Question: 1.18. For an arbitrary random walk, if :?>{Vn EN: Sn > O} > 0, then L :?>{Sn :s 0, Sn+l > O} < 00. n

1.18. For an arbitrary random walk, if :?>{Vn EN: Sn > O} > 0, then L :?>{Sn :s 0, Sn+l > O} < 00.

n Hence if in addition :?>{Vn EN: Sn :s O} > 0, then L 1:?>{Sn > O} -:?>{Sn+l > O}I < 00;

n and consequently L q>{Sn > O} < 00. n n

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[HINT: For the first series, consider the last time that Sn < 0; for the third series, apply Du Bois-Reymond's test. Cf. Theorem 8.4.4 below; this exercise will be completed in Exercise 15 of Sec. 8.5.]

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