arXiv subjects

Alexander Marynych

Publications and source records attributed to Alexander Marynych.

67 records · Page 4Linked to original sources

arXiv2015

Asymptotics of random processes with immigration II: convergence to stationarity

Let $X_1, X_2,\ldots$ be random elements of the Skorokhod space $D(\mathbb{R})$ and $ξ_1, ξ_2, \ldots$ positive random variables such that the pairs $(X_1,ξ_1), (X_2,ξ_2),\ldots$ are independent and identically distributed. We call the random process $(Y(t))_{t \in \mathbb{R}}$ defined by $Y(t):=\sum_{k \geq 0}X_{k+1}(t-ξ_1-\ldots-ξ_k)1_{\{ξ_1+\ldots+ξ_k\leq t\}}$, $t\in\mathbb{R}$ random process with immigration at the epochs of a renewal process. Assuming that $X_k$ and $ξ_k$ are independent and that the distribution of $ξ_1$ is nonlattice and has finite mean we investigate weak convergence of $(Y(t))_{t\in\mathbb{R}}$ as $t\to\infty$ in $D(\mathbb{R})$ endowed with the $J_1$-topology. The limits are stationary processes with immigration.

math.PR↗

arXiv2015

Asymptotics of random processes with immigration I: scaling limits

Let $(X_1, ξ_1), (X_2,ξ_2),\ldots$ be i.i.d.~copies of a pair $(X,ξ)$ where $X$ is a random process with paths in the Skorokhod space $D[0,\infty)$ and $ξ$ is a positive random variable. Define $S_k := ξ_1+\ldots+ξ_k$, $k \in \mathbb{N}_0$ and $Y(t) := \sum_{k\geq 0} X_{k+1}(t-S_k) 1_{\{S_k \leq t\}}$, $t\geq 0$. We call the process $(Y(t))_{t \geq 0}$ random process with immigration at the epochs of a renewal process. We investigate weak convergence of the finite-dimensional distributions of $(Y(ut))_{u>0}$ as $t\to\infty$. Under the assumptions that the covariance function of $X$ is regularly varying in $(0,\infty)\times (0,\infty)$ in a uniform way, the class of limiting processes is rather rich and includes Gaussian processes with explicitly given covariance functions, fractionally integrated stable Lévy motions and their sums when the law of $ξ$ belongs to the domain of attraction of a stable law with finite mean, and conditionally Gaussian processes with explicitly given (conditional) covariance functions, fractionally integrated inverse stable subordinators and their sums when the law of $ξ$ belongs to the domain of attraction of a stable law with infinite mean.

math.PR↗

arXiv2015

A note on convergence to stationarity of random processes with immigration

Let $X_1, X_2,\ldots$ be random elements of the Skorokhod space $D(\mathbb{R})$ and $ξ_1, ξ_2, \ldots$ positive random variables such that the pairs $(X_1,ξ_1), (X_2,ξ_2),\ldots$ are independent and identically distributed. The random process $Y(t):=\sum_{k \geq 0}X_{k+1}(t-ξ_1-\ldots-ξ_k)1_{\{ξ_1+\ldots+ξ_k\leq t\}}$, $t\in\mathbb{R}$, is called random process with immigration at the epochs of a renewal process. Assuming that the distribution of $ξ_1$ is nonlattice and has finite mean while the process $X_1$ decays sufficiently fast, we prove weak convergence of $(Y(u+t))_{u\in\mathbb{R}}$ as $t\to\infty$ on $D(\mathbb{R})$ endowed with the $J_1$-topology. The present paper continues the line of research initiated in Iksanov, Marynych and Meiners (2015+).

math.PR↗

arXiv2015

Renewal approximation for the absorption time of a decreasing Markov chain

We consider a Markov chain $(M_{n})_{n\ge 0}$ on the set $\mathbb{N}_{0}$ of nonnegative integers which is eventually decreasing, i.e. $\mathbb{P}\{M_{n+1}<M_{n}|M_{n}\ge a\}=1$ for some $a\in\mathbb{N}$ and all $n\ge 0$. We are interested in the asymptotic behaviour of the law of the stopping time $T=T(a):=\inf\{k\in\mathbb{N}_{0}: M_{k}<a\}$ under $\mathbb{P}_{n}:=\mathbb{P}(\cdot|M_{0}=n)$ as $n\to\infty$. Assuming that the decrements of $(M_{n})_{n\ge 0}$ given $M_{0}=n$ possess a kind of stationarity for large $n$, we derive sufficient conditions for the convergence in minimal $L^{p}$-distance of $\mathbb{P}_{n}((T-a_{n})/b_{n}\in\cdot)$ to some non-degenerate, proper law and give an explicit form of the constants $a_{n}$ and $b_{n}$.

math.PR↗

arXiv2014

Weak convergence of the number of zero increments in the random walk with barrier

