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Gyula Pap

Publications and source records attributed to Gyula Pap.

At least 73 records · Page 4Linked to original sources
arXiv2012

Testing stability in a spatial unilateral autoregressive model

Least squares estimator of the stability parameter $\varrho := |α| + |β|$ for a spatial unilateral autoregressive process $X_{k,\ell}=αX_{k-1,\ell}+βX_{k,\ell-1}+\varepsilon_{k,\ell}$ is investigated. Asymptotic normality with a scaling factor $n^{5/4}$ is shown in the unstable case, i.e., when $\varrho = 1$, in contrast to the AR(p) model $X_k=α_1 X_{k-1}+... +α_p X_{k-p}+ \varepsilon_k$, where the least squares estimator of the stability parameter $\varrho :=α_1 + ... + α_p$ is not asymptotically normal in the unstable, i.e., in the unit root case.

math.ST↗
arXiv2011

Parameter estimation in a spatial unit root autoregressive model

Spatial unilateral autoregressive model $X_{k,\ell}=αX_{k-1,\ell}+βX_{k,\ell-1}+γX_{k-1,\ell-1}+ε_{k,\ell}$ is investigated in the unit root case, that is when the parameters are on the boundary of the domain of stability that forms a tetrahedron with vertices $(1,1,-1), \ (1,-1,1),\ (-1,1,1)$ and $(-1,-1,-1)$. It is shown that the limiting distribution of the least squares estimator of the parameters is normal and the rate of convergence is $n$ when the parameters are in the faces or on the edges of the tetrahedron, while on the vertices the rate is $n^{3/2}$.

math.ST↗
arXiv2011

Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions

We consider a process $(X_t)_{t\in[0,T)}$ given by the SDE $dX_t = αb(t)X_t dt + σ(t) dB_t$, $t\in[0,T)$, with initial condition $X_0=0$, where $T\in(0,\infty]$, $α\in R$, $(B_t)_{t\in[0,T)}$ is a standard Wiener process, $b:[0,T)\to R\setminus\{0\}$ and $σ:[0,T)\to(0,\infty)$ are continuously differentiable functions. Assuming that $b$ and $σ$ satisfy a certain differential equation we derive an explicit formula for the joint Laplace transform of $\int_0^t\frac{b(s)^2}{σ(s)^2}(X_s)^2 ds$ and $(X_t)^2$ for all $t\in[0,T)$. As an application, we study asymptotic behavior of the maximum likelihood estimator of $α$ for $\sign(α-K)=\sign(K)$, $K\ne0$, and for $α=K$, $K\ne0$. As an example, we examine the so-called $α$-Wiener bridges given by SDE $dX_t = -\fracα{T-t}X_t dt + dB_t$, $t\in[0,T)$, with initial condition $X_0=0$.

math.PR↗
arXiv2010

Asymptotic behavior of unstable INAR(p) processes

In this paper the asymptotic behavior of an unstable integer-valued autoregressive model of order p (INAR(p)) is described. Under a natural assumption it is proved that the sequence of appropriately scaled random step functions formed from an unstable INAR(p) process converges weakly towards a squared Bessel process. We note that this limit behavior is quite different from that of familiar unstable autoregressive processes of order p. An application for Boston armed robberies data set is presented.

math.PR↗
arXiv2010

Outliers in INAR(1) models

In this paper the integer-valued autoregressive model of order one, contaminated with additive or innovational outliers is studied in some detail. Moreover, parameter estimation is also addressed. Supposing that the time points of the outliers are known but their sizes are unknown, we prove that the Conditional Least Squares (CLS) estimators of the offspring and innovation means are strongly consistent. In contrast, however, the CLS estimators of the outliers' sizes are not strongly consistent, although they converge to a random limit with probability 1. This random limit depends on the values of the process at the outliers' time points and on the values at the preceding time points and in case of additive outliers also on the values at the following time points. We also prove that the joint CLS estimator of the offspring and innovation means is asymptotically normal. Conditionally on the above described values of the process, the joint CLS estimator of the sizes of the outliers is also asymptotically normal.

math.PR↗
arXiv2009

A note on weak convergence of random step processes

First, sufficient conditions are given for a triangular array of random vectors such that the sequence of related random step functions converges towards a (not necessarily time homogeneous) diffusion process. These conditions are weaker and easier to check than the existing ones in the literature, and they are derived from a very general semimartingale convergence theorem due to Jacod and Shiryaev, which is hard to use directly. Next, sufficient conditions are given for convergence of stochastic integrals of random step functions, where the integrands are functionals of the integrators. This result covers situations which can not be handled by existing ones.

math.PR↗
arXiv2009

The accuracy of merging approximation in generalized St. Petersburg games

Merging asymptotic expansions of arbitrary length are established for the distribution functions and for the probabilities of suitably centered and normalized cumulative winnings in a full sequence of generalized St. Petersburg games, extending the short expansions due to Csörgő, S., Merging asymptotic expansions in generalized St. Petersburg games, \textit{Acta Sci. Math. (Szeged)} \textbf{73} 297--331, 2007. These expansions are given in terms of suitably chosen members from the classes of subsequential semistable infinitely divisible asymptotic distribution functions and certain derivatives of these functions. The length of the expansion depends upon the tail parameter. Both uniform and nonuniform bounds are presented.

