Summary
Consider a random walk S n on the integers, where the steps ξ i have mean 0 and variance σ2. Let T be the time of first self-intersection of the random walk. It is shown that, as σ→∞, T grows at rate σ2/3. More precisely, Tσ2/3 has a non-degenerate limit distribution which can be described in terms of Brownian motion local time.
References
Aldous, D.J.: Exchangeability and related topics, in: Ecole d'Ete St. Flour 1983, Springer Lecture Notes 1117. Berlin, Heidelberg, New York: Springer 1985a
Aldous, D.J.: Self-intersections of Random walks on discrete groups. Math. Proc. Camb. Philos. Soc. 98, 155–177 (1985b)
Arratia, R.: Limiting point processes for rescaling of coalescing and annihilating Random walks on Z d.Ann. Probab. 9, 909–936 (1981)
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
Billingsley, P.: Convergence of probability measures. New York: Wiley 1968
Borodin, A.N.: On the asymptotic behavior of local times of recurrent Random walks with finite variance. Theory Probab. Appl. 26, 758–772 (1981)
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)
Freed, K.F.: Polymers as self-avoiding walks. Ann. Probab. 9, 537–556 (1981)
Gnedenko, B.V., Kolmogorov, A.N.: Limit distributions for sums of independent Random variables. Reading-London: Addison-Wesley 1954
Knuth, D.E.: The Art of computer programming, Vol. 2, 2nd edition. Reading: Addison-Wesley 1981
Pavlov, Y.L.: Limit theorem for a characteristic of a Random mapping. Theory Probab. Appl. 26, 829–834 (1981)
Perkins, E.: Local time is a semimartingale. Z. Wahrscheinlichkeitstheor. Verw. Geb. 60, 79–117 (1982a)
Perkins, E.: Weak invariance principles for local time. Z. Wahrscheinlichktstheor. Verw. Geb. 60, 437–451 (1982b)
Pittel, B.: On the distributions related to transivive classes of Random finite mappings. Ann. Probab. 11, 428–441 (1983)
Pollard, D.: Convergence of stochastic processes. Berlin, Heidelberg, New York: Springer 1984
Pollard, J.M.: On not storing the path of a Random walk. BIT 19, 545–548 (1979)
Westwater, J.: On Edwards' model for polymer chains. Proc. of the 4th Bielefeld Conference on Mathematical Physics. Singapore: World Scientific Publishing 1984
Williams, D.: Path decomposition and continuity of local time for one-dimensional diffusions, I. Proc. Lond. Math. Soc. 28, 738–768 (1974)
Williams, D.: Diffusions, Markov processes and Martingales. New York: Wiley 1979
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Research supported by National Science Foundation Grant MCS80-02698.
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Aldous, D.J. Self-intersections of 1-dimensional random walks. Probab. Th. Rel. Fields 72, 559–587 (1986). https://doi.org/10.1007/BF00344721
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DOI: https://doi.org/10.1007/BF00344721
Keywords
- Stochastic Process
- Brownian Motion
- Random Walk
- Probability Theory
- Statistical Theory