Introduction

Let $R_{n}$ be the last time $X^n$ appeared in the past $$R_{n}(X^{n})=min\{ j\ge 0:X_{-j}^{-j+n-1}=X^{n} \}$$Let us define a new source alphabet $\mathcal{U}=\mathcal{X}^{n}$ and source $\{ U_{i} \}$ by setting $$U_{i}=X_{i}^{i+n-1}, \ \ i=0,\pm1,\pm2,\dots$$We see the new process is also stationary. >[!lem] Kac’s Lemma >For any $u\in \mathcal{U}$ such that $P(U_{1}=u)>0$ let $$Q_{u}(i)=P(U_{-i+1}=u,U_{j}\not=u\text{ for }-i+1<j<1|U_{1}=u)$$for $i=1,2,\dots$. Then $$E[R_{n}(X^{n})|X^{n}=x^{n}]=\frac{1}{P(X^{n}=x^{n})}$$for stationary ergodic sources.