No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Independent and Stationary Sequences of Random Variables
... Recall that for real α the random variable ξ is said to be α-stable if for all b 1 , b 2 > 0, there exist constants b > 0 and a ∈ R such that b 1 ξ 1 + b 2 ξ 2 d = bξ + a (here and below d = denotes coincidence by distribution), where ξ 1 , ξ 2 are independent copies of ξ. Classical one-dimensional stable distributions are well studied (see, e.g., [3,4,11]). ...
... where a ∈ R, C ≥ 0, 0 < α ≤ 2, and −1 ≤ κ ≤ 1, [1,6]) constructed α-stable distributions for α = 0. In papers [8,12], nonprobabilistic analogs of αstable distributions were constructed for α ∈ (2, 4) ∪ (4,6). Then M. V. Platonova generalized this method to α > 2. Using the constructed α-stable variables, probabilistic representations of solutions of the Cauchy problem for evolution equations with Riemann-Liouville operator were obtained. ...
... It is well known (see for example [4]) that then the distribution P belongs to the region of attraction of a one-dimensional stable random variable with parameters c 1 /β, (1)), i.e., there exists a sequence B n → ∞ such that the sequence of random variables ...
Complex-valued random variables that satisfy usual stability condition for a complex parameter α such that |α − 1/2| < 1/2, are constructed. The characteristic function of the obtained random variables is found and limit theorems for sums of independent identically distributed random variables are proved. The corresponding Lévy processes and semigroups of operators corresponding to these processes are constructed.
... The part of probability theory linked with limit theorems for sums of weakly dependent random variables was covered in the famous monograph [66] authored by Ibragimov and Linnik. Ibragimov also conjectured there (see [66], Chap. ...
... The part of probability theory linked with limit theorems for sums of weakly dependent random variables was covered in the famous monograph [66] authored by Ibragimov and Linnik. Ibragimov also conjectured there (see [66], Chap. XX) that the central limit theorem holds for stationary sequences satisfying the uniformly strong mixing condition, provided that the partial sums have finite variances which tend to infinity with the number of terms. ...
... Furthermore, when we restrict the jump uncertainty set to a singleton, our work complements the classical α-stable central limit theorems in the linear setting. Relevant references for the linear setting include [6,7,9,13,19,20,23] and the references cited therein. ...
... In this case, the symmetric finite measure μ in F μ is supported on {1, −1} with k := μ{1} = μ{−1}. Theorem 2.6.7 from Ibragimov and Linnik [19] indicates that ...
This article fills a gap in the literature by relaxing the integrability condition for the robust \(\alpha \)-stable central limit theorem under sublinear expectation. Specifically, for \(\alpha \in (0,1]\), we prove that the normalized sums of i.i.d. non-integrable random variables \(\big \{n^{-\frac{1}{\alpha }}\sum _{i=1}^{n}Z_{i}\big \}_{n=1}^{\infty }\) converge in law to \({\tilde{\zeta }}_{1}\), where \(({\tilde{\zeta }}_{t})_{t\in [0,1]}\) is a multidimensional nonlinear symmetric \(\alpha \)-stable process with jump uncertainty set \({\mathcal {L}}\). The limiting \(\alpha \)-stable process is further characterized by a fully nonlinear partial integro-differential equation (PIDE):
where
The approach used in this study involves the utilization of several tools, including a weak convergence approach to obtain the limiting process, a Lévy–Khintchine representation of the nonlinear \(\alpha \)-stable process and a truncation technique to estimate the corresponding \(\alpha \)-stable Lévy measures. In addition, the article presents a probabilistic method for proving the existence of a solution to the above fully nonlinear PIDE.
... Switching to the well established notation of [28] ξ(1) is S α (σ, −1, η) distributed for some σ > 0 and η ∈ R meaning that ( [28], Prop. 1.2.12) [20] is analytically exceptionally complicated and valid for the α-stable case only while our proof is just by insertion in (2.8). ...
... [28], equation 1.2.11, which seems to be because their formula comes from Zolotarev [31] which uses another parametrization of 1-stable distributions. Again, the treatment of [20] is analytically exceptionally complicated and valid for the 1-stable case only while our proof is just by insertion in (2.8). By Albin [1], Theorem 2, (1.5) holds with H = 1 for η = 0. ...
We study tail probabilities of superexponential innite divisible distributions as well as tail probabilities of suprema of Levy processes with superexponential marginal distributions over compact intervals. As an area of possible application we mention assessment of nancial risk.
... Moreover, similar bounds may be established for the beta-mixing coefficient on the basis of the recurrence properties; this is known to be quite useful in various limit theorems (cf. Ref. [16]) as well as in the extreme value theory [17]. However, the pursuit of this goal is not within the scope of this paper; certain applications will be studied in a separate publication. ...
