Summary
We prove an invariance principle for the random process (X n ) n≧1 given by
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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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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DOI: https://doi.org/10.1007/BF01321133