linear upper bound in $q$.

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arXiv2026

Ordered random walks and the Airy line ensemble

The Airy line ensemble is a random collection of continuous ordered paths that plays an important role within random matrix theory and the Kardar-Parisi-Zhang universality class. The aim of this paper is to prove a universality property of the Airy line ensemble. We study growing numbers of i.i.d. continuous-time random walks which are then conditioned to stay in the same order for all time using a Doob h-transform. We consider a general class of increment distributions; a sufficient condition is the existence of an exponential moment and a log-concave density. We prove that the top particles in this system converge in an edge scaling limit to the Airy line ensemble in a regime where the number of random walks is required to grow slower than a certain power (with a non-optimal exponent 3/50) of the expected number of random walk steps. Furthermore, in a similar regime we prove that the law of large numbers and fluctuations of linear statistics agree with non-intersecting Brownian motions.

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arXiv2026

Fluctuations of Discrete-Time Random Walks

These notes are devoted to fluctuations of one-dimensional random walks. We discuss various approaches to first-passage times and to the corresponding conditional distributions. After discussion of some classical methods, such as reflection principle for simple random walks and Wiener-Hopf factorisation, we proceed to the universality approach, which has been developed in recent past. Considering one-dimensional case allows us to avoid some technical obstacles and to present the core of this method in a more transparent way. It turns out that the universality method is much more robust than the Wiener-Hopf factorisation and allows one to consider walks with non-identically distributed or even dependent increments.

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arXiv2026

Corrected diffusion approximation for random walks conditioned to stay positive

Let $S_n$ be a random walk with i.i.d. increments which have zero mean and finite variance. For every $x\ge0$ we define the stopping time $τ_x:=\inf\{n\ge1:x+S_n\le0\}$ and consider the probabilities $\mathbb{P}(x+S_n\ge y,τ_x>n)$. We study the quality of the normal approximation for these probabilities and derive a Berry-Esseen-type inequality for $\mathbb{P}(x+S_n\ge y|τ_x>n)$. Our Theorem 1 is an extension of the results in our previous paper (arXiv:2412.08502) where we have considered the special case $x=0$. It is also worth mentioning that Theorem 1 complements the results of Siegmund and Yuh (1982) on the corrected diffusion approximation.

math.PR↗

arXiv2026

Persistence of AR($1$) sequences with Rademacher innovations and linear mod $1$ transforms

We study the probability that an AR(1) Markov chain $X_{n+1}=aX_n+ξ_{n+1}$, where $a\in(0,1)$ is a constant, stays non-negative for a long time. We find the exact asymptotics of this probability and the weak limit of $X_n$ conditioned to stay non-negative, assuming that the i.i.d.\ innovations $ξ_n$ take only two values $\pm1$ and $a \le \frac23$. This limiting distribution is quasi-stationary. It has no atoms and is singular with respect to the Lebesgue measure when $\frac12< a \le \frac23$, except for the case $a=\frac23$ and $\mathbb{P}(ξ_n=1)=\frac12$, where this distribution is uniform on the interval $[0,3]$. This is similar to the properties of Bernoulli convolutions. For $0 < a \le \frac12$, the situation is much simpler, and the limiting distribution is a $δ$-measure. To prove these results, we uncover a close connection between $X_n$ killed at exiting $[0, \infty)$ and the classical dynamical system defined by the piecewise linear mapping $x \mapsto \frac1a x + \frac12 \pmod 1$. Namely, the trajectory of this system started at $X_n$ deterministically recovers the values of the killed chain in reversed time. We use this fact to construct a suitable Banach space, where the transition operator of the killed chain has the compactness properties that allow us to apply a conventional argument of the Perron--Frobenius type.

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arXiv2026

Random walks with drift in the positive quadrant

We consider two dimensional random walks conditioned to stay in the positive quadrant. Assuming that the increments of the walk have finite second moments and that the drift vector is co-oriented with one of two axes, we construct positive harmonic functions for such walks and find tail asymptotics for the exit time from the positive quadrant. Moreover, we prove integral and local limit theorems. Finally, we apply our local limit theorems to singular lattice walks with steps $\{(1,-1),(1,1),(-1,1)\}$ and determine asymptotics for the number of walks of length $n$ which end on the line $\{(k,1),\ k\ge1\}$.

math.PR↗

arXiv2026

Harmonic polynomials and other exactly computable characteristics for $2$-dimensional random walks in cones

In this note we consider $2$-dimensional lattice random walks killed at leaving a wedge with opening $α\in(0,π]$. Assuming that the walk cannot jump over the boundary of the wedge we prove that there exists a harmonic polynomial if and only if $α=π/m$ with some integer $m$. Our proof is constructive and allows one to give exact expressions for harmonic polynomials for every integer $m$. Furthermore, we give exact expressions for all finite moments of the exit time, this result is valid for all angles $α$.

math.PR↗

arXiv2026

First passage times for decoupled random walks

Motivated by a connection to the infinite Ginibre point process, decoupled random walks were introduced in a recent article Alsmeyer, Iksanov and Kabluchko (2025). The decoupled random walk is a sequence of independent random variables, in which the $n$th variable has the same distribution as the position at time $n$ of a standard random walk with nonnegative increments. We prove distributional convergence in the Skorokhod space equipped with the $J_1$-topology of the running maxima and the first passage times of decoupled random walks. We show that there exist five different regimes, in which distinct limit theorems arise. Rather different functional limit theorems for the number of visits of decoupled standard random walk to the interval $[0,t]$ as $t\to\infty$ were earlier obtained in the aforementioned paper Alsmeyer, Iksanov and Kabluchko (2025). While the limit processes for the first passage times are inverse extremal-like processes, the limit processes for the number of visits are stationary Gaussian.