math.PR↗
arXiv2008

alpha-Wiener bridges: singularity of induced measures and sample path properties

Let us consider the process $(X_t^{(α)})_{t\in[0,T)}$ given by the SDE $dX_t^{(α)} = -\fracα{T-t}X_t^{(α)} dt+ dB_t$, $t\in[0,T)$, where $α\in R$, $T\in(0,\infty)$, and $(B_t)_{t\geq 0}$ is a standard Wiener process. In case of $α>0$ the process $X^{(α)}$ is known as an $α$-Wiener bridge, in case of $α=1$ as the usual Wiener bridge. We prove that for all $α,β\in R$, $α\neβ$, the probability measures induced by the processes $X^{(α)}$ and $X^{(β)}$ are singular on C[0,T). Further, we investigate regularity properties of $X_t^{(α)}$ as $t\uparrow T$.

math.PR↗
arXiv2008

Asymptotic behavior of maximum likelihood estimator for time inhomogeneous diffusion processes

We study asymptotic behavior of maximum likelihood estimator for a time inhomogeneous diffusion process given by a SDE $dX_t=αb(t)X_t dt + σ(t) dB_t$, $t\in[0,T)$, with a parameter $α\in R$, where $T\in(0,\infty]$ and $(B_t)_{t\in[0,T)}$ is a standard Wiener process. We formulate sufficient conditions under which the MLE of $α$ normalized by Fisher information converges to the limit distribution of Dickey-Fuller statistics. Next we study a SDE $dY_t=αb(t)a(Y_t) dt + σ(t) dB_t$, $t\in[0,T)$, with a perturbed drift satisfying $a(x)=x+O(1+|x|^γ)$ with some $γ\in[0,1)$. We give again sufficient conditions under which the MLE of $α$ normalized by Fisher information converges to the limit distribution of Dickey-Fuller statistics.

math.ST↗
arXiv2008

On the least squares estimator in a nearly unstable sequence of stationary spatial AR models

A nearly unstable sequence of stationary spatial autoregressive processes is investigated, when the sum of the absolute values of the autoregressive coefficients tends to one. It is shown that after an appropriate norming the least squares estimator for these coefficients has a normal limit distribution. If none of the parameters equals zero than the typical rate of convergence is n.

math.ST↗
arXiv2008

Weakly infinitely divisible measures on some locally compact Abelian groups

On the torus group, on the group of p-adic integers and on the p-adic solenoid, we give a construction of an arbitrary weakly infinitely divisible probability measure using a random element with values in a product of (possibly infinitely many) subgroups of the real numbers. As a special case of our results, we have a new construction of the Haar measure on the p-adic solenoid.

math.PR↗
arXiv2007

Poisson limit of an inhomogeneous nearly critical INAR(1) model

An inhomogeneous first--order integer--valued autoregressive (INAR(1)) process is investigated, where the autoregressive type coefficient slowly converges to one. It is shown that the process converges weakly to a Poisson or a compound Poisson distribution.

math.PR↗
arXiv2007

Limit theorems on locally compact Abelian groups

We prove limit theorems for row sums of a rowwise independent infinitesimal array of random variables with values in a locally compact Abelian group. First we give a proof of Gaiser's theorem, since it does not have an easy access and it is not complete. This theorem gives sufficient conditions for convergence of the row sums, but the limit measure can not have a nondegenerate idempotent factor. Then we prove necessary and sufficient conditions for convergence of the row sums, where the limit measure can be also a nondegenerate Haar measure on a compact subgroup. Finally, we investigate special cases: the torus group, the group of p-adic integers and the p-adic solenoid.

math.PR↗
arXiv2006

Fourier transform of a Gaussian measure on the Heisenberg group

An explicit formula is derived for the Fourier transform of a Gaussian measure on the Heisenberg group at the Schrodinger representation. Using this explicit formula, necessary and sufficient conditions are given for the convolution of two Gaussian measures to be a Gaussian measure.

math.PR↗
arXiv2006

A Delayed Black and Scholes Formula I

In this article we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic differential delay equation (sdde). We believe that the proposed model is sufficiently flexible to fit real market data, and is yet simple enough to allow for a closed-form representation of the option price. Furthermore, the model maintains the no-arbitrage property and the completeness of the market. The derivation of the option-pricing formula is based on an equivalent martingale measure.

math.PR↗
arXiv2006

A Delayed Black and Scholes Formula II

This article is a sequel to [A.H.M.P]. In [A.H.M.P], we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic delay equation with fixed delays in the drift and diffusion terms. In this article, we look at models of the stock price described by stochastic functional differential equations with variable delays. We present a class of examples of stock dynamics with variable delays that permit an explicit form for the option pricing formula. As in [A.H.M.P], the market is complete with no arbitrage. This is achieved through the existence of an equivalent martingale measure. In subsequent work, the authors intend to test the models in [A.H.M.P] and the present article against real market data.

math.PR↗
arXiv2006

Portmanteau theorem for unbounded measures

We prove an analogue of the portmanteau theorem on weak convergence of probability measures allowing measures which are unbounded on an underlying metric space but finite on the complement of any Borel neighbourhood of a fixed element.

math.PR↗