Abstract
Let \((X_n)_{n=1}^{\infty }\) be a sequence of independent identically distributed random variables. We study the normalized partial sums and the corresponding renormalization group flow in the space of probability densities. We prove the convergence to stable limit laws under suitable assumptions on the initial density.
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Notes
For simplicity in what follows we shall often call \(0<\alpha \le 2\) simply as the stable laws.
In the most general situation one can investigate \((\sum _{i=1}^n X_i -a_n)/b_n\) for suitable \(a_n\) and \(b_n\). Here for simplicity we just consider \(a_n=0\) and \(b_n=n^{1/\alpha }\). The condition \(a_n=0\) can be guaranteed by assuming \(\mathbb {E} X_i=0\). Later on this assumption will be included in the condition on the density \(\rho _X^{(0)}\). On the other hand the normalization factor \(b_n=n^{1/\alpha }\) is connected with some decay conditions on \(\rho _X^{(0)}\). For example, in the case \(\alpha =2\), one imposes the moment condition \(\mathbb {E} X_i^2<\infty \) and this leads to the usual central limit theorem.
To derive it one can use the independence of \(Y_m\) and \(Y_m^{\prime }\). Namely \(\widehat{\rho _Y^{(m+1)}}(\xi ) =\mathbb {E} e^{- i \xi \cdot Y_{m+1}} = \mathbb {E} e^{-i 2^{-\frac{1}{\alpha }} \xi \cdot Y_m } \mathbb {E} e^{-i 2^{-\frac{1}{\alpha }} \xi \cdot Y_m^{\prime }} = \biggl ( \widehat{\rho _Y^{(m)}}(2^{-1/\alpha } \xi ) \biggr )^2\).
We introduce two numbers \(\beta _0\), \(\beta \) so that one can easily run the iteration argument.
By an induction argument, easy to check that \(f_n\) can be expanded in terms of the basis functions \(g_{k,\alpha ,\gamma }\).
References
Li, D., Sinai, Y.G.: Blow ups of complex solutions of the 3D Navier-Stokes system and renormalization group method. J. Eur. Math. Soc. (JEMS) 10(2), 267–313 (2008)
Li, D., Sinai, Y.G.: Asymptotic behavior of generalized convolutions. Regul. Chaotic Dyn. 14(2), 248–262 (2009)
Christoph, G., Wolf, W.: Convergence Theorems with a Stable Limit Law. Akademie Verlag, Berlin (1992)
Gnedenko, B.V., Kolmogorov, A.N.: Sums of Independent Random Variables. Addison-Wesley, Reading, MA (1954)
De Haan, L., Peng, L.: Exact rates of convergence to a stable law. J. London Math. Soc. 59(2), 1134–1152 (1999)
Juozulynas, A., Paulauskas, V.: Some remarks on the rate of convergence to stable limit laws. Liet. Mat. Rink. 38(4), 439–455 (1998). (Translation in Lithuanian Math. J. 38 (1998), no. 4, 335–347 (1999)
Kuske, R., Keller, J.B.: Rate of convergence to a stable limit law. SIAM J. Appl. Math. 61(4), 1308–1323 (2000)
Feller, W.: An Introduction to Probability Theory and Its Applications, 2nd (ed.) vol. 2. John Wiley, New York (1971)
Acknowledgments
Y. G. Sinai was supported by NSF Grant DMS 1265547. D. Li was supported in part by an Nserc discovery grant. We thank the anonymous referees for very helpful comments and suggestions.
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Dedicated to the memory of K. Wilson.
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Li, D., Sinai, Y.G. An Application of the Renormalization Group Method to Stable Limit Laws. J Stat Phys 157, 915–930 (2014). https://doi.org/10.1007/s10955-014-1098-4
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DOI: https://doi.org/10.1007/s10955-014-1098-4