Adam Jakubowski

Nicolaus Copernicus University | umk · Department of Probability Theory and Stochastic Analysis

A submetric space is a topological space with continuous metrics, generating a metric topology weaker than the original one (e.g. a separable Hilbert space with the weak topology). We demonstrate that on submetric spaces there exists a theory of convergence in probability, in law etc. equally effective as the Probability Theory on metric spaces. In...

We give necessary and sufficient conditions for the existence of a phantom distribution function for a stationary random field on a regular lattice. We also introduce a less demanding notion of a directional phantom distribution, with potentially broader area of applicability. Such approach leads to sectorial limit properties, a phenomenon well-kno...

We consider a class of perpetuities which admit direct characterization of asymptotics of the key truncated moment. The class contains perpetuities without polynomial decay of tail probabilities thus not satisfying Kesten’s theorem. We show how to apply this result in deriving a new weak law of large numbers for solutions to stochastic recurrence e...

We give necessary and sufficient conditions for the Chebyshev integral inequality to be an equality.

We give necessary and sufficient conditions for the Chebyshev inequality to be an equality.

We consider a class of perpetuities which admit direct characterization of asymptotics of the key truncated moment. The class contains perpetuities without polynomial decay of tail probabilities and thus not satisfying Kesten's theorem. We show how to apply this result in deriving a new weak law of large numbers for solutions to stochastic recurren...

We give necessary and sufficient conditions for the existence of a phantom distribution function for a stationary random field on a regular lattice. We also introduce a less demanding notion of a directional phantom distribution, with potentially broader area of applicability. Such approach leads to sectorial limit properties, a phenomenon well-kno...

For a stationary sequence that is regularly varying and associated, we give conditions guaranteeing that partial sums of this sequence, under normalization related to the exponent of regular variation, converge in distribution to a stable non-Gaussian limit. The obtained limit theorem admits a natural extension to the functional convergence in Skor...

For a stationary sequence that is regularly varying and associated we give conditions which guarantee that partial sums of this sequence, under normalization related to the exponent of regular variation, converge in distribution to a stable, non-Gaussian limit. The obtained limit theorem admits a natural extension to the functional convergence in S...

In the paper we solve the limit problem for partial maxima of m-dependent stationary random fields and we extend the obtained solution to fields satisfying some local mixing conditions. New methods for describing the limitting distribution of maxima are proposed. A notion of a phantom distribution function for a random field is investigated. As an...

We study limit theorems for partial sums of instantaneous functions of a homogeneous Markov chain on a general state space. The summands are heavy-tailed and the limits are stable distributions. We show that if the transition operator of the chain is operator uniformly integrable and the chain is ρ-mixing, then the limit is the same as if the summa...

We study limit theorems for partial sums of instantaneous functions of a homogeneous Markov chain on a general state space. The summands are heavy-tailed and the limits are stable distributions. The conditions imposed on the transition operator $P$ of the Markov chain ensure that the limit is the same as if the summands were independent. Such a~sch...

We introduce a class of discrete time stationary trawl processes taking real or integer values and written as sums of past values of independent ‘seed’ processes on shrinking intervals (‘trawl heights’). Related trawl processes in continuous time were studied in Barndorff-Nielsen et al. (2011, 2014). In the case when the trawl function decays as a...

It is known that random walk Metropolis algorithms with heavy-tailed target densities can model atypical (slow) growth of maxima, which in general is exhibited by processes with the extremal index zero. The asymptotics of maxima of such sequences can be analyzed in terms of continuous phantom distribution functions. We show that in a large class of...

We provide an inequality which is a useful tool in studying both large deviation results and limit theorems for sums of random fields with "negligible" small values. In particular, the inequality covers cases of stable limits for random variables with heavy tails and compound Poisson limits of $0-1$ random variables.

The principle of conditioning is a well-known heuristic rule which allows constructing limit theorems for sums of dependent random variables from existing limit theorems for independent summands. We state a general limit theorem on convergence to stable laws, which is valid for stationary sequences and provides a link between the principle of condi...

