## An almost sure invariance principle for the range of planar random walks.(English)Zbl 1085.60018

Let $$S_n=X_1+\dots+X_n$$ be a random walk in $$Z^2$$, where $$X_1,X_2,\ldots$$ are symmetric i.i.d. vectors in $$Z^2$$. We assume that $$X_i$$ have $$2+\delta$$ moments for some $$\delta>0$$ and covariance matrix equal to the identity. We assume further that the random walk $$S_n$$ is strongly aperiodic in the sense of Spitzer. The range $$\mathcal R(n)$$ of the random walk $$S_n$$ is the set of sites visited by the walk up to step $$n$$: $$\mathcal R(n)=\{S_0,\ldots,S_{n-1}\}.$$ As usual, $$| \mathcal R(n)|$$ denotes the cardinality of the range up to step $$n$$. The authors show that for each $$k\geq1$$ $(\log n)^k\bigg[\frac1n| \mathcal R(n)| +\sum_{j=1}^k(-1)^j\bigg(\frac1{2\pi}\log n+c_X\bigg)^{-j}\gamma_{j,n}\bigg]\to0\qquad\text{a.s.},$ where $$W_t$$ is a Brownian motion, $$W_t^{(n)}=W_{nt}/\sqrt{n}$$, $$\gamma_{j,n}$$ is the renormalized intersection local time at time 1 for $$W^{(n)}$$ and $$c_X$$ is a constant depending on the distribution of the random walk.

### MSC:

 60F15 Strong limit theorems 60G50 Sums of independent random variables; random walks

### Keywords:

intersection local time; Wiener sausage
Full Text:

### References:

  Bass, R. and Chen, X. (2004). Self-intersection local time: Critical exponent, large deviations and law of the iterated logarithm. Ann. Probab. 32 3221–3247. · Zbl 1075.60097  Bass, R. and Khoshnevisan, D. (1993). Strong approximations to Brownian local time. In Seminar on Stochastic Processes 1992 43–65. Birkhäuser, Boston. · Zbl 0789.60062  Bass, R. and Khoshnevisan, D. (1993). Intersection local times and Tanaka formulas. Ann. Inst. H. Poincaré Probab. Statist. 29 419–451. · Zbl 0798.60072  Bass, R. F. and Kumagai, T. (2002). Laws of the iterated logarithm for the range of random walks in two and three dimensions. Ann. Probab. 30 1369–1396. · Zbl 1031.60031  Chen, X. and Rosen, J. (2002). Exponential asymptotics and law of the iterated logarithm for intersection local times of stable processes.  Dvoretzky, A. and Erdös, P. (1951). Some problems on random walk in space. Proc. Second Berkeley Symp. Math. Statist. 352–367. Univ. California Press, Berkeley. · Zbl 0044.14001  Dynkin, E. B. (1988). Self-intersection gauge for random walks and for Brownian motion. Ann. Probab. 16 1–57. JSTOR: · Zbl 0638.60081  Dynkin, E. B. (1988). Regularized self-intersection local times of planar Brownian motion. Ann. Probab. 16 58–74. JSTOR: · Zbl 0641.60085  Einmahl, U. (1987). A useful estimate in the multidimensional invariance principle. Probab. Theory Related Fields 76 81–101. · Zbl 0608.60029  Hamana, Y. (1998). An almost sure invariance principle for the range of random walks. Stochastic Process. Appl. 78 . 131–143. · Zbl 0934.60044  Jain, N. C. and Pruitt, W. E. (1970). The range of recurrent random walk in the plane. Z. Wahrsch. Verw. Gebiete 16 279–292. · Zbl 0194.49205  Le Gall, J.-F. (1986). Propriétés d’intersection des marches aléatoires, I. Convergence vers le temps local d’intersection. Comm. Math. Phys. 104 471–507. · Zbl 0609.60078  Le Gall, J.-F. (1990). Wiener sausage and self intersection local times. J. Funct. Anal. 88 299–341. · Zbl 0697.60081  Le Gall, J.-F. (1992). Some properties of planar Brownian motion. École d ’ Été de Probabilités de St. Flour XX 1990. Lecture Notes in Math. 1527 112–234. Springer, Berlin. · Zbl 0779.60068  Le Gall, J.-F. and Rosen, J. (1991). The range of stable random walks. Ann. Probab. 19 650–705. JSTOR: · Zbl 0729.60066  Marcus, M. and Rosen, J. (1994). Laws of the iterated logarithm for the local times of symmetric Lévy processes and recurrent random walks. Ann. Probab. 22 626–659. JSTOR: · Zbl 0815.60073  Marcus, M. and Rosen, J. (1994). Laws of the iterated logarithm for the local times of recurrent random walks on $$Z^2$$ and of Lévy processes and random walks in the domain of attraction of Cauchy random variables. Ann. Inst. H. Poincaré 30 467–499. · Zbl 0805.60069  Marcus, M. and Rosen, J. (1999). Renormalized Self-Intersection Local Times and Wick Power Chaos Processes . Mem. Amer. Math. Soc. 142 . Amer. Math. Soc., Providence, RI. · Zbl 1230.60005  Rosen, J. (1990). Random walks and intersection local time. Ann. Probab. 18 959–977. JSTOR: · Zbl 0717.60057  Rosen, J. (1996). Joint continuity of renormalized intersection local times. Ann. Inst. H. Poincaré Probab. Statist. 32 671–700. · Zbl 0867.60049  Rosen, J. (2001). Dirichlet processes and an intrinsic characterization for renormalized intersection local times. Ann. Inst. H. Poincaré 37 403–420. · Zbl 0981.60072  Rosenthal, H. P. (1970). On the subspaces of $$L^p$$ ($$p>2$$) spanned by sequences of independent random variables. Israel J. Math. 8 273–303. · Zbl 0213.19303  Spitzer, F. (1976). Principles of Random Walk . Springer, New York. · Zbl 0359.60003  Varadhan, S. R. S. (1969). Appendix to “Euclidian Quantum Field Theory” by K. Symanzyk. In Local Quantum Theory (R. Jost, ed.). Academic Press, New York.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.