×

A renewal theorem for relatively stable variables. (English) Zbl 1466.60184

Let \(X\) denote a real random variable with d.f. \(F(.)\). The Green (or renewal) measure of the random walk generated by \(F\) is given by \( U(I)=\sum_{n=0}^{\infty }P(S_{n}\in I)\). For decades many authors have studied the asymptotic form of \(U([x,x+h))\) as \(x\) tends to infinity. In this paper, the author presents exact asymptotic formulas in the case where \(X\) is relatively stable with \(EX=\infty \). Among other results, he proves the following theorem. Suppose that \(X\) is non-negative and non-arithmetic and that \( L(x)=\int_{0}^{x}(1-F(t))dt\) is slowly varying. Then for each \(h>0\), it holds that \(L(x)U([x,x+h))\rightarrow h\) as \(x\rightarrow \infty \). In the case where \(X\) is real, the author states conditions to insure that \( m(x)U([x,x+h))\rightarrow hC_{+}\) as \(x\rightarrow \infty \) and \( m(-x)U([x,x+h))\rightarrow hC_{-}\) as \(x\rightarrow -\infty \), where \(m(x)\) is a slowly varying function (depending on \(F\)) and the constants \( C_{+},C_{-}\)are given. The proofs are based on a detailed asymptotic analysis of the real and the imaginary part of \(1/(1-\phi (z))\), where \(\phi (z)\) is the characteristic function of \(X\).

MSC:

60K05 Renewal theory
60G20 Generalized stochastic processes
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Q.Berger, ‘Notes on random walks in the Cauchy domain of attraction’, Probab. Theory Related Fields175 (2019) 1-44. · Zbl 1479.60086
[2] N. H.Bingham, G. M.Goldie and J. L.Teugels, Regular variation (Cambridge University Press, Cambridge, 1989). · Zbl 0667.26003
[3] D.Blackwell, ‘A renewal theorem’, Duke Math. J.15 (1948) 145-150. · Zbl 0030.20102
[4] F.Caravenna and R. A.Doney, ‘Local large deviations and the strong renewal theorem’, Electron. J. Probab.24 (2019) 48, paper no. 72. · Zbl 1467.60068
[5] P.Erds̎, W.Feller and H.Pollard, ‘A property of power series with positive coefficients’, Bull. Amer. Math. Soc.55 (1949) 201-204. · Zbl 0032.27802
[6] K. B.Erickson, ‘Strong renewal theorems with infinite mean’, Trans. Amer. Math. Soc.151 (1970) 263-291. · Zbl 0212.51601
[7] K. B.Erickson, ‘The strong law of large numbers when the mean is undefined’, Trans. Amer. Math. Soc.185 (1973) 371-381. · Zbl 0304.60016
[8] W.Feller, An introduction to probability theory and its applications, vol. 2, 2nd edn (John Wiley and Sons, New York, 1971). · Zbl 0158.34902
[9] W.Feller and S.Orey, ‘A renewal theorem’, J. Math. Mech.10 (1961) 619-624. · Zbl 0096.33401
[10] A.Garsia and J.Lamperti, ‘A discrete renewal theorem with infinite mean’, Comment. Math. Helv.37 (1962/3) 221-234. MR \(26 \sharp 5630\). · Zbl 0114.08803
[11] H.Kesten and R. A.Maller, ‘Infinite limits and infinite limit points of random walks and trimmed sums’, Ann. Probab.22 (1994) 1473-1513. · Zbl 0816.60067
[12] H.Kesten and R. A.Maller, ‘Stability and other limit laws for exit times of random walks from a strip or a half line’, Ann. Inst. Henri Poincaré35 (1999) 685-734. · Zbl 0940.60064
[13] R. A.Maller, ‘Relative stability and the strong law of large numbers’, Z. Wahrsch. verw. Geb.43 (1978) 141-148. · Zbl 0366.60035
[14] R. A.Maller, ‘Relative stability, characteristic functions and stochastic compactness’, J. Aust. Math. Soc. (Series A)28 (1979) 499-509. · Zbl 0396.60020
[15] D.Ornstein, ‘Random walks’, Trans. Amer. Math. Soc.138 (1969) 1-60. · Zbl 0181.44501
[16] E. J. G.Pitman, ‘On the behaviour of the characteristic function of a probability distribution in the neighbourhood of the origin’, J. Aust. Math. Soc.8 (1968) 422-443. · Zbl 0164.48502
[17] B. A.Rogozin, ‘On the distribution of the first ladder moment and height and fluctuations of a random walk’, Theory Probab. Appl.16 (1971) 575-595. · Zbl 0269.60053
[18] B. A.Rogozin, ‘Relatively stable walks’, Theory Probab. Appl.21 (1976) 375-379. · Zbl 0381.60043
[19] F.Spitzer, Principles of random walks (Van Nostrand, Princeton, NJ, 1964). · Zbl 0119.34304
[20] C. J.Stone, ‘On the potential operator for one dimensional recurrent random walks’, Trans. Amer. Math. Soc.136 (1969) 427-445. · Zbl 0215.53104
[21] K.Uchiyama, ‘On the ladder heights of random walks attracted to stable laws of exponent 1’, Electron. Commun. Probab.23 (2018) 1-12. · Zbl 1390.60170
[22] K.Uchiyama, ‘Estimates of potential functions of random walks on \(\mathbb{Z}\) with zero mean and infinite variance and their applications’, Preprint, 2020, http://arxiv.org/abs/1802.09832.
[23] K.Uchiyama, ‘A note on the exit problem from an interval for random walks oscillating on \(\mathbb{Z}\) with infinite variance’, Preprint, 2019, http://arxiv.org/abs/1908.00303.
[24] R.Wei, ‘On the long‐range directed polymer model’, J. Stat. Phys.165 (2016) 320-350. · Zbl 1355.82070
[25] J. A.Williamson, ‘Random walks and Riesz kernels’, Pacific J. Math.25 (1968) 393-415. · Zbl 0239.60066
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.