Strong invariance principle for geometrically ergodic chains

Put $P_0=2+\delta$, $Q_0=2+\delta +\epsilon$, and $f=g-\pi g$. The given moment makes $\pi \mathrm{\mid }g\mathrm{\mid }<\mathrm{\infty }$, since $\mathrm{\mid }g\mathrm{\mid }\le 1+\mathrm{\mid }g{\mathrm{\mid }}^{Q_0}$, and the finite-sum power inequality gives $\pi \mathrm{\mid }f{\mathrm{\mid }}^{Q_0}<\mathrm{\infty }$. Choose $P_0<p<q<Q_0$ and set $b=1\mathrm{/}p<1\mathrm{/}{P}_0<1\mathrm{/}2$. To obtain exactly the requested observation indices, apply [F1] to the chain starting at $X_1$, with its actual initial law, and observable $f$. This gives a standard-Borel factor retaining the full path $(f(X_1),f(X_2),\dots )$, the stationary path and its moments, and an everywhere finite common geometric bound. Work on this factor until the original-space lift at the end. Apply [F2] there, retaining its finite prefix and integer delay $D$. Define $F(t)$ to be the sum of the first $\lfloor t\rfloor$ centered observations of the retained tail beginning at calendar position $D$, with the empty sum zero. This real-time tail process has cycle lengths $L_j$, centered rewards $Z_j$, envelopes $H_j$, and relative boundaries $T_0=0$, $T_k={\sum }_{j<k}{L}_j$. Then $F(T_k)={\sum }_{j<k}{Z}_j$, $\mathbb{E}{H}_0^q<\mathrm{\infty }$, and $\mu =\mathbb{E}{L}_0\in [1,\mathrm{\infty })$. The pair $(Z_j,L_j-\mu )$ is a centered adjacent-iid vector in $L^q$; exponential length moments and $\mathrm{\mid }{Z}_j\mathrm{\mid }\le H_j$ verify its two coordinate moments. No iid assertion about the full tours is used.

Apply [F3] jointly to $(Z_j,L_j-\mu )$. Write its long-run covariance as $\mathrm{\Gamma }=\left(\begin{array}{cc}u& c\\ c& s^2\end{array}\right)\ge 0$. Since the lengths are iid, $s^2=\mathrm{Var}(L_0)$; the reward and cross entries still include their actual lag-one terms. The Gaussian approximation gives a continuous centered Gaussian vector process $A$ of covariance $\min (t,r)\mathrm{\Gamma }$ such that ${\max }_{k\le n}\mathrm{\mid }F(T_k)-A_1(k)\mathrm{\mid }=o(n^b)$ and ${\max }_{k\le n}\mathrm{\mid }{T}_k-\mu k-A_2(k)\mathrm{\mid }=o(n^b)$. For now suppose $s>0$. Put $a=c\mathrm{/}{s}^2$ and $v^2=u-c^2\mathrm{/}{s}^2\ge 0$. Then $B_0=A_2\mathrm{/}s$ is standard Brownian. If $v>0$, $B=(A_1-a{A}_2)\mathrm{/}v$ is standard Brownian independent of the entire process $B_0$: each finite stack is jointly Gaussian and every cross-covariance is $(c-a{s}^2)\min (t,r)=0$, so [F10] gives independence on finite rational-time cylinders and continuity extends it to all times. If $v=0$, the residual Gaussian process is zero on rational times almost surely and hence identically zero; adjoin an independent unused $B$. Thus, with independent $B,B_0$, we have ${\max }_{k\le n}\mathrm{\mid }F(T_k)-a(T_k-\mu k)-vB(k)\mathrm{\mid }=o(n^b)$ and ${\max }_{k\le n}\mathrm{\mid }{T}_k-\mu k-s{B}_0(k)\mathrm{\mid }=o(n^b)$. This regression uses the full $c={\mathrm{\Gamma }}_{12}$, not just the covariance inside a single cycle.

