MathDB
Miklos Schweitzer 1977_10

Source: random walk

January 25, 2009
limitprobability and stats

Problem Statement

Let the sequence of random variables {Xm,  m≥0 },  X0=0 \{ X_m, \; m \geq 0\ \}, \; X_0=0, be an infinite random walk on the set of nonnegative integers with transition probabilities pi=P(Xm+1=i+1∣Xm=i)>0,  i≥0  p_i=P(X_{m+1}=i+1 \mid X_m=i) >0, \; i \geq 0 \, qi=P(Xm+1=i−1∣Xm=i)>0,  i>0. q_i=P(X_{m+1}=i-1 \mid X_m=i ) >0, \; i>0. Prove that for arbitrary k>0 k >0 there is an αk>1 \alpha_k > 1 such that Pn(k)=P(max⁡0≤j≤nXj=k) P_n(k)=P \left ( \max_{0 \leq j \leq n} X_j =k \right) satisfies the limit relation lim⁡L→∞1L∑n=1LPn(k)αkn<∞. \lim_{L \rightarrow \infty} \frac 1L \sum_{n=1}^L P_n(k) \alpha_k ^n < \infty. J. Tomko
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