Limit Behavior of a Compound Poisson Process with Switching and Dominated Summands
- PDF / 242,376 Bytes
- 11 Pages / 594 x 792 pts Page_size
- 10 Downloads / 190 Views
LIMIT BEHAVIOR OF A COMPOUND POISSON PROCESS WITH SWITCHING AND DOMINATED SUMMANDS A. N. Borodin∗
UDC 519.2
The paper deals with limit behavior of a compound Poisson process with switching and dominated summands. The switching is provided by Bernulli’s random variables and a Markov chain. Under suitable normalization the limit process is a Brownian motion with switching variance and jumps. Bibliography: 4 titles.
This investigation continues the topic of paper [1]. In the present paper, a special class of compound Poisson processes with switchings and dominated summands is considered. The limit behavior of such processes is studied as a → ∞, where a is a certain parameter included in the structure of these processes. There is a sequence of i.i.d. two-dimensional random vectors. Each of them is a mixture of 1 two vectors. With probability 1 − , these vectors have zero average and a finite variance, and a √ 1 they are arbitrary but equipped with coefficient a characterizing their with probability a dominance. Switchings from one coordinate of a random vector to another occurs in moments determined by independent Bernoulli variables independent of the original vectors and the Poisson process. The same variables determine the mixture of the coordinates. In this case if one falls, then the switching does not occur, whereas if zero falls, then the switching is carried out. Thus the switching process starts as a classic compound Poisson process with set of the first coordinates of the random vectors. Then in the geometrically distributed time the summands are replaced by the variables from another coordinate of i.i.d. vectors and in accordance with these variables the further development of the process takes place. After an independent geometrically distributed time, the summands are replaced by variables from the original√coordinate, and so on. When switching occurs, the sum includes summands with coefficient a, that leads to the presence of rare dominated summands. The present paper studies the limit behavior of a compound Poisson process with switching when the Poisson time parameter increases in a times. With appropriate normalization, the limit process is a Brownian motion with switching variance and jumps. One can also consider more general Poisson processes with switchings, in which the selection of summands comes from three or more coordinates of the identically distributed multidimensional random vectors. However, a general approach to the study of distributions behavior of compound Poisson processes with switchings and jumps proposed in the present paper does not change, but leads to certain complications. 1. Compound Poisson process with switching and dominated summands The Poisson process N (t), t ≥ 0, with intensity λ1 > 0 can be represented as follows: l τk ≤ t 1[0,t] (τ1 ), N (t) := max l : k=1 ∗
St.Petersburg Department of Steklov Mathematical Institute; St.Petersburg State University, St.Petersburg, Russia, e-mail: [email protected].
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 474, 2018,
Data Loading...
