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Strong law and central limit theorem for a process between maxima and sums (1991)

Hollander, F. ; Hooghiemstra, G. ; Keane, M. ; [et al.]

Springer

Probability theory and related fields 90 (1991), S. 37-55

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ISSN: 1432-2064

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: 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.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF01321133

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