Abstract
The local limit theorem describes the behavior of the convolution powers of a probability distribution supported on \(\mathbb {Z}\). In this work, we explore the role played by positivity in this classical result and study the convolution powers of the general class of complex valued functions finitely supported on \(\mathbb {Z}\). This is discussed as de Forest’s problem in the literature and was studied by Schoenberg and Greville. Extending these earlier works and using techniques of Fourier analysis, we establish asymptotic bounds for the sup-norm of the convolution powers and prove extended local limit theorems pertaining to this entire class. As the heat kernel is the attractor of probability distributions on \(\mathbb {Z}\), we show that the convolution powers of the general class are similarly attracted to a certain class of analytic functions which includes the Airy function and the heat kernel evaluated at purely imaginary time.
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References
Bakhoom, N.G.: Expansions of the function \(F_k(x)=\int _0^{\infty }e^{-u^k+xu}\, du\). Proc. Lond. Math. Soc. 35(2), 83–100 (1933)
Barbatis, G., Davies, E.B.: Sharp bounds on heat kernels of higher order uniformly elliptic operators. J. Oper. Theory 36, 197–198 (1995)
Brenner, P., Thomée, V., Wahlbin, L.: Besov Spaces and Applications to Difference Methods for Initial Value Problems. Springer, Berlin (1975)
Burwell, W.R.: Asymptotic expansions of generalized hypergeometric functions. Proc. Lond. Math. Soc. 22(2), 57–72 (1924)
Davies, E.B.: Uniformly elliptic operators with measurable coefficients. J. Funct. Anal. 132, 141–169 (1995)
Diaconis, P., Saloff-Coste, L.: Convolution powers of complex functions on Z. Math. Nachr. 287(10), 1106–1130 (2014)
Dungey, N.: Higher order operators and Gaussian bounds on Lie groups of polynomial growth. J. Oper. Theory 46(1), 45–61 (2002)
Eidelman, S.D.: Parabolic Systems. North-Holland Publishg Company and Wolters-Nordhoff Publishing, Groningen (1969)
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)
Greville, T.N.E.: On stability of linear smoothing formulas. SIAM J. Numer. Anal. 3(1), 157–170 (1966)
Hardy, G.H., Littlewood, J.E. Some problems of “Partitio Numerorum”: I. A new solution of Waring’s problem. Ges. Wiss. Göttingen., 33–54 (1920)
Hersh, R.: A class of “Central Limit Theorems” for convolution products of generalized functions. Trans. Am. Math. Soc. 140, 71–85 (1969)
Hochberg, K.J.: A signed measure on path space related to Wiener measure. Ann. Probab. 6(3), 433–458 (1978)
Hochberg, K.J.: Limit theorem for signed distributions. Proc. Am. Math. Soc. 79(2), 298–302 (1980)
Hochberg, K., Orsingher, E.: Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations. J. Theor. Probab. 9(2), 511–532 (1996)
Krylov, V.Y.: Some properties of the distributions corresponding to the equation \(\partial u/\partial t=(-1)^{q+1}\partial ^{2q}u/\partial x^{2q}\). Soviet Math. Dok. 1, 760–763 (1960)
Lachal, A.: First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation \(\frac{\partial }{\partial t}=\pm \frac{\partial ^N}{\partial x^N}\). Electron. J. Probab. 12(29), 300–353 (2007)
Lachal, A.: A survey on the pseudo-process driven by the high-order heat-type equation \(\frac{\partial }{\partial t}=\pm \frac{\partial ^N}{\partial x^N}\) concerning the hitting and sojourn times. Methodol. Comput. Appl. Probab. 14(3), 549–566 (2012)
Lawler, G., Limic, V.: Random Walk: A Modern Introduction. Cambridge University Press, New York (2010)
Levin, D., Lyons, T.: A signed measure on rough paths associated to a PDE of high order: results and conjectures. Rev. Mat. Iberoam. 25(3), 971–994 (2009)
Nishioka, K.: Monopoles and dipoles in biharmonic pseudo process. Proc. Jpn. Acad. Ser. A Math. Sci. 72(3), 47–50 (1996)
Robinson, D.W.: Elliptic Operators and Lie Groups. Oxford University Press, Oxford (1991)
Robinson, D.W.: Elliptic differential operators on Lie groups. J. Func. Anal. 97, 373–402 (1991)
Sato, S.: An approach to the biharmonic pseudo process by a random walk. J. Math. Kyoto. Univ. 42(3), 403–422 (2002)
Schoenberg, I.J.: On smoothing operations and their generalized functions. Bull. Am. Math. Soc. 59, 229–230 (1953)
Spitzer, F.: Principle of Random Walk, 2nd edn. Springer, New York (1976)
Thomée, V.: Stability of difference schemes in the maximum-norm. J. Differ. Equ. 1, 273–292 (1965)
Thomée, V.: Stability theory for partial difference operators. SIAM Rev. 11, 152–195 (1969)
Zhukov, A.I.: A limit theorem for difference operators. J. Uspekhi Math. Nauk. 14(3), 129–136 (1959)
Acknowledgments
The authors would like to thank the referee for many useful suggestions and comments. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-0707428 and by the National Science Foundation under Grant No. DMS-1004771.
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Communicated by David Walnut.
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Randles, E., Saloff-Coste, L. On the Convolution Powers of Complex Functions on \(\mathbb {Z}\) . J Fourier Anal Appl 21, 754–798 (2015). https://doi.org/10.1007/s00041-015-9386-1
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DOI: https://doi.org/10.1007/s00041-015-9386-1