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Strong law and central limit theorem for a process between maxima and sums
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  • Published: March 1991

Strong law and central limit theorem for a process between maxima and sums

  • F. den Hollander1,
  • G. Hooghiemstra1,
  • M. Keane1 &
  • …
  • J. Resing1 

Probability Theory and Related Fields volume 90, pages 37–55 (1991)Cite this article

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Summary

We prove an invariance principle for the random process (X n ) n≧1 given by

$$\left\{ \begin{gathered} X_1 = x \in \mathbb{R} \hfill \\ X_{n + 1} = \max (X_{n,} \alpha _n X_n + Y_n ),{\text{ }}n \geqq 1 \hfill \\ \end{gathered} \right.$$

where (Y n ) n≧1 are i.i.d. random variables and (α n ) n≧ are nonrandom numbers tending upward to 1 (both in ℝ). This process interpolates between maxima (α n ≡0) and sums (α n ≡1). Depending on the distribution ofY n and on the rate at which α n →1 the scaling behaviour exhibits different regimes. Our techniques are flexible and are applicable to more general types of iterative schemes.

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References

  1. Billingsley, P.: Convergence of probability measures. New York: Wiley 1968

    Google Scholar 

  2. Breiman, L.: Probability. Reading, Mass: Addison-Wesley 1968

    Google Scholar 

  3. Csörgö, S., Mason, D.M.: The asymptotic distribution of sums of extreme values from a regularly varying distribution. Ann. Probab.14, 974–983 (1986)

    Google Scholar 

  4. Daley, D.J., Haslett, J.: A thermal energy storage process with controlled input. Adv. Appl. Probab.14, 257–271 (1982)

    Google Scholar 

  5. Gnedenko, B.: Sur la distribution limite du term maximum d'une série aléatoire. Ann. Math.44, 423–453 (1943)

    Google Scholar 

  6. Gnedenko, B., Kolmogorov, A.N.: Limit distributions for sums of independent random variables. Reading, Mass: Addison-Wesley 1968

    Google Scholar 

  7. Greenwood, P.E., Hooghiemstra, G.: On the domain of attraction of an operator between supremum and sum. Probab. Th. Rel. Fields89, 201–210 (1991)

    Google Scholar 

  8. Haan, L. de: On regular variation and its application to the weak convergence of sample extremes. Mathematical Centre Tracts 32, CWI, Amsterdam 1970

    Google Scholar 

  9. Haslett, J.: Problems in the storage of solar thermal energy. In: Jacobs, O.L.R. et al. (eds). Analysis and optimization of stochastic systems. London: Academic Press 1980

    Google Scholar 

  10. Haslett, J.: New bounds for the thermal energy storage process with stationary input. J. Appl. Probab.19, 894–899 (1982)

    Google Scholar 

  11. Hooghiemstra, G., Keane, M.: Calculation of the equilibrium distribution for a solar energy storage model. J. Appl. Probab.22, 852–864 (1985)

    Google Scholar 

  12. Hooghiemstra, G., Scheffer, C.L.: Some limit theorems for an energy storage model. Stochastic Processes Appl.22, 121–128 (1984)

    Google Scholar 

  13. McLeish, D.L.: Dependent central limit theorems and invariance principle. Ann. Probab.2, 620–628 (1974)

    Google Scholar 

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Authors and Affiliations

  1. Delft University of Technology, Mekelweg 4, 2628 CD, Delft, The Netherlands

    F. den Hollander, G. Hooghiemstra, M. Keane & J. Resing

Authors
  1. F. den Hollander
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  2. G. Hooghiemstra
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  3. M. Keane
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  4. J. Resing
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den Hollander, F., Hooghiemstra, G., Keane, M. et al. Strong law and central limit theorem for a process between maxima and sums. Probab. Th. Rel. Fields 90, 37–55 (1991). https://doi.org/10.1007/BF01321133

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  • Received: 05 July 1990

  • Revised: 08 March 1991

  • Issue Date: March 1991

  • DOI: https://doi.org/10.1007/BF01321133

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Keywords

  • Stochastic Process
  • Probability Theory
  • Limit Theorem
  • Random Process
  • Mathematical Biology
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