# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Recurrence properties of a special type of heavy-tailed random walk. (English) Zbl 1223.60021

Consider the heavy tail distribution

${p}_{k}:=\frac{1}{2\zeta \left(3\right)}\phantom{\rule{0.166667em}{0ex}}{k}^{-3},\phantom{\rule{2.em}{0ex}}k\in ℤ\setminus \left\{0\right\},$

on the integer lattice and let $Q={\left({Q}_{n}\right)}_{n=0,1,2,\cdots }$ be the corresponding integer-valued random walk. Furthermore, let $S={\left({S}_{n}\right)}_{n=0,1,2,\cdots }$ be the random walk on the two-dimensional integer lattice with step distribution

${p}_{\left(0,k\right)}^{\text{'}}={p}_{\left(k,0\right)}^{\text{'}}=\frac{1}{2}{p}_{k}·$

The author derives local limit theorems for $Q$ and $S$ as well as the asymptotics for the time of first return to the origin, and the number of visits to the origin in the first $n$ steps.

##### MSC:
 60F05 Central limit and other weak theorems 60G50 Sums of independent random variables; random walks
##### Keywords:
random walk; local limit theorem; recurrence; heavy tail
##### References:
 [1] Bunimovich, L.A., Sinai, Y.G.: Statistical properties of Lorentz gas with periodic configuration of scatterers. Commun. Math. Phys. 78, 479–497 (1980/1981) · Zbl 0459.60099 · doi:10.1007/BF02046760 [2] Bunimovich, L.A., Sinai, Y.G., Chernov, N.I.: Statistical properties of two-dimensional hyperbolic billiards. Usp. Mat. Nauk 46, 43–92 (1991) (in Russian). English translation in Russ. Math. Surv. 46, 47–106 (1991) [3] Chernov, N.: Advanced statistical properties of dispersing billiards. J. Stat. Phys. 122, 1061–1094 (2006) · Zbl 1098.82020 · doi:10.1007/s10955-006-9036-8 [4] Chernov, N., Dolgopyat, D.: Anomalous current in periodic Lorentz gases with infinite horizon. Usp. Mat. Nauk 64(4), 73–124 (2009) (in Russian). English translation in Russ. Math. Surv. 64(4), 651–699 (2009) [5] Darling, D.A., Kac, M.: On occupation times for Markov processes. Trans. Am. Math. Soc. 84, 444–458 (1957) · doi:10.1090/S0002-9947-1957-0084222-7 [6] de Haan, L., Peng, L.: Slow convergence to normality: an Edgeworth expansion without third moment. Prob. Math. Stat. 17(2), 395–406 (1997) [7] Dolgopyat, D., Szász, D., Varjú, T.: Recurrence properties of planar Lorentz process. Duke Math. J. 142 (2008) [8] Dolgopyat, D., Szász, D., Varjú, T.: Limit theorems for locally perturbed planar Lorentz process. Duke Math. J. 148 (2009) [9] Donsker, M.D.: An invariance principle for certain probability limit theorems. Mem. Am. Math. Soc. 6 (1951) [10] Dvoretzky, A., Erdos, P.: Some problems on random walk in space. In: Proc. 2nd Berkeley Sympos. Math. Statis. Probab., pp. 353–367 (1951) [11] Erdos, P., Taylor, S.J.: Some problems concerning the structure of random walk paths. Acta Math. Hung. (1960) [12] Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 2, 2nd edn. Wiley, New York (1971) [13] Ibragimov, I.A., Linnik, Yu.V.: Independent and Stationary sequences of random variables (1971). ISBN 90 01 41885 6 [14] Juozulynas, A., Paulauskas, V.: Some remarks on the rate of convergence to stable laws. Lith. Math. J. 38, 335–347 (1998) · Zbl 0984.60031 · doi:10.1007/BF02465818 [15] Nándori, P.: Number of distinct sites visited by a random walk with internal states. Prob. Theory Relat. Fields (2010). doi: 10.1007/s00440-010-0277-8 [16] Paulin, D., Szász, D.: Locally perturbed random walks with unbounded jumps. J. Stat. Phys. 141, 1116–1130 (2010) · Zbl 1205.82083 · doi:10.1007/s10955-010-0078-6 [17] Rvaceva, E.: On the domains of attraction of multidimensional distributions. Sel. Trans. Math. Stat. Prob. 2, 183–207 (1962) [18] Spohn, H.: Long time tail for spacially inhomogeneous random walks. In: Mathematical Problems in Theoretical Physics. Lecture Notes in Physics, p. 116. Springer, Berlin (1980) [19] Szász, D., Telcs, A.: Random walk in an inhomogeneous medium with local impurities. J. Stat. Phys. 26(3) (1981) [20] Szász, D., Varjú, T.: Limit laws and recurrence for the planar Lorentz process with infinite horizon. J. Stat. Phys. 129, 59–80 (2007) · Zbl 1128.82011 · doi:10.1007/s10955-007-9367-0