Let $(X_n:n\in\mathbb{N})$ be a sequence of independent, identically distributed random variables of finite mean $m$ and finite variance $\sigma^2$. Set $S_0=0$ and $S_n=X_1+\dots+X_n$ for $n\in\mathbb{N}$. Consider the linear interpolation $(S_t)_{t\geq0}$ given by $$S_{n+t}=(1-t)S_n+tS_{n+1}$$ for $n\in\mathbb{Z}^+$ and $t\in[0,1]$. Set $S_t^{(N)}=N^{-1}S_{Nt}$ and write $\mu_N$ for the distribution of $(S_t^{(N)})_{t\geq0}$ on $C([0,\infty),\mathbb{R})$. Show that the sequence $(\mu_N)_{N\in\mathbb{N}}$ converges weakly on $C([0,\infty),\mathbb{R})$ and determine its limit.
My thoughts: I expect that Donsker's Invariance Principle is expected to be used here, but this (as far as my lectures have taught) is for random walks with steps of zero mean. This on the other hand is a biased random walk, and additionally the scaling is not as it is in Donsker's principle (the process for convergence to wiener measure is $(N^{-1/2}S_{Nt})_{t\geq0}$, so I am quite unsure of how to proceed. Any advice would be greatly appreciated!
I think here (unless there is a typo), since you are looking at the process $$\frac{1}{N} S_{Nt}= \frac{Nt}{N}\frac{1}{Nt} S_{Nt}$$ you will not need any "big guns" like Donsker's theorem or the CLT. For each fixed $t$ not zero (if $t=0$ then $S_0=0$) $S_t^{(N)}$ is tending a.s. to $mt$ by the strong law of large numbers. It also holds that for any $t_1,...,t_K\in [0,\infty)$, $(S_{t_1}^{(N)},....,S_{t_K}^{(N)}) \to (m t_1,...,m t_K)$ a.s.. So the finite dimensional distributions of the process all converge to $m t$. The only remaining thing to show in view of Prohorov's theorem is tightness of the process $S_t^{(N)}$. How would you show that?