... whereV(y) = V(y) + (d − 1) ln y 2 , y > 0 (see (16)). By the first theorem of the calculus (also known as the Newton-Leibniz theorem) we have, ...
In this paper, polynomial recurrence bounds for a class of stochastic differential equations with a rotational symmetric gradient type drift and an additive Wiener process are established, as well as certain a priori moment inequalities for solutions. The key feature of this paper is that the approach does not use Lyapunov functions because it is not clear how to construct them. The method based on Dynkin’s (nonrandom) chain of equations is applied instead. Another key feature is that the asymptotic conditions on the potential near infinity are assumed as inequalities—which allows for more flexibility compared to a single limit at infinity, making it less restrictive.
... Here B(Λ) stands for the Borel σ-algebra of the sets of Λ. For definition and properties of orthogonal stochastic measures and stochastic integral in (2.4) we refer, e.g., Cramér and Leadbetter [14], Ibragimov and Linnik [38], and Shiryaev [66]. ...
... We will consider here stationary processes possessing spectral density functions. For the following results we refer to Ibragimov and Linnik [38], Sect. 16 ...
... Makhno in 21 his articles [13,14] and in the monograph [15]; more precisely, in [13][14][15] the drift is of the 22 usual form, but the assumptions are stated for the integrated drift, which does correspond 23 to our setting after an easy reformulation. Here we extend, and relax, and also correct some 24 of the assumptions from [22], with the main aim to replace the assumptions of the limit 25 type (see (7) in what follows) to the asymptotic inequalities (see (9) in what follows). 26 It is known that the rate of convergence to the invariant distribution as well as the 27 rates for certain mixing coefficients may be derived from the estimates of the type ...
... 33 Moreover, similar bounds may be established for the beta-mixing coefficient on the basis of 34 the recurrence properties; this is known to be quite useful in various limit theorems (cf. [7]) 35 as well as in the extreme value theory, cf. [11]. ...
Polynomial recurrence bounds for a class of stochastic differential equations with a gradient type drift and an additive Wiener process are studied without Lyapunov functions.
... By Assumption 4.4, we have ∥(X i − z)K h (X i − z)∥ ≤ C 3 , so that |ξ i | ≤ Ch −w/2 . Then, by Theorem 17.2.1 of Ibragimov et al. [1971], we have that |Cov(ξ 1 , ξ i )| ≤ Ch −w α(i − 1). ...
Conformal prediction has been a popular distribution-free framework for uncertainty quantification. In this paper, we present a novel conformal prediction method for time-series, which we call Kernel-based Optimally Weighted Conformal Prediction Intervals (KOWCPI). Specifically, KOWCPI adapts the classic Reweighted Nadaraya-Watson (RNW) estimator for quantile regression on dependent data and learns optimal data-adaptive weights. Theoretically, we tackle the challenge of establishing a conditional coverage guarantee for non-exchangeable data under strong mixing conditions on the non-conformity scores. We demonstrate the superior performance of KOWCPI on real time-series against state-of-the-art methods, where KOWCPI achieves narrower confidence intervals without losing coverage.
... For light-tailed Z, we recall from Broutin, Devroye, Lugosi, Oliveira [6] that r n → 1/EZ. For moderate and heavy tails, however, we have r n → 0. When α ∈ (0, 1), then Z is in the domain of attraction of the extremal stable law with parameter α, which itself has a tail distribution function that decays at the rate 1/n α (see, e.g., Ibragimov and Linnik [11], Zolotarev [15] or Malevich [13]). One can then deduce that r n decays at the rate 1/n 1−α . ...
Recommendation systems are pivotal in aiding users amid vast online content. Broutin, Devroye, Lugosi, and Oliveira proposed Subtractive Random Forests (\textsc{surf}), a model that emphasizes temporal user preferences. Expanding on \textsc{surf}, we introduce a model for a multi-choice recommendation system, enabling users to select from two independent suggestions based on past interactions. We evaluate its effectiveness and robustness across diverse scenarios, incorporating heavy-tailed distributions for time delays. By analyzing user topic evolution, we assess the system's consistency. Our study offers insights into the performance and potential enhancements of multi-choice recommendation systems in practical settings.
... Extensions to stationary sequences were given by [41,42] For a derivation of the Edgeworth expansion for a sample mean from the Gram-Charlier expansion, see [5,43] for the univariate and vector cases. These showed for the first time that the coefficients in these expansions were Bell polynomials in the cumulants. ...
