Blog
About

17
views
0
recommends
+1 Recommend
0 collections
    0
    shares
      • Record: found
      • Abstract: found
      • Article: found
      Is Open Access

      Asymptotics of random processes with immigration II: convergence to stationarity

      Preprint

      , ,

      Read this article at

      Bookmark
          There is no author summary for this article yet. Authors can add summaries to their articles on ScienceOpen to make them more accessible to a non-specialist audience.

          Abstract

          Let \(X_1, X_2,\ldots\) be random elements of the Skorokhod space \(D(\mathbb{R})\) and \(\xi_1, \xi_2, \ldots\) positive random variables such that the pairs \((X_1,\xi_1), (X_2,\xi_2),\ldots\) are independent and identically distributed. We call the random process \((Y(t))_{t \in \mathbb{R}}\) defined by \(Y(t):=\sum_{k \geq 0}X_{k+1}(t-\xi_1-\ldots-\xi_k)1_{\{\xi_1+\ldots+\xi_k\leq t\}}\), \(t\in\mathbb{R}\) random process with immigration at the epochs of a renewal process. Assuming that \(X_k\) and \(\xi_k\) are independent and that the distribution of \(\xi_1\) is nonlattice and has finite mean we investigate weak convergence of \((Y(t))_{t\in\mathbb{R}}\) as \(t\to\infty\) in \(D(\mathbb{R})\) endowed with the \(J_1\)-topology. The limits are stationary processes with immigration.

          Related collections

          Most cited references 10

          • Record: found
          • Abstract: not found
          • Article: not found

          Some Useful Functions for Functional Limit Theorems

           Ward Whitt (1980)
            Bookmark
            • Record: found
            • Abstract: not found
            • Article: not found

            Stability of perpetuities

              Bookmark
              • Record: found
              • Abstract: found
              • Article: not found

              Weak convergence of probability measures and random functions in the function space D[0,∞)

              This paper extends the theory of weak convergence of probability measures and random functions in the function space D[0,1] to the case D [0,∞), elaborating ideas of C. Stone and W. Whitt. 7)[0,∞) is a suitable space for the analysis of many processes appearing in applied probability.
                Bookmark

                Author and article information

                Journal
                27 November 2013
                2015-10-09
                Article
                1311.6923

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                Custom metadata
                20 pages, accepted for publication in Bernoulli
                math.PR

                Comments

                Comment on this article