Abstract
Let W be a non-negative random variable with EW=1, and let {W i } be a family of independent copies of W, indexed by all the finite sequences i=i 1⋅⋅⋅i n of positive integers. For fixed r and n the random multiplicative measure μ n r has, on each r-adic interval \(A_{i_1 ...i_n }^r \) at nth level, the density \(W_{i_1 } \cdot \cdot \cdot W_{i_1 \ldots i_n } \) with respect to the Lebesgue measure on [0,1]. If EW log W<log r, the sequence {μ n r } n converges a.s. weakly to the Mandelbrot measure μ ∞ r . For each fixed 1≤n≤∞, we study asymptotic properties for the sequence of random measures {μ n r } r as r→∞. We prove uniform laws of large numbers, functional central limit theorems, a functional law of iterated logarithm, and large deviation principles. The function-indexed processes \(\{ \mu _r^n (f),\;f \in G\}\) is a natural extension to a tree-indexed process at nth level of the usual smoothed partial-sum process corresponding to n=1. The results extend the classical ones for {μ 1 r } r , and the recent ones for the masses of {μ ∞ r } r established in Ref. 23.
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References
Alexander, S. A., and Pyke, R. (1986). A uniform central limit theorem for set-indexed partial-sum processes with finite variance. Ann. Probab. 14, 582–597.
Barral, J. (1999). Moments, continuité, et analyse multifractale des martingales de Mandelbrot. Probab. Theory Related Fields 113, 535–569.
Bass, R. F. (1985). Law of iterated logarithm for set-indexed partial sum processes with finite variance. Z. Wahrsch. Verw. Gebiete 70, 591–608.
Bass, R. F., and Pyke, R. (1984). A strong law of large numbers for partial-sum processes indexed by sets. Ann. Probab. 12, 268–271.
Biggins, J. D. (1992). Uniform convergence of martingales in the branching random walk. Ann. Probab. 20, 137–151.
Breiman, L. (1992). Probability. Classics in Applied Mathematics 7, SIAM 1992.
Choi, B. D., and Sung, S. H. (1987). Almost sure convergence theorems of weighted sums of random variables. Stochastic Anal. Appl. 5, 365–377.
Chow, Y. S., and Teicher, H. (1978). Probability: Independence, Interchangeability, and Martingales, Springer, New York.
Durrett, R., and Liggett, T. (1983). Fixed points of the smoothing transformation. Z. Wahrsch. Verw. Gebeite 64, 275–301.
Dembo, A., and Zajic, T. (1995). Uniform large and moderate deviations for functional empirical processes. Stochastic Process. Appl. 67, 195–211.
Dembo, A., and Zeitouni, O. (1998). Large Deviations Techniques and Applications, Springer, New York.
Eichelsbacher, P., and Schmock, U. (1998). Exponential approximations in completely regular topological spaces and extensions of Sanov's theorem. Stochastic Process. Appl. 77, 233–251.
Etemadi, N. (1981). An elementary proof of the strong law of large numbers. Z. Wahrsch. Verw. Gebiete 55, 119–122.
Gaenssler, P., and Ziegler, K. (1994). On function-indexed partial-sum processes. In Grigelionis, B., et al. (eds.), Prob. Theory and Math. Stat., VSP/TEV Vilnius, pp. 285–311.
Gaenssler, P., and Ziegler, K. (1994a). A uniform law of large-numbers for set-indexed processes with applications to empirical and partial-sum processes. In Probability in Banach Spaces, Vol.9, Birkhäuser, Boston, pp. 385–400.
Guivarc'h, Y. (1990). Sur une extension de la notion de loi semi-stable. Ann. IHP 26, 261–285.
Kahane, J. P., and Peyrière, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. Math. 22, 131–145.
Lacey, M. (1989). Law of the iterated logarithm for partial sum processes indexed by functions. J. Theoret. Probab. 2, 377–398.
Ledoux, M., and Talagrand, M. (1991). Probability in Banach Spaces, Springer-Verlag, New York.
Liu, Q. (1998). Fixed points of a generalized smoothing transformation and applications to branching random walks. Adv. Appl. Probab. 30, 85–112.
Liu, Q. (2000). On generalized multiplicative cascades. Stochastic Process. Appl. 86, 263–286.
Liu, Q. (2001). Asymptotic properties and absolute continuity of laws stable by random weighted mean. Stochastic Process. Appl. 95, 83–107.
Liu, Q., and Rouault, A. (2000). Limit theorems for Mandelbrot's multiplicative cascades. Ann. Appl. Probab. 10, 218–231.
Mandelbrot, B. (1974a). Multiplications aléatoires et distributions invariantes par moyenne pondérée aléatoire. C. R. Acad. Sci. Paris Sér. I Math. 278, 289–292 and 355–358.
Mandelbrot, B. (1974b). Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–333.
Massart, P. (2000). About the constants in Talagrand's concentration inequalities for empirical processes. Ann. Probab. 28, 863–884.
Najim, J. (2002). A Cramér type theorem for weighted empirical means. Electron. J. Probab. 7(4), 1–32.
Petrov, V. V. (1975). Sums of Independent Random Variables, Springer-Verlag, Berlin.
Pollard, D. (1984). Convergence of Stochastic Processes, Springer, Berlin.
Rogers, L. C. G., and Williams, D. (1994). Diffusions, Markov Processes, and Martingales, Vol.1, John Wiley and Sons, Chichester. Reprint of the second edition (1994), Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2000.
Strassen, V. (1964). An invariance principle for the law of iterated logarithm. Z. Wahrsch. Verw. Gebiete 3, 211–226.
Talagrand, M. (1996). New concentration inequalities in product spaces. Invent. Math. 126, 503–563.
van der Vaart, A. W., and Wellner, J. A. (1996). Weak Convergence and Empirical Processes, Springer, New York.
Waymire, E. C., and Williams, D. (1996). A cascade decomposition theory with applications to Markov and exchangeable cascades. Trans. Amer. Math. Soc. 348, 585–632.
Wu, L. (1994). Large deviations, moderate deviations, and LIL for empirical processes. Ann. Probab. 22, 17–27.
Yurinskii (1977). On the error of the Gaussian approximation for convolutions. Theory Probab. Appl. XXII, 236–247.
Ziegler, K. (1997). Functional central limit theorems for triangular arrays of function-indexed processes under uniformly integrated entropy conditions. J. Multivariate Anal. 62, 233–272.
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Liu, Q., Rio, E. & Rouault, A. Limit Theorems for Multiplicative Processes. Journal of Theoretical Probability 16, 971–1014 (2003). https://doi.org/10.1023/B:JOTP.0000012003.49768.f6
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DOI: https://doi.org/10.1023/B:JOTP.0000012003.49768.f6