Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA formula for hidden regular variation behavior for symmetric stable distributions
116
0
0.0
(
0
)
Added by
Malin Pal\\\"o Forsstr\\\"om
Publication date
2019
fields
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We develop a formula for the power-law decay of various sets for symmetric stable random vectors in terms of how many vectors from the support of the corresponding spectral measure are needed to enter the set. One sees different decay rates in different directions, illustrating the phenomenon of hidden regular variation. We give several examples and obtain quite varied behavior, including sets which do not have exact power-law decay.
rate research
Read More
We survey the connections between extreme-value theory and regular variation, in one and higher dimensions, from the algebraic point of view of our recent work on Popa groups.
The L2-approximation of occupation and local times of a symmetric $alpha$-stable L{e}vy process from high frequency discrete time observations is studied. The standard Riemann sum estimators are shown to be asymptotically efficient when 0 < $alpha$ $le$ 1, but only rate optimal for 1 < $alpha$ $le$ 2. For this, the exact convergence of the L2-approximation error is proven with explicit constants.
General sufficient conditions are given for absolute continuity and convergence in variation of the distributions of the unctionals on a probability space, generated by a Poisson point measure. The phase space of the Poisson point measure is supposed to be of the form (0,infty)times U, and its intensity measure to be equal dtPi(du). We introduce the family of time stretching transformations of the configurations of the point measure. The sufficient conditions for absolute continuity and convergence in variation are given in the terms of the time stretching transformations and the relative differential operators. These conditions are applied to solutions of SDEs driven by Poisson point measures, including an SDEs with non-constant jump rate.
A regenerative random composition of integer $n$ is constructed by allocating $n$ standard exponential points over a countable number of intervals, comprising the complement of the closed range of a subordinator $S$. Assuming that the L{e}vy measure of $S$ is infinite and regularly varying at zero of index $-alpha$, $alphain(0,,1)$, we find an explicit threshold $r=r(n)$, such that the number $K_{n,,r(n)}$ of blocks of size $r(n)$ converges in distribution without any normalization to a mixed Poisson distribution. The sequence $(r(n))$ turns out to be regularly varying with index $alpha/(alpha+1)$ and the mixing distribution is that of the exponential functional of $S$. The result is derived as a consequence of a general Poisson limit theorem for an infinite occupancy scheme with power-like decay of the frequencies. We also discuss asymptotic behavior of $K_{n,,w(n)}$ in cases when $w(n)$ diverges but grows slower than $r(n)$. Our findings complement previously known strong laws of large numbers for $K_{n,,r}$ in case of a fixed $rinmathbb{N}$. As a key tool we employ new Abelian theorems for Laplace--Stiletjes transforms of regularly varying functions with the indexes of regular variation diverging to infinity.
We derive an Ito-formula for the Dawson-Watanabe superprocess, a well-known class of measure-valued processes, extending the classical Ito-formula with respect to two aspects. Firstly, we extend the state-space of the underlying process $(X(t))_{tin [0,T]}$ to an infinite-dimensional one - the space of finite measure. Secondly, we extend the formula to functions $F(t,X_t)$ depending on the entire paths $X_t=(X(swedge t))_{s in [0,T]}$ up to times $t$. This later extension is usually called functional Ito-formula. Finally we remark on the application to predictable representation for martingales associated with superprocesses.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
