Closed-class stationary laws and initial-law-dependent Cesàro weights

The finite graph's strongly connected components form a partition, and its component graph has no directed cycle: a cycle would make its components mutually reachable and hence one component. Following outgoing edges from any component must therefore terminate at a sink, since the component graph is finite and acyclic. A sink is exactly a closed component by [F2]. From any state outside the closed classes, follow a component path to a sink. Each component edge is realized by a positive transition, and strong connectivity supplies finite positive paths from the current state to the source of each such transition and from its endpoint to the next transition's source. Concatenating these paths gives a positive-transition path into a closed class. Before its first entry, this path stays in U because a closed class has no outgoing edge. If U is nonempty, [F3] applies to T. Writing k=aL+r as in [F3] gives $\mathrm{\parallel }{T}^k{\mathrm{\parallel }}_{\mathrm{\infty }}\to 0$, so the Neumann series $N={\sum }_{k\ge 0}{T}^k$ converges and is $(I-T{)}^{-1}$.

For each closed class $C_j$, the restriction $K_{C_j,C_j}$ is a finite irreducible row-stochastic matrix. By [F5] it has a unique invariant probability ${\pi }_j$. Extending ${\pi }_j$ by zero outside $C_j$ preserves invariance because no transition leaves the class. Thus every convex combination in the statement is invariant.

Let $v$ be any invariant probability row vector and write its coordinates as $(v_C,v_U)$. Since closed classes have no transitions to U, the stationary equation on the transient coordinates is $v_U=v_{U}T$. If U is nonempty, [F3] gives $T^L\mathbf{1}\le (1-\epsilon )\mathbf{1}$ for some $\epsilon >0$, and therefore $\mathrm{\parallel }{v}_U{\mathrm{\parallel }}_1=\mathrm{\parallel }{v}_U{T}^L{\mathrm{\parallel }}_1\le {\sum }_{i\in U}(v_U{)}_i{\sum }_{k\in U}(T^L{)}_{ik}\le (1-\epsilon )\mathrm{\parallel }{v}_U{\mathrm{\parallel }}_1$. Hence $v_U=0$. Since no closed class has transitions to another closed class, the stationary equation now restricts separately to each block $C_j$. Thus each $v_{C_j}$ is invariant on its class, so [F5] makes it ${\alpha }_j{\pi }_j$ with ${\alpha }_j\ge 0$; their masses sum to one. This proves the asserted full set of invariant probabilities.

For each transient state i, the probability weight of first entering class $C_j$ after k transient transitions is $(e_i^{\mathsf{T}}{T}^k{b}_j)$. These disjoint first-entry weights sum, by [F4], to $((I-T{)}^{-1}{b}_j{)}_i$. Initial mass already in $C_j$ remains there, so the total class weight from u is ${\alpha }_j=u(C_j)+u_U(I-T{)}^{-1}{b}_j$. Moreover ${\sum }_j{b}_j=(I-T)\mathbf{1}$, since every transition from a transient state either remains in U or enters exactly one of the closed classes; thus ${\sum }_j{\alpha }_j=u(C_1\cup \cdots \cup C_s)+u_U\mathbf{1}=1$. If U is empty, the same formula reduces directly to $u(C_j)$.

Define ${\bar{u}}_N=N^{-1}{\sum }_{k=0}^{N-1}u{K}^k$. Every coordinate lies in [0,1]. From any subsequence, apply [F7] to its first coordinate, restrict to that subsequence, and repeat for the finitely many remaining coordinates; the final subsequence converges coordinatewise. Also ${\bar{u}}_{N}K-{\bar{u}}_N=(u{K}^N-u)\mathrm{/}N$, whose ℓ1 norm is at most $2\mathrm{/}N$. Hence every subsequential limit is an invariant probability.

The mass $a_n=u{K}^n(C_j)$ is nondecreasing because $C_j$ is closed; its increment at each step is the first-entry weight from step 1.4. The partial sums of these weights tend to $u_U(I-T{)}^{-1}{b}_j$ by [F3]–[F4], and the unabsorbed transient mass tends to zero, so $a_n\to {\alpha }_j$. For any η>0, choose N₀ so |uKⁿ(Cⱼ)−αⱼ|<η for n≥N₀; the average error is at most the finite sum of the first N₀ errors divided by N, plus η, and hence tends to at most η. Since η is arbitrary, the Cesàro class mass tends to αⱼ. Thus every subsequential limit $\bar{u}$ has $\bar{u}(C_j)={\alpha }_j$ for each j. By step 1.3 this invariant probability must be ${\sum }_j{\alpha }_j{\pi }_j$.

Every convergent subsequence of $({\bar{u}}_N)$ has the same limit from step 2.1. If the full sequence did not converge to that limit, there would be ε>0 and a subsequence staying at least ε away. Its finitely many bounded coordinates admit a convergent further subsequence by repeated finite applications of [F7]; step 2.1 forces that further limit to be the stated mixture, a contradiction. Thus the full Cesàro sequence converges to the initial-law-dependent mixture stated.