Bernoulli

  • Bernoulli
  • Volume 23, Number 2 (2017), 1279-1298.

Asymptotics of random processes with immigration II: Convergence to stationarity

Alexander Iksanov, Alexander Marynych, and Matthias Meiners

Full-text: Open access

Enhanced PDF (206 KB)

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 $Y(t):=\sum_{k\geq0}X_{k+1}(t-\xi_{1}-\cdots-\xi_{k})\mathbf{1}_{\{\xi_{1}+\cdots+\xi_{k}\leq t\}}$, $t\in\mathbb{R}$ 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.

Article information

Source
Bernoulli, Volume 23, Number 2 (2017), 1279-1298.

Dates
Received: June 2015
First available in Project Euclid: 4 February 2017

Permanent link to this document
https://projecteuclid.org/euclid.bj/1486177399

Digital Object Identifier
doi:10.3150/15-BEJ777

Mathematical Reviews number (MathSciNet)
MR3606766

Zentralblatt MATH identifier
1377.60046

Keywords
random point process renewal shot noise process stationary renewal process weak convergence in the Skorokhod space

Citation

Iksanov, Alexander; Marynych, Alexander; Meiners, Matthias. Asymptotics of random processes with immigration II: Convergence to stationarity. Bernoulli 23 (2017), no. 2, 1279--1298. doi:10.3150/15-BEJ777. https://projecteuclid.org/euclid.bj/1486177399


Export citation

References

  • [1] Alsmeyer, G., Iksanov, A. and Meiners, M. (2015). Power and exponential moments of the number of visits and related quantities for perturbed random walks. J. Theoret. Probab. 28 1–40.
  • [2] Athreya, K.B. and Ney, P.E. (1972). Branching Processes. New York: Springer. Die Grundlehren der mathematischen Wissenschaften, Band 196.
  • [3] Billingsley, P. (1968). Convergence of Probability Measures. New York: Wiley.
    Mathematical Reviews (MathSciNet): MR233396
    Zentralblatt MATH: 0172.21201
  • [4] Goldie, C.M. and Maller, R.A. (2000). Stability of perpetuities. Ann. Probab. 28 1195–1218.
  • [5] Iksanov, A. (2013). Functional limit theorems for renewal shot noise processes with increasing response functions. Stochastic Process. Appl. 123 1987–2010.
  • [6] Iksanov, A., Marynych, A. and Meiners, M. (2014). Limit theorems for renewal shot noise processes with eventually decreasing response functions. Stochastic Process. Appl. 124 2132–2170.
  • [7] Iksanov, A., Marynych, A. and Meiners, M. (2015). Asymptotics of random processes with immigration I: Scaling limits. Bernoulli 23 1233–1278.
  • [8] Jagers, P. (1968). Age-dependent branching processes allowing immigration. Teor. Verojatnost. i Primenen 13 230–242.
  • [9] Kaplan, N. (1975). Limit theorems for a $GI/G/\infty $ queue. Ann. Probab. 3 780–789.
  • [10] Karlin, S. and Taylor, H.M. (1975). A First Course in Stochastic Processes, 2nd ed. New York: Academic Press.
  • [11] Konstantopoulos, T. and Lin, S.-J. (1998). Macroscopic models for long-range dependent network traffic. Queueing Systems Theory Appl. 28 215–243.
  • [12] Lindvall, T. (1973). Weak convergence of probability measures and random functions in the function space $D(0,\infty)$. J. Appl. Probab. 10 109–121.
  • [13] Lindvall, T. (1992). Lectures on the Coupling Method. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley.
  • [14] Mikosch, T. and Resnick, S. (2006). Activity rates with very heavy tails. Stochastic Process. Appl. 116 131–155.
  • [15] Miller, D.R. (1974). Limit theorems for path-functionals of regenerative processes. Stochastic Process. Appl. 2 141–161.
  • [16] Pakes, A.G. and Kaplan, N. (1974). On the subcritical Bellman-Harris process with immigration. J. Appl. Probab. 11 652–668.
  • [17] Resnick, S. (2002). Adventures in Stochastic Processes. Boston, MA: Birkhäuser.
    Mathematical Reviews (MathSciNet): MR1181423
    Zentralblatt MATH: 0762.60002
  • [18] Resnick, S.I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. New York: Springer.
  • [19] Thorisson, H. (2000). Coupling, Stationarity, and Regeneration. Probability and Its Applications. New York: Springer.
  • [20] Whitt, W. (1980). Some useful functions for functional limit theorems. Math. Oper. Res. 5 67–85.
  • [21] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer Series in Operations Research. New York: Springer.

Bernoulli Society for Mathematical Statistics and Probability

Bernoulli

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more