Stochastic processes with convergent fi-di distributions but no tightness

I am looking for natural and interesting examples of stochastic processes with convergent finite-dimensional distributions but no tightness. As for the space, it is either $$C[0,1]$$ equipped with the uniform norm or $$D[0,1]$$ equipped with the Skorokhod topology $$M_2$$. Here I've chosen the topology $$M_2$$ since it is weaker than the Skorokhod topologies $$J_1$$ and $$M_1$$, so tightness in $$M_2$$ is the least restrictive.

What I am up to are examples of interest by their own and not specifically crafted to answer the question. Say, some type of stochastic processes that was the subject of a reasonable paper, where one of the main results proves only convergence of fi-di distributions, at least in some cases.

UPD: My best example is on partial sums of moving averages of heavy-tailed i.i.d. random variables, where the coefficients have different signs. The main point of this example is that it combines naturally few modes of tightness and non-tightness in various Skorokhod topologies.

Let $$(X_n)_{n \ge 0}$$ be i.i.d. r.v.'s from the domain of normal (for simplicity) attraction of an $$\alpha$$-stable law with $$\alpha \in (0,2)$$. Consider the sequence of moving averages $$Y_n:=a_0 X_n + a_1 X_{n-1}$$ and the processes of their partial sums $$S_n(t):= \sum_{k=1}^{[nt]} Y_k$$, where $$t \in [0,1]$$. Put $$a:=a_0 + a_1$$ and assume that $$a \neq 0$$.

Then $$S_n(t) / n^{1/\alpha}$$ converges weakly to $$a S(t)$$ for every $$t$$, where $$S$$ is the limiting $$\alpha$$-stable process. The same clearly holds for all fi-di distributions. Then:

-If $$a_0 \neq 0, a_1= 0$$, then $$S_n / n^{1/\alpha}$$ converges weakly to $$aS$$ in $$(D, J_1)$$.

-If $$a_0 \neq 0, a_1 \neq 0$$, then the distributions of $$S_n / n^{1/\alpha}$$ are not $$J_1$$-tight.

-However, if $$a_0 a_1 \ge 0$$, then $$S_n / n^{1/\alpha}$$ converge weakly to $$aS$$ in $$(D, M_1)$$ (Avram and Taqqu, 1992).

-If $$a_0 a_1 <0$$, then the distributions of $$S_n / n^{1/\alpha}$$ are not even $$M_2$$-tight.

There are examples of moving averages of higher orders where there is $$M_2$$-convergence (with proofs due to Basrak and Krizmanic, 2014).

• Here's a couple of "naturally occurring" situations when no tightness is known, but alas in a more complicated setting. One is loop ensemble in the double dimer model, see arXiv:1809.00690; the tightness is expected to hold but the proof is missing. Another is the energy field in the critical 2d Ising model archive-ouverte.unige.ch/unige:18163; the scaling limits of correlations blow up so badly on the diagonal that no distribution-valued continuous field with such correlations exists. Commented Jun 9, 2020 at 7:12

How about $$\min\{n\cdot \mathrm{dist}(t,X),1\}$$, where $$X$$ is your favorite point process, say, the Poisson process?
For an example with Markov property, consider a continuous time Markov chain with two states $$0,1$$ and transition intensities $$1$$ and $$n$$.
In a somewhat different flavor, take the process that is given by independent Bernoulli at the points of $$n^{-1}\mathbb{Z}$$ and extended to be constant on each $$\left[\frac{k-\frac{1}{2}}{n};\frac{k+\frac{1}{2}}{n}\right)$$.
• Did you mean something like $\max\{ n - n \cdot dist(t, X), 0\}$? Otherwise your sequence, divided by $n$ (which is a natural thing to do) converges to $dist(t,X)$. Am I wrong? Commented Jun 8, 2020 at 11:36