When \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{C}\), one may use complex variables
techniques in order to study realizability. Take for simplicity the case
\(m=p=1\) (otherwise one argues with each entry). If we can write
\(W_{\mathcal{A}}(s)=P(s) / q(s)\) with \(\operatorname{deg} P<\)
\(\operatorname{deg} q\), pick any positive real number \(\lambda\) that is
greater than the magnitudes of all zeros of \(q\). Then, \(W_{\mathcal{A}}\) must
be the Laurent expansion of the rational function \(P / q\) on the annulus
\(|s|>\lambda\). (This can be proved as follows: The formal equality \(q
W_{\mathcal{A}}=P\) implies that the Taylor series of \(P / q\) about \(s=\infty\)
equals \(W_{\mathcal{A}}\), and the coefficients of this Taylor series are those
of the Laurent series on \(|s|>\lambda\). Equivalently, one could substitute
\(z:=1 / s\) and let
$$
\widetilde{q}(z):=z^{d} q(1 / z), \widetilde{P}(z):=z^{d} P(1 / z),
$$
with \(d:=\operatorname{deg} q(s)\); there results the equality
\(\widetilde{q}(z) W(1 / z)=\widetilde{P}(z)\) of power series, with
\(\widetilde{q}(0) \neq 0\), and this implies that \(W(1 / z)\) is the Taylor
series of \(\widetilde{P} / \widetilde{q}\) about 0 . Observe that on any other
annulus \(\lambda_{1}<|s|<\lambda_{2}\) where \(q\) has no roots the Laurent
expansion will in general have terms in \(s^{k}\), with \(k>0\), and will
therefore be different from \(W_{\mathcal{A} .)}\) Thus, if there is any
function \(g\) which is analytic on \(|s|>\mu\) for some \(\mu\) and is so that
\(W_{\mathcal{A}}\) is its Laurent expansion about \(s=\infty\), realizability of
\(\mathcal{A}\) implies that \(g\) must be rational, since the Taylor expansion at
infinity uniquely determines the function. Arguing in this manner it is easy
to construct examples of nonrealizable Markov sequences. For instance,
$$
\mathcal{A}=1, \frac{1}{2}, \frac{1}{3 !}, \frac{1}{4 !}, \ldots
$$
is unrealizable, since \(\mathcal{W}_{\mathcal{A}}=e^{1 / s}-1\) on \(s \neq 0\).
As another example, the sequence
$$
\mathcal{A}=1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots
$$
cannot be realized because \(\mathcal{W}_{\mathcal{A}}=-\ln
\left(1-s^{-1}\right)\) on \(|s|>1\).
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