Q:

Convergence of stochastic process sample path to deterministic function

Suppose we have a sequence of stochastic processes $\{X_{n,t}; t\geq 0\}_{n\in\mathbb N}$ such that $\mathsf E(X_{n,t})=f_n(t)=ng(t)$ where $n$ is some parameter and $g$ is just a function of $t$ (i.e. no dependence on $n$ for $g$). Furthermore assume that $\mathsf{Var}(X_{n,t}/n)\rightarrow 0$ as $n\rightarrow\infty.$ So we have that, for each fixed $t$, $X_{n,t}/n$ converges to $g(t)$ in probability as $n\rightarrow\infty.$ If it helps, we can also assume that there is a bound $|X_{n,t}|<M$ surely for all $n$ and $t.$ 
Question: Can we in any sense justify saying that: 
$$``X_{n,t} \text{ converges to } f_n(t)"$$
even if $X_{n,t}$ and $f_n(t)$ grow unboundedly for fixed $t$ values?
Maybe 
$$d(X_{n,t},f_n(t))=\frac{\sup_{t\geq0}|X_{n,t}-f_n(t)|}{n}$$ 
or 
$$d(X_{n,t},f_n(t))=\int_0^\infty \frac{(X_{n,t}-f_n(t))^2}{n^2} \ dt$$ 
or something else can serve as a metric so that $d(X_{n,t},f_n(t))$ converges to zero in probability? Is this idea of mixing metrics and probabilistic convergence a bit absurd?
The idea is that the graphs of $X_{n,t}$ and $f_n(t)$ as functions of $t$ are quite distinct for small $n$ and are visibly indistinguishable for large $n$, even though the maximal extent of the range of both graphs grows unboundedly in $n$.
The motivating example is radioactive decay, where $n$ is the initial number of particles that decay at rate $\alpha$. Let $X_{n,t}$ represent this stochastic process (for fixed $n$ and $t$, $X_{n,t}$ is just a binomial for $n$ trials and probability of success $e^{-\alpha t}$). In some sense, for a large number of initial particles $X_{n,t}$ is 'close' to $ne^{-\alpha t}.$ Is there an appropriate way to define this 'closeness' or is scaling by $n$ the only reasonable way to go?

A:

The idea of convergence in distribution of random functions is standard.  The classical example is Donsker's theorem which  probably does not contain the exact solution to your problem but does give a taste of the kind of thing involved.  There are two interlocking key steps.  You have to pick a function space for your random elements live in, and you have to check a condition called "tightness", which involves (loosely speaking) a probabilistic bound on how wiggly the random functions can be. The details are technical, and vary from problem to problem. A classical (and readable) book on the subject is Billingsley's Convergence of Probability Measures.
That being said, your notions about scaling are naive.  You seem to be  looking for a result of form "$a_n$ converges to $b_n$" when you really want a result of form "$a_n/f(n)$ has such-and-such a limit", or "$(a_n-b_n)/g(n)$ has such-and-such a limit".

