# zbMATH — the first resource for mathematics

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

##### MSC:
 60G50 Sums of independent random variables; random walks
Full Text:
##### References:
 [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
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.