Reflection Principle

   Let X1, a seqence of independent random variables.


  random walk at time k, the position of the random walk after n step is given by

   Hitting Times (First passage times)
   Let Tk=min{t|St=k} be first hitting times at k.
  Let us define functions that satisfy the following condition.

   Put (t0.k0)(t1.k1)={s|s=the pass with St1-t0=k1,X0=k0}
2>Tk3,(t0.k0)(t1.k1))={s|s=the pass with St1-t0=k1,X0=k0,Tk2>Tk3}
  The following relation is satisfied.

   Let us find the total number of following path.
   The following condition is satisfied.
   The total number of pass  (Tm>T-m,(0.0)(n.m)) is given by