Consider the following random walk $(y\_t)\_{t \in \mathbb Z\_+}$:


$$y\_t = y\_{t-1} + u\_t,\quad (u\_t)\_{t \in \mathbb Z\_+} \overset{iid}{\sim} N(0,1), \quad (t \in \mathbb Z\_+)$$


where $y\_0, u\_1, u\_2,...$ are independent.


We know that the process is not stationary and non-ergodic. On the other hand, if $|a|< 1$, we have:  

$$y\_t = a y\_{t-1} + u\_t,\quad (u\_t)\_{t \in \mathbb Z\_+} \overset{iid}{\sim} N(0,1), \quad (t \in \mathbb Z\_+)$$
is stationary and ergodic. Consider then the following sequence of stochastic processes: let $(a\_n)\_{n \in \mathbb N}$ be a sequence of real numbers such that, $|a\_n|< 1$ for all $n \in \mathbb N$ and $a\_n \to 1$, as $n \to \infty$. So, for every $n \in \mathbb N$, define the process $(y\_t^n)\_{t \in \mathbb Z\_+}$ as:
$$y^n\_t = a\_n y^n\_{t-1} + u^n\_t,\quad (u^n\_t)\_{t \in \mathbb Z\_+} \overset{iid}{\sim} N(0,1), \quad (t \in \mathbb Z\_+)$$


So, I would like to know which metric we can use to determine whether the sequence of stochastic processes $(y^n\_t)\_{t \in \mathbb{Z}\_+}$ converges to the random walk $(y\_{t})\_{t \in \mathbb{Z}\_+}$ defined above.



A question about convergence of stochastic processes converging to a random walk
Assume, naturally, that for each $n$ we have $y\_0^n\to y\_0$ (as $n\to\infty$) in distribution and $y\_0^n$ is independent of $(u^n\_t)$.


Then for each $T=0,1,\dots$ we have $Y^n\_T\to Y\_T$ in distribution, where $Y^n\_T:=(y^n\_0,\dots,y^n\_T)$ and $Y^n\_T:=(y\_0,\dots,y\_T)$. This follows because (say) for all $t=0,1,\dots$
$$y^n\_t=a\_n^t y^n\_0+\sum\_{k=1}^t a\_n^{t-1-k}u\_k$$
(which latter can be proved by induction on $t$), so that for all $T=0,1,\dots$
$$Y^n\_T=(y^n\_0,\dots,y^n\_T)\to\Big(y\_0,y\_0+u\_1,\dots,y\_0+\sum\_{k=1}^T u\_k\Big)
=(y\_0,\dots,y\_T)=Y\_T$$
in distribution.


The latter convergence is equivalent to the convergence of $d(P\_{Y^n},P\_Y)$ to $0$, where
$$d(P\_{Y^n},P\_Y):=\sum\_{T=0}^\infty c\_T\,
\frac{d\_{LP}(P\_{Y^n\_T},P\_{Y\_T})}{1+d\_{LP}(P\_{Y^n\_T},P\_{Y\_T})},$$
where (i) $(c\_T)\_{T=0}^\infty$ is any summable sequence of positive numbers and
(ii) $P\_{Y^n\_T}$ and $P\_{Y\_T}$ are, respectively, the distributions of $Y^n\_T$ and $Y\_T$ in $\mathbb R^{T+1}$ and $d\_{LP}$ is the [Lévy–Prokhorov distance](https://en.wikipedia.org/wiki/L%C3%A9vy%E2%80%93Prokhorov_metric#Properties) between such distributions.


Any metrics on the sets of all probability distributions over $\mathbb R^T$ can be used here in place of $d\_{LP}$. If $y\_0$ and the $y\_0^n$'s are Gaussian, an especially convenient metric seems to be the Wasserstein $W\_2$ metric, for which there is a rather simple [explicit expression](https://projecteuclid.org/journals/michigan-mathematical-journal/volume-31/issue-2/A-class-of-Wasserstein-metrics-for-probability-distributions/10.1307/mmj/1029003026.full) in the Gaussian case.