Define $\gamma =s^2\mathrm{/}\mu$ and $\lambda ={\mu }^2\mathrm{/}{s}^2$, so $\lambda \gamma =\mu$ and $\gamma \sqrt{\lambda }=s$. Choose any fixed $r>p$ and $r<a_1<q_1<\mathrm{\infty }$. Apply [F4] to the centered iid scalar Poisson variables $C_i-\lambda$, embedded as $(C_i-\lambda ,0)$, obtaining a joint law with a standard Brownian driver $B_0^{\mathrm{\prime }}$ and ${\max }_{k\le n}\mathrm{\mid }{\sum }_{i=1}^k{C}_i-\lambda k-\sqrt{\lambda }{B}_0^{\mathrm{\prime }}(k)\mathrm{\mid }=o(n^{1\mathrm{/}r})$. Their moments of every order follow directly from the Poisson exponential series in [F7]. By [F8], disintegrate this joint law to a kernel for the entire count sequence given the integer driver sequence, and evaluate it at the already present $(B_0(k){)}_{k\ge 0}$. This preserves all original variables and reproduces the asserted count/driver joint law. The kernel depends only on $B_0$ and is sampled using new randomness independent of everything earlier, so $B$ remains independent of the joint pair $(B_0,(C_i{)}_i)$. Conditional on $C_i=m$, insert $m$ independent uniform positions in $(i-1,i]$, independently over unit intervals and of earlier variables given the counts. By [F7] these are the exact finite-window Poisson laws. More explicitly, partitioning a unit interval into lengths $d_j$ gives multinomial conditional counts; summing their mass against $\mathrm{Pois}(\lambda )$ yields the product ${\prod }_j{e}^{-\lambda d_j}(\lambda d_j{)}^{k_j}\mathrm{/}{k}_j!$. Hence the resulting locally finite simple right-continuous clock $N$ has independent Poisson increments of rate $\lambda$, $N(k)={\sum }_{i\le k}{C}_i$, and remains independent of $B$ jointly with $B_0$.

Two uniform bounds control the remaining calendar errors. By the common $q$th envelope moment and [F9], $\mathbb{P}({\max }_{j\le C{2}^h}{H}_j>\eta 2^{h\mathrm{/}p})\le C^{\mathrm{\prime }}{\eta }^{-q}{2}^{h(1-q\mathrm{/}p)}$, summable in $h$. Intersecting Borel–Cantelli events over positive rational $\eta$ and integer $C$ yields ${\max }_{j\le Cn}{H}_j=o(n^b)$ without any independence requirement. For any standard Brownian motion and fixed $A,C>0$, cover $[0,C{2}^h]$ by overlapping intervals of length $2A\cdot 2^{hb}$. Every pair of times at distance at most $A\cdot 2^{hb}$ lies in one such interval, and [F10] bounds the oscillation tail at $K\sqrt{2^{hb}h}$ by $4e^{-c{K}^{2}h}$. There are at most $C^{\mathrm{\prime }}{2}^{h(1-b)}+2$ intervals. Taking $K$ large and using [F9] gives the uniform modulus $O(n^{b\mathrm{/}2}\sqrt{\log n})=o(n^b)$ for windows of length $A{n}^b$; for fixed window length the same proof gives $O(\sqrt{\log n})$. All fixed integer $A,C$ can be placed on one event. Finally [F6] applies to the centered one-dependent rewards $Z_j$, so their partial-sum increments over at most $A{n}^b$ consecutive cycle indices, among the first $Cn$ indices, are uniformly $o(n^b)$. A finite random bound for $A$ or $C$ is covered by their countably many integer choices.