We introduce a class of discrete time stationary trawl processes taking real or integer values and written as sums of past values of independent `seed' processes on shrinking intervals (`trawl heights'). Related trawl processes in continuous time were studied in Barndorff-Nielsen (2011) and Barndorff-Nielsen et al. (2014), however in our case, the...

The $S$ topology on the Skorokhod space was introduced by the author in 1997 and since then it proved to be a useful tool in several areas of the theory of stochastic processes. The paper brings complementary information on the $S$ topology. It is shown that the convergence of sequences in the $S$ topology admits a compact description, exhibiting t...

The notion of a phantom distribution function (phdf) was introduced by O'Brien (1987). We show that the existence of a phdf is a quite common phenomenon for stationary weakly dependent sequences. It is proved that any $\alpha$-mixing stationary sequence with continuous marginals admits a continuous phdf. Sufficient conditions are given for stationa...

A cylindrical Levy process does not enjoy a cylindrical version of the semi-martingale decomposition which results in the need to develop a completely novel approach to stochastic integration. In this work, we introduce a stochastic integral for random integrands with respect to cylindrical Levy processes in Hilbert spaces. The space of admissible...

We study convergence in law of partial sums of linear processes with heavy-tailed innovations. In the case of summable coefficients necessary and sufficient conditions for the finite dimensional convergence to an $\alpha$-stable L\'evy Motion are given. The conditions lead to new, tractable sufficient conditions in the case $\alpha \leq 1$. In the...

In this paper we explore the problem of reconstruction of RGB images with additive Gaussian noise. In order to solve this problem we use Feynman-Kac formula and non local means algorithm. Expressing the problem in stochastic terms allows us to adapt to anisotropic diffusion the concept of similarity patches used in non local means. This novel look...

In this article, we prove a new functional limit theorem for the partial sum sequence $S_{[nt]}=\sum_{i=1}^{[nt]}X_i$ corresponding to a linear sequence of the form $X_i=\sum_{j \in \bZ}c_j \xi_{i-j}$ with i.i.d. innovations $(\xi_i)_{i \in \bZ}$ and real-valued coefficients $(c_j)_{j \in \bZ}$. This weak convergence result is obtained in space $\b...

This paper is motivated by relations between association and independence of random variables. It is well known that, for real random variables, independence implies association in the sense of Esary, Proschan and Walkup (1967), while, for random vectors, this simple relationship breaks. We modify the notion of association in such a way that any ve...

This paper is motivated by relations between association and independence of random variables. It is well-known that for real random variables independence implies association in the sense of Esary, Proschan and Walkup, while for random vectors this simple relationship breaks. We modify the notion of association in such a way that any vector-valued...

We provide a device, called the local predictor, which extends the idea of the predictable compensator. It is shown that a fBm with the Hurst index greater than 1/2 coincides with its local predictor while fBm with the Hurst index smaller than 1/2 does not admit any local predictor. The local predictor of a martingale (in particular: Brownian motio...

The aim of this paper is to provide conditions which ensure that the affinely transformed partial sums of a strictly stationary process converge in distribution to an infinite variance stable distribution. Conditions for this convergence to hold are known in the literature. However, most of these results are qualitative in the sense that the parame...

A class of stochastic processes, called "weak Dirichlet processes", is introduced and its properties are investigated in detail. This class is much larger than the class of Dirichlet processes. It is closed under C1-transformations and under absolutely continuous changes of measure. If a weak Dirichlet process has finite energy, as defined by Grave...

Both the Doob-Meyer and the Graversen-Rao decomposition theorems can be proved following an approach based on predictable compensators of discretizations and weak-L1 technique, which was developed by K.M. Rao. It is shown that any decomposition obtained by Rao's method gives predictability of compensators without additional assumptions (like submar...

We construct the Doob-Meyer decomposition of a submartingale as a pointwise superior limit of decompositions of discrete submartingales suitably built upon discretizations of the original process. This gives, in particular, a direct proof of predictability of the increasing process in the Doob-Meyer decomposition.

We consider the Cauchy problem for a semilinear stochastic differential inclusion in a Hilbert space. The linear operator generates a strongly continuous semigroup and the nonlinear term is multivalued and satisfies a condition which is more heneral than the Lipschitz condition. We prove the existence of a mild solution to this problem. This soluti...

