We study large-sample properties of higher-order Markov chains on a finite alphabet $\Sigma$ when the order $m_n$ is allowed to grow with the sequence length $n$. By embedding the process into a first-order chain on $\Sigma^{m_n}$ and exploiting return-time decompositions, we establish a central limit theorem for additive functionals $\sum_{t}\! g_n(Y_t^{(n)})$ under natural ergodicity and sparsity conditions. The normalization involves the stationary return time to a suitably chosen state and accommodates triangular arrays with $m_n\!\to\!\infty$ and $m_n/n\!\to\!0$. We further illustrate the assumptions in a binary variable length Markov chain (VLMC), deriving explicit lower bounds on stationary masses that yield a concrete growth regime (e.g., $m_n\log m_n/n \to 0$) ensuring the CLT. These results provide asymptotic foundations for inference in sparse/partitioned higher-order models; including VLMCs and sparse Markov models (SMMs) where the effective dimensionality grows with the sample size.
We develop Gaussian approximation bounds in higher-order Wasserstein distance $W_p$, $p\geq2$, for sums of multivariate martingale differences generated by a uniformly ergodic Markov chain. Under an $L^{(2+\eta)p}$-moment condition with $\eta>0$, we establish the explicit bound $$ O\left( p^3 \|A\|_4^2 + pd^{1/4}\|A\|_...
Let $P(t)$ be the transition function of an absorbing continuous-time Markov chain on a countable state space. We study asymptotic relations of the form $p_{ij}(t)\sim a_{ij} L(t)$, $t\to\infty$, where $L$ is independent of $i$ and $j$. We obtain general conditions under which the coefficient matrix $A=(a_{ij})$ has ra...
We establish conditional limit theorems, pointwise in the initial state, for absorbing Markov chains on a compact metric space $M$. We assume $L^1(M,\rho)$-continuous transition densities, irreducibility and aperiodicity. For the observable $f_\beta(x)=d_M(x,x_0)^{-\beta}$, with suitable $x_0$ satisfying $\rho(B_r(x_0)...
Bernat Bassols Cornudella, Matheus M. Castro· 0 citations
We consider the probability that a discrete random matrix $M_n(\xi)$ is \emph{strongly non-singular}, meaning all its leading principal submatrices are non-singular. This property is equivalent to the existence of an LU factorization. We show that for any discrete random variable $\xi$ with finite support and $|\xi|_\i...
S. Mateo, John Urschel, Nicholas West· 2 citations
Let $Y_1,\ldots,Y_n$ be independent symmetric random variables with log-concave tails. We give a dimension-free characterization of the expected supremum of the canonical process $X_x=\sum_{i=1}^n x_iY_i$ without any $\Delta_2$ or regular-growth assumption on the coordinate tails. The characterization is governed by sc...
We study the smallest nonzero eigenvalue of the sample correlation matrix $\mathbf{R}_n$ formed from a $p_n \times n$ data matrix with i.i.d. real entries $\xi$ of mean zero and unit variance, in the high-dimensional regime $p_n / n \to \phi \in (0, \infty) \setminus \{1\}$. In the tall regime $\phi>1$, we prove almost...
Ze-Qin Lin, Guamgming Pan, Hao-Zhu Zhao et al.· 1 citation
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