We record the uniform estimates used for this clock and for Brownian motion. Summing the Poisson mass series gives $\mathbb{E}{e}^{\theta (N(v)-N(u)-\lambda (v-u))}=\exp (\lambda (v-u)(e^{\theta }-1-\theta ))$. The exponential series and $k!\ge 2\cdot 3^{k-2}$ for $k\ge 2$ give $e^{\theta }-1-\theta \le {\theta }^2\mathrm{/}[2(1-\mathrm{\mid }\theta \mathrm{\mid }\mathrm{/}3)]$ for $\mathrm{\mid }\theta \mathrm{\mid }<3$. Exponential Markov with parameter $x\mathrm{/}(\lambda (v-u)+x\mathrm{/}3)$ and either sign therefore bounds the increment deviation tail by $2\exp [-c_{\lambda }\min \{x^2\mathrm{/}(v-u),x\}]$. On dyadic horizons $n$, take $x=K(\sqrt{(v-u)\log (n+2)}+\log (n+2))$ and union over the at most $(Cn+1{)}^2$ integer intervals in $[0,Cn]$. A sufficiently large deterministic $K$ makes the probabilities summable, so [F9] proves this bound for all such intervals eventually almost surely, for every positive integer $C$ by countable intersection. In particular unit counts are $O(\log n)$ uniformly, intervals of integer length at least $n^b$ have count increase at least $\lambda \mathrm{/}2$ times their length eventually, and $N(t)\mathrm{/}t\to \lambda$. Also [F10], a union bound and [F9] make all within-unit Brownian oscillations up to time $Cn$ be $O(\sqrt{\log n})$. These estimates require no independence between evaluation times and the Brownian motion. Consequently the integer coupling in step 3.1 extends to ${\sup }_{z\le n}\mathrm{\mid }\gamma N(z)-\mu z-s{B}_0(z)\mathrm{\mid }=o(n^b)$, and step 2.1 gives ${\max }_{k\le n}\mathrm{\mid }{T}_k-\gamma N(k)\mathrm{\mid }=o(n^b)$.

Let $m(t)=\max \{k:T_k\le t\}$ and $y(t)=\inf \{z:N(z)>t\mathrm{/}\gamma \}$. Brownian reflection on dyadic horizons gives ${\sup }_{z\le n}\mathrm{\mid }{B}_0(z)\mathrm{\mid }=O(\sqrt{n\log n})=o(n)$; thus step 2.1 implies $T_k\mathrm{/}k\to \mu$. Together with $N(z)\mathrm{/}z\to \lambda$, this gives $m(t),y(t)=O(t)$, absorbing their finite initial values. Put $k=\lfloor y(t)\rfloor$. Monotonicity gives $\gamma N(k)\le t<\gamma N(k+1)$, except for an integer endpoint convention which is covered by shifting at most two indices. The unit-count bound in step 4.1 therefore gives $\mathrm{\mid }\gamma N(k)-t\mathrm{\mid }=O(\log n)$ uniformly for $t\le n$. On every required range of $O(n)$ indices, $\mathrm{\mid }{T}_j-\gamma N(j)\mathrm{\mid }\le n^b$ eventually by its little-o bound. Choose an integer $J$ with $\gamma \lambda J\mathrm{/}2>4$. The lower-increment bound in step 4.1 makes a change of $\lceil J{n}^b\rceil$ indices change $\gamma N$ by more than $2n^b+O(\log n)$ eventually. The corresponding $T$ values bracket $t$, using index zero at the lower boundary. Thus ${\sup }_{t\le n}\mathrm{\mid }m(t)-y(t)\mathrm{\mid }=O(n^b)$. The countable estimates for all deterministic horizon multipliers cover a finite random multiplier as well. This is a direct monotone inverse comparison, not an imported renewal approximation.

