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On moments of ladder height variables. (English) Zbl 0598.60079
Let \(X_ 1,X_ 2,...,X_ n,..\). be i.i.d. random variables. Define \(S_ 0=0\), \(S_ n=X_ 1+...+X_ n\), \(N=\inf \{n\geq 1:\) \(S_ n\leq 0\}\). The author gives necessary and sufficient conditions in order that \(E| S_ N|^ p<\infty\) for \(p\geq 1\). He points out some problems. In particular, what is the necessary and sufficient condition for \(E| S_ N|^ p<\infty\) when \(0<p<1\).
Reviewer: V.M.Kruglov

60G50 Sums of independent random variables; random walks
Full Text: DOI
[1] Chow, Y.S; Lai, T.L, On the maximal excess in boundary crossings of random walks related to fluctuation theory and laws of large numbers, Bull. inst. math. acad. sinica, 7, 271-289, (1979) · Zbl 0413.60060
[2] Chow, Y.S; Lai, T.L, Moments of ladder variables for driftless random walks, Z. wahrsch. verw. gebeite, 48, 253-257, (1978) · Zbl 0416.60078
[3] Chow, Y.S; Teicher, H, Probability theory, (1978), Springer-Verlag Berlin
[4] Doney, R.A, Moments of ladder heights in random walks, J. appl. probab., 17, 248-252, (1980) · Zbl 0424.60072
[5] Erickson, K.B, The SLLN when the Mean is undefined, Trans. amer. math. soc., 185, 371-381, (1973)
[6] Feller, W, ()
[7] Lai, T.L, Asymptotic moments of random walks with applications to ladder variables and renewal theory, Ann. probab., 4, 51-66, (1976) · Zbl 0351.60062
[8] Hogan, M, Moments of the minimum of a random walk and complete convergence, Stanford univ., dept. of statistics, tech. report, (1984)
[9] Robbins, H, Mixture of distributions, Ann. math. statist., 19, 360-369, (1948) · Zbl 0037.36301
[10] Spitzer, F, A Tauberian theory and its probability interpretation, Trans. amer. math. soc., 94, 150-169, (1960) · Zbl 0216.21201
[11] Spitzer, F, Principles of random walks, (1964), Van Nostrand Princeton, N.J · Zbl 0119.34304
[12] Wolff, R.W, Conditions for finite ladder height and delay moments, Oper. res., 32, 909-916, (1984) · Zbl 0552.60089
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