Self-intersections of 1-dimensional random walks. (English) Zbl 0602.60055

Suppose that \(S_ n\), \(n\geq 0\) is a random walk on the integers and that its steps have mean 0 and variance \(\sigma^ 2\). It is shown that the time \(T_ 1\) of first self-intersection of the random walk grows at rate \(\sigma^{2/3}\) as \(\sigma\to \infty\). In fact the following limit theorem is proved for the entire process \((T_ i)\) of self- intersections, where \(T_ i=\min \{n>T_{i-1}:S_ n=S_ m\) for some \(m<n\}\), \(i\geq 2\). Let W(t), \(t\geq 0\) be Brownian motion and define \(0<U_ 1<U_ 2<..\). by stipulating that, conditioned on W, the times \((U_ i)\) are the points of a non-homogeneous Poisson process of rate L(t,W(t)), \(t\geq 0\) where L(t,x) is the local time of W. It is shown that if \(S^*(t)=\sigma^{-4/3}S_{[t\sigma^{2/3}]}\), \(t\geq 0\) and \(T_ i^*=\sigma^{-2/3}T_ i\), then under certain technical conditions \[ (S^*,T^{*}_{1},T^{*}_{2},...)\to^{{\mathcal D}}(W,U_ 1,U_ 2,...) \] as \(\sigma\to \infty\), in the sense of weak convergence on D[0,\(\infty)\times [0,\infty)\times [0,\infty)\times..\). It is further shown that \(\sigma^{-2/3}ET_ 1\to EU_ 1=2^{2/3}\Gamma (5/3)E(Y^{-2/3})\approx 1.43\), where \(Y=\int^{\infty}_{-\infty}L^ 2(1,x)dx\).
Reviewer: F.Papangelou


60G50 Sums of independent random variables; random walks
60F05 Central limit and other weak theorems
60J65 Brownian motion
Full Text: DOI


[1] Aldous, D.J.: Exchangeability and related topics, in: Ecole d’Ete St. Flour 1983, Springer Lecture Notes 1117. Berlin, Heidelberg, New York: Springer 1985a
[2] Aldous, D.J.: Self-intersections of Random walks on discrete groups. Math. Proc. Camb. Philos. Soc. 98, 155-177 (1985b) · Zbl 0566.60066
[3] Arratia, R.: Limiting point processes for rescaling of coalescing and annihilating Random walks on Z d .Ann. Probab. 9, 909-936 (1981) · Zbl 0496.60098
[4] Barlow, M.T.: L(B t , t) is Not a Semimartingale, in: Seminaire de Probabilites XVI, 209-211, Springer Lecture Notes in Mathematics 920. Berlin, Heidelberg, New York: Springer 1982 · Zbl 0479.60082
[5] Billingsley, P.: Convergence of probability measures. New York: Wiley 1968 · Zbl 0172.21201
[6] Borodin, A.N.: On the asymptotic behavior of local times of recurrent Random walks with finite variance. Theory Probab. Appl. 26, 758-772 (1981) · Zbl 0488.60078
[7] Borodin, A.N.: Distribution of integral functions of the Brownian motion process. LOMI 119, 19-38 (1982) (Russian. English translation in J. Soviet Math. 27, 3005-3022) · Zbl 0491.60082
[8] Freed, K.F.: Polymers as self-avoiding walks. Ann. Probab. 9, 537-556 (1981) · Zbl 0468.60097
[9] Gnedenko, B.V., Kolmogorov, A.N.: Limit distributions for sums of independent Random variables. Reading-London: Addison-Wesley 1954 · Zbl 0056.36001
[10] Knuth, D.E.: The Art of computer programming, Vol. 2, 2nd edition. Reading: Addison-Wesley 1981 · Zbl 0477.65002
[11] Pavlov, Y.L.: Limit theorem for a characteristic of a Random mapping. Theory Probab. Appl. 26, 829-834 (1981) · Zbl 0488.60019
[12] Perkins, E.: Local time is a semimartingale. Z. Wahrscheinlichkeitstheor. Verw. Geb. 60, 79-117 (1982a) · Zbl 0465.60065
[13] Perkins, E.: Weak invariance principles for local time. Z. Wahrscheinlichktstheor. Verw. Geb. 60, 437-451 (1982b) · Zbl 0465.60065
[14] Pittel, B.: On the distributions related to transivive classes of Random finite mappings. Ann. Probab. 11, 428-441 (1983) · Zbl 0515.60015
[15] Pollard, D.: Convergence of stochastic processes. Berlin, Heidelberg, New York: Springer 1984 · Zbl 0544.60045
[16] Pollard, J.M.: On not storing the path of a Random walk. BIT 19, 545-548 (1979) · Zbl 0418.65002
[17] Westwater, J.: On Edwards’ model for polymer chains. Proc. of the 4th Bielefeld Conference on Mathematical Physics. Singapore: World Scientific Publishing 1984 · Zbl 0583.60066
[18] Williams, D.: Path decomposition and continuity of local time for one-dimensional diffusions, I. Proc. Lond. Math. Soc. 28, 738-768 (1974) · Zbl 0326.60093
[19] Williams, D.: Diffusions, Markov processes and Martingales. New York: Wiley 1979 · Zbl 0402.60003
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.