Let $E_i$ be the interarrival times of $N$. Their iid exponential law follows from [F7]: for $0<t_1<\cdots <t_k\le T$, sum its ordered density over all total counts $m\ge k$ and integrate the last $m-k$ ordered points above $t_k$. The simplex volume is $(T-t_k{)}^{m-k}\mathrm{/}(m-k)!$, so the first-$k$ density is $e^{-\lambda T}{\lambda }^k{\sum }_{j\ge 0}(\lambda (T-t_k){)}^j\mathrm{/}j!={\lambda }^k{e}^{-\lambda t_k}$. Increasing $T$ covers the whole ordered simplex. The spacing change of variables has determinant one and gives ${\prod }_{i=1}^k\lambda e^{-\lambda e_i}$ on positive spacings. Hence $\mathbb{E}{E}_i=1\mathrm{/}\lambda$, $\mathrm{Var}(E_i)=1\mathrm{/}{\lambda }^2$ and all moments are finite, by elementary exponential integration and integration by parts for integer moments, bounding other positive powers by the next integer power plus one. Apply [F4] to $(E_i-1\mathrm{/}\lambda ,0)$ with the same auxiliary exponent $r>p$, and glue its Brownian driver $V$ to the entire already realized arrival sequence by [F8], using a fresh kernel depending only on that sequence. Then ${\max }_{k\le n}\mathrm{\mid }{\sum }_{i=1}^k{E}_i-k\mathrm{/}\lambda -V(k)\mathrm{/}\lambda \mathrm{\mid }=o(n^{1\mathrm{/}r})$. Since $y(t)={\sum }_{i\le \lfloor t\mathrm{/}\gamma \rfloor +1}{E}_i$, set $V_1(t)=\sqrt{\gamma }V(t\mathrm{/}\gamma )$ and use step 4.1's within-unit Brownian bound to obtain ${\sup }_{t\le n}\mathrm{\mid }y(t)-t\mathrm{/}\mu -V_1(t)\mathrm{/}(\lambda \sqrt{\gamma })\mathrm{\mid }=o(n^b)$. The clock sigma-algebra generated by $B_0,N,V_1$ and their fresh clock randomness remains independent of $B$: each new kernel depended on $B_0$ or on the clock alone and used randomness independent of earlier variables.

Apply [F5] to $B$, $N$ and this entire clock sigma-algebra. It supplies a standard Brownian motion $W^{\ast }$ independent of that sigma-algebra, hence independent of $V_1$, and ${\sup }_{z\le n}\mathrm{\mid }B(z)-{\lambda }^{-1\mathrm{/}2}{W}^{\ast }(N(z))\mathrm{\mid }=O({\log }^2(n+2))$. Define $G(t)=(v\mathrm{/}\sqrt{\lambda })W^{\ast }(t\mathrm{/}\gamma )-(a\mu \mathrm{/}(\lambda \sqrt{\gamma }))V_1(t)$. It is continuous centered Gaussian with independent stationary increments and variance per unit time $v^2\mathrm{/}(\lambda \gamma )+a^2{\mu }^2\mathrm{/}({\lambda }^2\gamma )=(v^2+a^2{s}^2)\mathrm{/}\mu =u\mathrm{/}\mu$. Thus, when $u>0$, $G=\kappa W$ for a standard Brownian $W$ and $\kappa =\sqrt{u\mathrm{/}\mu }$. When $u=0$, positive semidefiniteness forces $a=v=0$, so $G=0$ and adjoin an unused standard $W$. The case $v=0$ needs no residual normalization or nonzero Brownian coefficient.

For $y=y(t)$, the following exact decomposition has eight terms: $F(t)-G(t)={\sum }_{i=1}^8{\mathrm{\Phi }}_i(t)$, where ${\mathrm{\Phi }}_1=F(t)-F(T_{m(t)})$, ${\mathrm{\Phi }}_2=F(T_{m(t)})-F(T_{\lfloor y\rfloor })$, ${\mathrm{\Phi }}_3=F(T_{\lfloor y\rfloor })-a{T}_{\lfloor y\rfloor }+a\mu y-vB(y)$, ${\mathrm{\Phi }}_4=a(T_{\lfloor y\rfloor }-\gamma N(y))$, ${\mathrm{\Phi }}_5=-a\mu (y-t\mathrm{/}\mu -V_1(t)\mathrm{/}(\lambda \sqrt{\gamma }))$, ${\mathrm{\Phi }}_6=v(B(y)-{\lambda }^{-1\mathrm{/}2}{W}^{\ast }(N(y)))$, ${\mathrm{\Phi }}_7=(v\mathrm{/}\sqrt{\lambda })(W^{\ast }(N(y))-W^{\ast }(t\mathrm{/}\gamma ))$, and ${\mathrm{\Phi }}_8=a(\gamma N(y)-t)$. Adding these expressions cancels every intermediate reward, clock and Gaussian evaluation and leaves exactly $F(t)-G(t)$. This verifies the signs and deterministic drift, including the bounded last term.