A class of stochastic processes, called "weak Dirichlet processes", is introduced and its properties are investigated in detail. This class is much larger than the class of Dirichlet processes. It is closed under C^1$-transformations and under absolutely continuous change of measure. If a weak Dirichlet process has finite energy, as defined by Grav...

We study the rate at which the difference $X^n_t=X_t-X_{[nt]/n}$ between a process $X$ and its time-discretization converges. When $X$ is a continuous semimartingale it is known that, under appropriate assumptions, the rate is $\sqrt{n}$, so we focus here on the discontinuous case. Then $\alpha_nX^n$ explodes for any sequence $\alpha_n$ going to in...

It is proved that in Hilbert spaces a single Hilbert–Schmidt operator radonifies cylindrical semimartingales to strong semimartingales. This improves a result due to Badrikian and Üstünel (also L. Schwartz), who needed composition of three Hilbert–Schmidt operators.

It is shown that in a large class of topological spaces every uniformly tight sequence of random elements contains a subsequence which admits the usual almost sure (a.s.) Skorokhod representation on the Lebesgue interval.

It is shown that in a large class of topological spaces every uniformly tight sequence of random elements contains a subsequence which admits the usual almost sure (a.s.) Skorokhod representation on the Lebesgue interval. In his famous paper [11], Skorokhod proved that there exist X -valued random elements Y 0 , Y 1 , Y 2 , . . . , defined on the u...

It is shown that in a large class of topological spaces every uniformly tight sequence of random elements contains a subsequence which admits the usual a.s. Skorohod representation on the Lebesgue interval. 1.1 The a.s. Skorohod representation Let (X ; ae) be a Polish space and let X 1 ; X 2 ; : : : be random elements taking values in X and converg...

Motivated by original Skorokhod's ideas, a new topology has been defined on the space P(X ) of tight probability distributions on a topological space (X ; ). The only topological assumption imposed on (X ; ) is that some countable family of continuous functions separates points of X . This new sequential topology, defined by means of a variant of t...

A new topology (called S) is defined on the space D of functions x:[0,1]→R 1 which are right-continuous and admit limits from the left at each t>0. Although S cannot be metricized, it is quite natural and shares many useful properties with the traditional Skorokhod’s topologies J 1 and M 1 . In particular, on the space P(D) of laws of stochastic pr...

A p-stable limit theorem holds for partial sums Sn of a stationary sequence, if Sn/Bn --> g[mu] for some 1/p-regularly varying sequence and some non-denegerate strictly p-stable law [mu]. The case 0 < p < 2 is investigated in detail and simplified necessary and sufficient conditions are given. The conditions consist of a mixing condition and polyno...

The large deviation problem for sums of independent identically distributed random vectors with values from Zd is considered. It is assumed that the underlying distribution has the heavy tails. The special attention is paid to the role which is played by the density of the distribution support at infinity. The proven local and integral theorems gen...

Necessary and sufficient conditions are given for multidimensional p-stable limit theorems (i.e. theorems on convergence of normalized partial sums S n /b n of a stationary sequence of random vectors to a non-degenerate strictly p-stable limiting law μ, with 1/p-regularly varying normalizing sequence b n ). It is proved that similarly as in the one...

There exists a satisfactory limit theory for stochastic integrals driven by finite dimensional semimartingales. The paper is an attempt to construct a corresponding theory in the case of processes with values in infinite dimensional Hilbert spaces. Limit theorems are given for scalar-and tensor-product integrals and for Hilbert-Schmidt integrands....

We generalize existing limit theory for stochastic integrals driven by semimartingales and with left-continuous integrands. Joint Skorohod convergence is replaced with joint finite dimensional convergence plus an assumption excluding the case when oscillations of the integrand appear immediately before oscillations of the integrator. Integrands may...

The following question is examined: Suppose one-dimensional marginals of a strictly stationary sequence belong to the domain of attraction of a stable law. Do all finite-dimensional marginals possess this property as well? A negative is given within the class of one-dependent sequences.