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arXiv2025

Critical multitype branching: a functional limit theorem and ancestral processes

We consider the long-term behaviour of critical multitype branching processes conditioned on non-extinction, both with respect to the forward and the ancestral processes. Forward in time, we prove a functional limit theorem in the space of trajectories of the linearly-scaled $h$-transformed process; the change of measure allows us to work on the same probability space for all times. Backward in time, we trace the lines of descent of individuals sampled from a (stationary) population and analyse the ancestral type distribution, that is, the type distribution of the ancestors in the distant past, as well as the type process along the ancestral line.

math.PR↗

arXiv2025

Co-existence of branching populations in random environment

In this paper we consider two branching processes living in a joint random environment. Assuming that both processes are critical we address the following question: What is the probability that both populations survive up to a large time $n$? We show that this probability decays as $n^{-θ}$ with $θ>0$ which is determined by the random environment. Furthermore, we prove the corresponding conditional limit theorem. One of the main ingredients in the proof is a qualitative bound for the entropic repulsion for two-dimensional random walks conditioned to stay in the positive quadrant. We believe that this bound is also of independent interest.

math.PR↗

arXiv2024

Asymptotic expansions for normal deviations of random walks conditioned to stay positive

We consider a one-dimensional random walk $S_n$ having i.i.d. increments with zero mean and finite variance. We continue our study of asymptotic expansions for local probabilities $\mathbf P(S_n=x,τ_0>n)$, which has been started in \cite{DTW23}. Obtained there expansions make sense in the zone $x=o(\frac{\sqrt{n}}{\log^{1/2} n})$ only. In the present paper we derive an alternative expansion, which deals with $x$ of order $\sqrt{n}$.

math.PR↗

arXiv2024

Berry-Esseen inequality for random walks conditioned to stay positive

We consider random walks conditioned to stay positive. When the mean of increments is zero and variance is finite it is known that they converge to the Rayleigh distribution. In the present paper we derive a Berry-Esseen type estimate and show that the rate of convergence is of order $n^{-1/2}$.

math.PR↗

arXiv2024

Stable random walks in cones

In this paper we consider a multidimensional random walk killed on leaving a right circular cone with a distribution of increments belonging to the normal domain of attraction of an $α$-stable and rotationally-invariant law with $α\in (0,2)\setminus \{1\}$. Based on Bogdan et al. (2018) describing the tail behaviour of the exit time of $α$-stable process from a cone and using some properties of Martin kernel of the isotropic $α$-stable process, in this paper we construct a positive harmonic function of the discrete time random walk under consideration. Then we find the asymptotic tail of the distribution of the exit time of this random walk from the cone. We also prove the corresponding conditional functional limit theorem.

math.PR↗

arXiv2024

Lower deviations for branching processes with immigration

Let $\{Y_{n}$, $n \geq 1\}$ be a critical branching process with immigration having finite variance for the offspring number of particles and finite mean for the immigrating number of particles. In this paper, we study lower deviation probabilities for $Y_{n}$. More precisely, assuming that $k,n \to \infty$ such that $k=o\left(n \right)$, we investigate the asymptotics of $\mathbf{P}\left(Y_{n} \leq k \right)$ and $\mathbf{P}\left(Y_{n} = k \right)$. Our results clarify the role of the moment conditions in the local limit theorem for $Y_n$ proven by Mellein.

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arXiv2024

Multinomial random combinatorial structures and $r$-versions of Stirling, Eulerian and Lah numbers

We introduce multinomial and $r$-variants of several classic objects of combinatorial probability, such as the random recursive and Hoppe trees, random set partitions and compositions, the Chinese restaurant process, Feller's coupling, and some others. Just as various classic combinatorial numbers - like Stirling, Eulerian and Lah numbers - emerge as essential ingredients defining the distributions of the mentioned processes, the so-called $r$-versions of these numbers appear in exact distributional formulas for the multinomial and $r$-counterparts. This approach allows us to offer a concise probabilistic interpretation for various identities involving $r$-versions of these combinatorial numbers, which were either unavailable or meaningful only for specific values of the parameter $r$. We analyze the derived distributions for fixed-size structures and establish distributional limit theorems as the size tends to infinity. Utilizing the aforementioned generalized Stirling numbers of both kinds, we define and analyze $(r,s)$-Lah distributions, which have arisen in the existing literature on combinatorial probability in various contexts.

math.PR↗

arXiv2024

Expansions for random walks conditioned to stay positive

We consider a one-dimensional random walk $S_n$ with i.i.d. increments with zero mean and finite variance. We study the asymptotic expansion for the tail distribution $\mathbf P(τ_x>n)$ of the first passage times $τ_x:=\inf\{n\ge1:x+S_n\le0\}$ for $\ x\ge0.$ We also derive asymptotic expansion for local probabilities $\mathbf P(S_n=x,τ_0>n)$. Studying the asymptotic expansions we obtain a sequence of discrete polyharmonic functions and obtain analogues of renewal theorem for them.

math.PR↗

arXiv2023

Probabilistic approach to risk processes with level-dependent premium rate

We study risk processes with level dependent premium rate. Assuming that the premium rate converges, as the risk reserve increases, to the critical value in the net-profit condition, we obtain upper and lower bounds for the ruin probability. In contrast to existing in the literature results, our approach is purely probabilistic and based on the analysis of Markov chains with asymptotically zero drift.

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