On the summability of Wold Form coefficients

Question

Let $y_{t}$ be a scalar stationary stochastic process with its Wold representation

$y_{t} = \mu_{t} + \sum_{k = 0}^{\infty} \phi_{k}\epsilon_{t-k}$.

I understand that the theorem says that $\sum_{k=0}^{\infty} \phi_{k}^2$ is finite. Although, we if assume that $Var(\epsilon_{t}) = \sigma^2, \forall t$, can we state that $\sum_{k=0}^{\infty} |\phi_{k}|$ is also finite?

I don't think so, but I would like some clarification. For example, if we let $\mu_{t} = 0, \forall t$ and $\phi_{k} = 1/k$ for every $k$, then the above conjecture fails. What do you guys think? Is this a valid counterexample? Thanks in advance!

Answer 1

Yes, that's a valid counterexample.

It might be fun/useful to think about this more broadly. An equivalent question to yours might be, "Are there sequences in $ℓ_2$, the space of square summable sequences, that aren't in $ℓ_1$, the space of absolutely convergent sequences?

Those sums you have are basically the p-norm.

The p-norm and its generalizations:

• In a finite, $n$ dimensional vector space, the the p-norm is: $$ \|\phi \|_p = \left( \sum_{k=1}^n |\phi_k|^p \right) ^\frac{1}{p}$$
• In a countably infinite $ℓ_p$ space, the p-norm is: $$ \|\phi \|_p = \left( \sum_{k=1}^\infty |\phi_k|^p \right) ^\frac{1}{p}$$
• In $L_p$ spaces, the generalization for the p-norm is: $$ \|f\|_p = \left( \int_S |f|^p d \mu\right)^\frac{1}{p}$$

As described in this answer, $f(x) = \frac{1}{x}$ for $x >0$ and $f(0)=0$ is an example of a function that is in $L_2$ but not $L_1$.