ASYMPTOTIC EXPECTED NUMBER OF PASSAGES OF A RANDOM WALK THROUGH AN INTERVAL

作者：Kella Offer^*; Stadje Wolfgang

来源：Journal of Applied Probability, 2013, 50(1): 288-294.

• Random walk
• passage
• generalized renewal theorem
• two-sided renewal theorem

In this note we find a new result concerning the asymptotic expected number of passages of a finite or infinite interval (x, x + h] as x -%26gt; infinity for a random walk with increments having a positive expected value. If the increments are distributed like X then the limit for 0 %26lt; h %26lt; infinity turns out to have the form E min(vertical bar X vertical bar, h)/EX, which unexpectedly is independent of h for the special case where vertical bar X vertical bar %26lt;= b %26lt; infinity almost surely and h %26gt; b. When h = infinity, the limit is E max(X, 0)/EX. For the case of a simple random walk, a more pedestrian derivation of the limit is given.

• 出版日期2013-3