Uniformly for $t\le n$, the first term is $o(n^b)$ by the incomplete-cycle bound in [F2] and step 3.2. The second is $o(n^b)$ by the inverse comparison in step 5.1 and [F6]. The third is $o(n^b)$ by step 2.1 at $\lfloor y\rfloor$, the bounded fractional-part drift $a\mu (y-\lfloor y\rfloor )$, and the within-unit Brownian bound. The fourth is $o(n^b)$ by the length/Poisson comparison at $\lfloor y\rfloor$ and the $O(\log n)$ count increase between that point and $y$. The fifth is $o(n^b)$ by step 6.1, and the sixth is $O({\log }^{2}n)=o(n^b)$ by step 7.1. The strict inverse of the simple Poisson clock satisfies $N(y(t))=\lfloor t\mathrm{/}\gamma \rfloor +1$, so its distance from $t\mathrm{/}\gamma$ is at most one; step 3.2's fixed-window Brownian bound makes the seventh term $O(\sqrt{\log n})=o(n^b)$, while $\mathrm{\mid }{\mathrm{\Phi }}_8\mathrm{\mid }\le \mathrm{\mid }a\mathrm{\mid }\gamma$. The indices $m(t),y(t)$ are $O(n)$ by step 5.1, so every approximation was evaluated inside its verified range. Therefore ${\sup }_{t\le n}\mathrm{\mid }F(t)-\kappa W(t)\mathrm{\mid }=o(n^b)$ for $s>0$.

If $s^2=0$, the iid lengths have variance zero, hence $L_j=\mu$ for every $j$ simultaneously almost surely. In particular $T_k=\mu k$; positive semidefiniteness of $\mathrm{\Gamma }$ implies $c=0$. The joint approximation [F3] gives $F(T_k)=\sqrt{u}B(k)+o(n^b)$ uniformly for $k\le n$, with a standard Brownian $B$ when $u>0$, and an unused one when $u=0$. For $t\le n$, $m(t)=\lfloor t\mathrm{/}\mu \rfloor$, so step 3.2's fixed-window modulus replaces $B(m(t))$ by $B(t\mathrm{/}\mu )$ at error $o(n^b)$. The incomplete-tour bound is again $o(n^b)$. Thus ${\sup }_{t\le n}\mathrm{\mid }F(t)-\sqrt{u}B(t\mathrm{/}\mu )\mathrm{\mid }=o(n^b)$. The process $W(t)=\sqrt{\mu }B(t\mathrm{/}\mu )$ is standard Brownian by its continuous Gaussian covariance and increments, and the coefficient is the same $\kappa =\sqrt{u\mathrm{/}\mu }$. This proves the tail approximation in every covariance case, including a zero residual or zero full Gaussian covariance, without inverting a singular matrix.

