Assume $X_{n}$ is an i.i.d (independent, identical distribution) sequence of random variables with the distribution function F(x) and non-negative expected value, $EX_{n}=\mu>0$ . Let $Y_n=0,Y_n=\sum_{k=1}^{n}X_k, n>=1; N(t)=sup[n:Y_n<=t]$. Find the distribution of N(t) and EN(t).
Now if $X_{n}$s were non-negative, $N_t$ would be a renewal process. but, here, the only info we have (that I don't know where to use it) is that $EX_{n}=\mu>0$. so my first question is
1-N(t) is a counting process anyway, regardless of $X_n$ being positive or negative, right? what is its distribution function?
2-I have read this almost related answer: https://math.stackexchange.com/questions/1654305/renewal-process-and-renewal-function
but as I said I'm not sure if it is a renewal process at all. If yes, I am not sure if the lower limit of the integral $(F \star F)(t)=\int_0^t F(t-u)F(du)$ should stay $0$ or it should be - infinity?
3- Is the following way to compute $EN(t)=m_t$ correct? $m_t=\sum_{n=0}^{\inf}nP(N(T)=n)=\sum_{n=1}^{\inf}P(N(T)>=n)=\sum_{n=1}^{\inf}P(Y_n<=t)=\sum_{n=1}^{\inf}F_n(t)$ where $F_n(t)$ is the dirstibution function of $Y_n$