arXiv subjects

Kamil Marcin Kosiński

Publications and source records attributed to Kamil Marcin Kosiński.

7 recordsLinked to original sources

arXiv2018

Sample path properties of reflected Gaussian processes

We consider a stationary queueing process $Q_X$ fed by a centered Gaussian process $X$ with stationary increments and variance function satisfying classical regularity conditions. A criterion when, for a given function $f$, $\mathbb P (Q_{X}(t) > f(t)\, \text{ i.o.})$ equals 0 or 1 is provided. Furthermore, an Erdös-Révész type law of the iterated logarithm is proven for the last passage time $ξ(t) = \sup\{s:0\le s\le t, Q_{X}(s)\ge f(s)\}$. Both of these findings extend previously known results that were only available for the case when $X$ is a fractional Brownian motion.

math.PR↗

arXiv2015

Extremes of multidimensional Gaussian processes

This paper considers extreme values attained by a centered, multidimensional Gaussian process $X(t)= (X_1(t),\ldots,X_n(t))$ minus drift $d(t)=(d_1(t),\ldots,d_n(t))$, on an arbitrary set $T$. Under mild regularity conditions, we establish the asymptotics of \[\log\mathbb P\left(\exists{t\in T}:\bigcap_{i=1}^n\left\{X_i(t)-d_i(t)>q_iu\right\}\right),\] for positive thresholds $q_i>0$, $i=1,\ldots,n$, and $u\to\infty$. Our findings generalize and extend previously known results for the single-dimensional and two-dimensional cases. A number of examples illustrate the theory.

math.PR↗

arXiv2015

Lévy-driven polling systems and continuous-state branching processes

In this paper we consider a ring of $N\ge 1$ queues served by a single server in a cyclic order. After having served a queue (according to a service discipline that may vary from queue to queue), there is a switch-over period and then the server serves the next queue and so forth. This model is known in the literature as a \textit{polling model}. Each of the queues is fed by a non-decreasing Lévy process, which can be different during each of the consecutive periods within the server's cycle. The $N$-dimensional Lévy processes obtained in this fashion are described by their (joint) Laplace exponent, thus allowing for non-independent input streams. For such a system we derive the steady-state distribution of the joint workload at embedded epochs, i.e. polling and switching instants. Using the Kella-Whitt martingale, we also derive the steady-state distribution at an arbitrary epoch. Our analysis heavily relies on establishing a link between fluid (Lévy input) polling systems and multi-type Jiřina processes (continuous-state discrete-time branching processes). This is done by properly defining the notion of the \textit{branching property} for a discipline, which can be traced back to Fuhrmann and Resing. This definition is broad enough to contain the most important service disciplines, like exhaustive and gated.

math.PR↗

arXiv2015

Asymptotics for sums of a function of normalized independent sums

We derive a central limit theorem for sums of a function of independent sums of independent and identically distributed random variables. In particular we show that previously known result from Rempa\la and Weso\lowski (Statist. Probab. Lett. 74 (2005) 129--138), which can be obtained by applying the logarithm as the function, holds true under weaker assumptions.

math.PR↗

arXiv2015

On the functional limits for sums of a function of partial sums

We derive a functional central limit theorem (fclt) for normalised sums of a function of the partial sums of independent and identically distributed random variables. In particular, we show, using a technique presented in Huang and Zhang (Electron. Comm. Probab. 12 (2007), 51--56), that the result from Qi (Statist. Probab. Lett. 62 (2003), 93--100), for normalised products of partial sums, can be generalised in this fashion to a fclt.

math.PR↗

arXiv2015

On the functional limits for partial sums under stable law

For the partial sums $(S_n)$ of independent random variables we define a stochastic process $s_n(t):=(1/d_n)\sum_{k \le [nt]} ({S_k}/{k}-μ)$ and prove that $$(1/{\log N})\sum_{n\le N}(1/n)\mathbf {I}\left\{s_n(t)\le x\right\} \to G_t(x)\quad \text{a.s.}$$ if and only if $(1/{\log N})\sum_{n\le N} (1/n)\mathbb{P}\left(s_n(t)\le x\right) \to G_t(x)$, for some sequence $(d_n)$ and distribution $G_t$. We also prove an almost sure functional limit theorem for the product of partial sums of i.i.d. positive random variables attracted to an $α$-stable law with $α\in (1,2]$.

math.PR↗

arXiv2014

On the infimum attained by the reflected fractional Brownian motion

Let $\{B_H(t):t\ge 0\}$ be a fractional Brownian motion with Hurst parameter $H\in(\frac{1}{2},1)$. For the storage process $Q_{B_H}(t)=\sup_{-\infty\le s\le t} \left(B_H(t)-B_H(s)-c(t-s)\right)$ we show that, for any $T(u)>0$ such that $T(u)=o(u^\frac{2H-1}{H})$, \[\mathbb P (\inf_{s\in[0,T(u)]} Q_{B_H}(s)>u)\sim\mathbb P(Q_{B_H}(0)>u),\quad\text{as}\quad u\to\infty.\] This finding, known in the literature as the strong Piterbarg property, is in line with previously observed properties of storage processes with self-similar and infinitely divisible input without Gaussian component.

math.PR↗