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An asymptotic expansion for the distribution of the supremum of a random walk

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Let ${S_n}$ be a random walk drifting to -∞. We obtain an asymptotic expansion for the distribution of the supremum of ${S_n}$ which takes into account the influence of the roots of the equation $1-∫_ℝe^{sx}F(dx)=0,F$ being the underlying distribution. An estimate, of considerable generality, is given for the remainder term by means of submultiplicative weight functions. A similar problem for the stationary distribution of an oscillating random walk is also considered. The proofs rely on two general theorems for Laplace transforms.
Twórcy
  • Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk 90, 630090 Russia, sgibnev@math.nsc.ru
Bibliografia
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  • [9] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, Colloq. Publ. 31, Amer. Math. Soc., Providence, 1957.
  • [10] S. Janson, Moments for first passage times, the minimum, and related quantities for random walks with positive drift, Adv. Appl. Probab. 18 (1986), 865-879.
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  • [13] V. I. Lotov, Asymptotics of the distribution of the supremum of consecutive sums, Mat. Zametki 38 (1985), 668-678 (in Russian).
  • [14] B. A. Rogozin and M. S. Sgibnev, Banach algebras of measures on the line, Siberian Math. J. 21 (1980), 265-273.
  • [15] M. S. Sgibnev, On the distribution of the supremum of sums of independent summands with negative drift, Mat. Zametki 22 (1977), 763-770 (in Russian).
  • [16] M. S. Sgibnev, Submultiplicative moments of the supremum of a random walk with negative drift, Statist. Probab. Lett. 32 (1997), 377-383.
  • [17] M. S. Sgibnev, Equivalence of two conditions on singular components, ibid. 40 (1998), 127-131.
  • [18] M. S. Sgibnev, On the distribution of the supremum in the presence of roots of the characteristic equation, Teor. Veroyatnost. i Primenen. 43 (1998), 383-390 (in Russian).
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  • [20] N. Veraverbeke, Asymptotic behavior of Wiener-Hopf factors of a random walk, Stochastic Process. Appl. 5 (1977), 27-37.
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