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Let $A_t=t-S_{N_t -1}$ with $N_t$ a renewal process

1)Show $A_t$ checks the Markov property

my proof: $S_{N_t}=X_1+\cdots+X_{N_t}$ and the increments are independents $$P(S_{N_t-1}=t-y\mid S_{N_{u_1}-1}=u_1-x_1,\ldots,S_{N_{u_n}-1}=u_n-x_n)=P(S_{N_t-1}-S_{N_{u_n}-1}=t-y-(u_n-x_n))=P(S_{N_t-1}=t-y\mid S_{N_{u_n}-1}=u_n-x_n)$$

Is it correct?

Thank you

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