We continue the line of research of random walks with barrier initiated by Iksanov and M{ö}hle (2008). Assuming that the tail of the step of the underlying random walk has a power-like behavior at infinity with exponent $-α$, $α\in(0,1)$, we prove that the number $V_n$ of zero increments in the random walk with barrier, properly centered and normalized, converges weakly to the standard normal law. This refines previously known weak law of large numbers for $V_n$ proved in Iksanov and Negadailov (2008).

math.PR↗

arXiv2013

Weak convergence of finite-dimensional distributions of the number of empty boxes in the Bernoulli sieve

The Bernoulli sieve is a random allocation scheme obtained by placing independent points with the uniform [0,1] law into the intervals made up by successive positions of a multiplicative random walk with factors taking values in the interval (0,1). Assuming that the number of points is equal to n we investigate the weak convergence, as n tends to infinity, of finite-dimensional distributions of the number of empty intervals within the occupancy range. A new argument enables us to relax the constraints imposed in previous papers on the distribution of the factor of the multiplicative random walk.

math.PR↗

arXiv2012

On asymptotics of the beta-coalescents

We show that the total number of collisions in the exchangeable coalescent process driven by the beta $(1,b)$ measure converges in distribution to a 1-stable law, as the initial number of particles goes to infinity. The stable limit law is also shown for the total branch length of the coalescent tree. These results were known previously for the instance $b=1$, which corresponds to the Bolthausen--Sznitman coalescent. The approach we take is based on estimating the quality of a renewal approximation to the coalescent in terms of a suitable Wasserstein distance. Application of the method to beta $(a,b)$-coalescents with $0<a<1$ leads to a simplified derivation of the known $(2-a)$-stable limit. We furthermore derive asymptotic expansions for the moments of the number of collisions and of the total branch length for the beta $(1,b)$-coalescent by exploiting the method of sequential approximations.

math.PR↗

arXiv2012

Exponential-Uniform Identities Related to Records

We consider a rectangular grid induced by the south-west records from the planar Poisson point process in $R^2_+$. A random symmetry property of the matrix whose entries are the areas of tiles of the grid implies cute multivariate distributional identities for certain rational functions of independent exponential and uniform random variables.

math.PR↗

arXiv2012

A generalization of the Erdős-Turán law for the order of random permutation

We consider random permutations derived by sampling from stick-breaking partitions of the unit interval. The cycle structure of such a permutation can be associated with the path of a decreasing Markov chain on $n$ integers. Under certain assumptions on the stick-breaking factor we prove a central limit theorem for the logarithm of the order of the permutation, thus extending the classical Erdős-Turán law for the uniform permutations and its generalization for Ewens' permutations associated with sampling from the PD/GEM$(θ)$ distribution. Our approach is based on using perturbed random walks to obtain the limit laws for the sum of logarithms of the cycle lengths.

math.PR↗

arXiv2011

On the asymptotics of moments of linear random recurrences

We propose a new method of analyzing the asymptotics of moments of certain linear random recurrences which is based on the technique of iterative functions. By using the method, we show that the moments of the number of collisions and the absorption time in the Poisson-Dirichlet coalescent behave like the powers of the "log star" function which grows slower than any iteration of the logarithm, and thereby prove a weak law of large numbers. Finally, we discuss merits and limitations of the method and give several examples related to beta coalescents, recursive algorithms and random trees.

math.PR↗

arXiv2011

Lambda-coalescents with dust component

We consider the lambda-coalescent processes with positive frequency of singleton clusters. The class in focus covers, for instance, the beta$(a,b)$-coalescents with $a>1$. We show that some large-sample properties of these processes can be derived by coupling the coalescent with an increasing L{é}vy process (subordinator), and by exploiting parallels with the theory of regenerative composition structures. In particular, we discuss the limit distributions of the absorption time and the number of collisions.

math.PR↗

arXiv2010

The Bernoulli sieve: an overview

The Bernoulli sieve is a version of the classical balls-in-boxes occupancy scheme, in which random frequencies of infinitely many boxes are produced by a multiplicative random walk, also known as the residual allocation model or stick-breaking. We give an overview of the limit theorems concerning the number of boxes occupied by some balls out of the first $n$ balls thrown, and present some new results concerning the number of empty boxes within the occupancy range.

math.PR↗

arXiv2010

Limit theorems for the number of occupied boxes in the Bernoulli sieve

The Bernoulli sieve is a version of the classical `balls-in-boxes' occupancy scheme, in which random frequencies of infinitely many boxes are produced by a multiplicative renewal process, also known as the residual allocation model or stick-breaking. We focus on the number $K_n$ of boxes occupied by at least one of $n$ balls, as $n\to\infty$. A variety of limiting distributions for $K_n$ is derived from the properties of associated perturbed random walks. Refining the approach based on the standard renewal theory we remove a moment constraint to cover the cases left open in previous studies.

math.PR↗