Let $X_1,X_2,\ldots$ be mutually independent random variables, and for each $X_i$, there is $P(X_i=1)=p$ and $P(X_i=0)=1-p$.
Let $Y_1,Y_2,\ldots$ be binary random variables defined by \begin{equation} \begin{split} Y_1&=X_1,\\ Y_t&=\begin{cases} 1,&\mbox{if $\frac{1}{t}\sum^t_{j=1}X_j\ge p+(1-2p)\frac{1}{t-1}\sum^{t-1}_{i=1}Y_i$}\\ 0,&\mbox{otherwise} \end{cases}. \end{split} \end{equation}
We'd like to prove that \begin{equation} E(Y_t)=Pr(Y_t=1)\le p, \ \ \ \forall\, t=1,2,\ldots \end{equation}
Could you please tell me how to prove it? Thanks a lot.