Restore the finite initial segment of [F2]. Let $A$ be its finite centered reward and $D$ its finite integer length. For $n\ge D$, the actual sum is $A+F(n-D)$. Keep the same standard Brownian motion $W$ and use $A+F(n-D)-\kappa W(n)=A+[F(n-D)-\kappa W(n-D)]+\kappa [W(n-D)-W(n)]$. The fixed-window Brownian estimates in step 3.2 hold on one event for every positive integer window length. On that event choose an integer at least the finite random $D$; then ${\max }_{D\le k\le n}\mathrm{\mid }W(k-D)-W(k)\mathrm{\mid }=O(\sqrt{\log n})$, regardless of any dependence of $D$ and $W$. Finitely many earlier indices contribute a finite maximum. Steps 9.1 and 10.1 therefore give the complete-path bound ${\max }_{k\le n}\mathrm{\mid }{\sum }_{i=1}^{k}f(X_i)-\kappa W(k)\mathrm{\mid }=o(n^b)$ in the factor construction. Neither a moment of the initial law nor an assertion that a randomly shifted Brownian path is standard has been used.

Identify $\kappa$ without making an unsupported second-moment inference from the almost-sure approximation. Use the same factor and fixed post-delay tour law under invariant initial law, as supplied in [F1]–[F2]. Steps 2.1, 3.1, 3.2, 4.1, 5.1, 6.1, 7.1, 8.1, 9.1, 10.1, 11.1 then give a CLT with variance ${\kappa }^2=u\mathrm{/}\mu$: the approximation error divided by $\sqrt{n}$ tends to zero almost surely since $b<1\mathrm{/}2$, and $\kappa W(n)\mathrm{/}\sqrt{n}$ has exactly the centered normal law of variance ${\kappa }^2$ for each $n$; apply [F12]'s converging-together result. Independently, [F11] gives exponential stationary strong mixing of the original observations. Their $Q_0$th moment, with $Q_0>2$, satisfies [F12]'s moment regime because every positive power of an exponential rate is summable. Hence [F12] gives the stationary CLT with its absolutely convergent covariance-sum variance ${\sigma }_g^2$, the asymptotic variance in the Statement's convention. The stationary observation-path equality in [F1] makes these two limits refer to the same law. Uniqueness of centered normal characteristic functions, $e^{-t^2{\kappa }^2\mathrm{/}2}=e^{-t^2{\sigma }_g^2\mathrm{/}2}$, gives ${\kappa }^2={\sigma }_g^2$. The assumed positivity therefore gives $\kappa ={\sigma }_g>0$. This argument uses no uniform-integrability assumption on the approximation error.

Finally preserve the actual original chain probability space. The factor construction gives a joint law of its real observation sequence and the rational-time coordinates of the final standard Brownian motion $W$. Both are standard-Borel real sequence spaces by [F8]. Use [F1]'s lifting assertion, or disintegrate this joint law and apply its conditional kernel at the actual sequence $(f(X_1),f(X_2),\dots )$ by [F8]; this preserves the entire original probability space and reproduces that observation/Gaussian-coordinate joint law. The approximation event uses only these observations and integer driver coordinates, so it still has probability one. The rational coordinates extend to a continuous Brownian path on a common measurable full-measure event: for each positive integers $M,k$, some integer $l$ makes all rational pairs in $[0,M]$ at distance below $1\mathrm{/}l$ have coordinate difference below $1\mathrm{/}k$. This is a Borel condition and has probability one under the original Brownian rational law; completeness of $\mathbb{R}$ gives a unique consistent continuous extension on compact intervals. Every fixed real-time evaluation is the limit along a deterministic rational sequence, hence measurable; on the null failure event set the path to zero. Its finite-dimensional laws agree with standard Brownian laws by continuity, so [F10] identifies genuine standard Brownian motion on the extension. With $B=W$, steps 11.1, 12.1 give error $o(n^{1\mathrm{/}p})$, and $p>P_0$ implies $n^{1\mathrm{/}p}\mathrm{/}(n^{1\mathrm{/}{P}_0}\log n)\to 0$. Choosing the finite eventual bound and starting index in this almost-sure comparison gives precisely the $C(\omega ),N(\omega )$ in the Statement. AC was used through the stated local Gaussian, factor, tour and extension suppliers; the full moment range, arbitrary initial law, positive-variance convention and original observation indices are retained.