We consider a stationary sequence of associated real random variables and state conditions which guarantee that partial sums of this sequence, when properly normalized, converge in distribution to a stable, non-Gaussian limit. Limit theorems for jointly stable and associated random variables are investigated in detail. In the general case we assume...

Let X1, X2,... be a stationary sequence of random variables. Denote by M(k)n the kth largest value of X1, X2, ..., Xn. We find necessary and sufficient conditions for the existence of an (r- 1)-dependent stationary sequence X1,X2, ...(determined by a distribution function G and numbers [beta]1,[beta]2,...,[beta]r[greater-or-equal, slanted] 0,[Sigma...

Let $\{X_k\}_{k \in \mathbb{N}}$ be a nonstationary sequence of random variables. Sufficient conditions are found for the existence of an independent sequence $\{\tilde{X}_k\}_{k \in \mathbb{N}}$ such that $\sup_{x \in \mathbb{R}^1}|P(M_n \leq x) - P(\tilde{M}_n \leq x)| \rightarrow 0$ as $n \rightarrow \infty$, where $M_n$ and $\tilde{M}_n$ are $n...

Let be a stationary sequence of random variables with partial sums Sn. Necessary and sufficient conditions are found for weak convergence , where [mu] is strictly p-stable and Bn --> [infinity] is 1/p-regularly varying. Limit theorems involving centering are also discussed.

The partial-sum processes defined by a quadratic form in independent random variables are martingales. For such processes, using suitable tools of the martingale limit theory, we obtain both sufficient and necessary conditions for the functional central limit theorem to hold. Quadratic forms with nulls on the diagonal are considered only.

The relative extremal index of a stationary sequence {Xn} with respect to a stationary sequence {X'n} is defined as a number 0<[theta]<[infinity] such that for every sequence un, . This concept generalizes both the notion of the extremal index and a phantom distribution function. Criteria of existence are given. The main tools are asymptotic multip...

A necessary and sufficient condition for the weak convergence of partial sums of strongly mixing random sequences towards p-stable distributions is established and applied to recent work of Davis (1983) and Jakubowski and Kobus (1987).

Several α-stable limit theorems for sums of dependent random vectors are proved via point processes theory; p-mixing, m-dependence, and the type of mixing treated within the extreme value theory are considered.

For sequences of stochastic Integrals K n .X n , functional limit Theorems are presented, these results hold under simple natural conditions.

Suppose that {pH) is a sequenoe of random probabili- ty measures on a real and separable Hilbert space such that, for each neN, is a pointwisely convergent convolution of some sequence iP,,,Jke N) of random measures. The sequence {A) is said to be sh$- tight if one can find random vectors {~,l such that the "centered" sequence (,~,*6-,~~) is tight....

Let $\{X_{nk}: k \in \mathbb{N}, n \in \mathbb{N}\}$ be a double array of random variables adapted to the sequence of discrete filtrations $\{\{\mathscr{F}_{nk}: k \in \mathbb{N} \cup \{0\}\}: n \in \mathbb{N}\}$. It is proved that for every weak limit theorem for sums of independent random variables there exists an analogous limit theorem which is...

A special (extended) kind of convergence in distribution of processes with filtration is considered. Recent theorems on the functional convergence of semimartingales are improved by showing that their assumptions imply the extended convergence of semimartingales to continuous in probability processes with independent increments.

Let {Xnk}, k = 1,2 kn; n = 1,2,..., be an array of random variables defined on a common probability space (Ω, F, P). If {Xnk} are row-wise independent, then there exists a quite satisfactory theory of the weak convergence of sums \(S_n ^\prime = \sum\limits_{k = 1}^{k_n } {X_{nk} .} \) One of the most reasonable trends in the analogous theory for d...

We discuss limiting procedures which support the interpretation of stochastic diierential equations.

We provide a heuristics for managing local dependencies in limit theorems for sums of weakly dependent random variables. The recipe is motivated by the case of m-dependent random variables and allows to calculate parameters of the limit on the base of asymptotic properties of laws of finite sums. The theory is illustrated with examples of limit the...