The only cases where exact distributions of estimates are known is for samples from exponential families, and then only for special functions of the parameters. So statistical inference was traditionally based on the asymptotic normality of estimates. To improve on this we need the Edgeworth expansion for the distribution of the standardised estimate. This is an expansion in n−1/2 about the normal distribution, where n is typically the sample size. The first few terms of this expansion were originally given for the special case of a sample mean. In earlier work we derived it for any standard estimate, hugely expanding its application. We define an estimate w^ of an unknown vector w in Rp, as a standard estimate, if Ew^→w as n→∞, and for r≥1 the rth-order cumulants of w^ have magnitude n1−r and can be expanded in n−1. Here we present a significant extension. We give the expansion of the distribution of any smooth function of w^, say t(w^) in Rq, giving its distribution to n−5/2. We do this by showing that t(w^), is a standard estimate of t(w). This provides far more accurate approximations for the distribution of t(w^) than its asymptotic normality.
... n /1 ƒ.n;Â / D E Recall that we are assuming (15). According to Theorem 2.6.5 of Ibragimov and Linnik [21], ' is regularly varying at 0 with exponent˛, and since˛< 2, the Potter bounds (see [12,Theorem 1.5.6]) show that for some constant C , nj'.Âa 1 n . ...
... where Y α has the asymmetric α-stable law, which we recall has a density d α . In fact, if the random variable X is integer-valued (and not supported on a non-trivial arithmetic progression), a more precise version of the above convergence known as local limit theorem, is true, see [20,Theorem 4 ...
... As the random variables E j are bounded, we deduce from covariance inequality in [52] that ...
The problem of estimating the spatial-functional expectile regression for a given spatial mixing structure $\left(X_{\mathbf{i}}, Y_{\mathbf{i}}\right) \in \mathcal{F} \times \mathbb{R}$, when $\mathbf{i} \in \mathbb{Z}^N, N \geq 1$ and $\mathcal{F}$ is a metric space, is investigated. We have proposed the $M$-estimation procedure to construct the Spatial Local Linear (SLL) estimator of the expectile regression function.
The main contribution of this study is the establishment of the asymptotic properties of the SLL expectile regression estimator. Precisely, we establish the almost complete convergence with rate.
This result is proved under some mild conditions on the model in the mixing framework. The implementation of the SLL estimator is evaluated using an empirical investigation.
A COVID-19 data application is performed allowing to highlight the substantial superiority of the SLL-expectile over SLL-quantile in risk exploration
... (3)] for a more general statement 3 . Then the local limit theorem for densities [IL71,Thm. 4.3.1] gives that if f n is bounded for some n ≥ 1 (which is given by Assumption 1), then f n (0) ∼ 1 an f α (0). ...
We study the wetting model, which considers a random walk constrained to remain above a hard wall but with additional pinning potential for each contact with the wall. This model is known to exhibit a wetting phase transition, from a localized phase (with trajectories pinned to the wall) to a delocalized phase (with unpinned trajectories). First of all, we take the opportunity to present an overview of the model, collecting and complementing well-known and other folkore results. Then, we investigate a version with elevated boundary conditions, which can alternatively be seen as a wetting model in a square well. This has been studied from a mathematical point of view by Lacoin and Texeira (Electron. J. Probab., vol 20, 2015), who describe both some of its equilibrium and dynamical properties, when the model is based on the simple symmetric random walk. In the present paper, we complement these results, focusing on the equilibrium properties of the model, for a general underlying random walk (in the domain of attraction of a stable law). First, we compute the free energy and give some properties of the phase diagram; interestingly, we find that, in addition to the wetting transition, a so-called saturation phase transition may occur. Then, in the so-called Cram\'er's region, we find an exact asymptotic equivalent of the partition function, together with a (local) central limit theorem for the fluctuations of the left-most and right-most pinned points, jointly with the number of contacts at the bottom of the well.
... There exists a vast literature providing different types of LCLT results (or local stable limit theorems) in the stable setting with explicit and implicit convergence rates, e.g. [5,6,7,10,20,25,32,35,37]. ...
... The following auxiliary result is a variation of Theorem 17.2.2 from the renowned book by Ibragimov and Linnik [9]. In the interest of self-contained explanation and for future reference, we have chosen to present this result here in the required form. ...
We study the mixing properties of an important optimization algorithm of machine learning: the stochastic gradient Langevin dynamics (SGLD) with a fixed step size. The data stream is not assumed to be independent hence the SGLD is not a Markov chain, merely a Markov chain in a random environment , which complicates the mathematical treatment considerably. We derive a strong law of large numbers and a functional central limit theorem for SGLD.
... Then one can show that if we change the time by , the continuous coalescent becomes the one introduced in [6]. Besides, if we assume condition (18), then the law of the coalescent is fully characterized by the density 0, by Proposition 6.2, which is nothing else but the time-change of the density (2) in [6]. ...
We investigate scaling limits of trees built by uniform attachment with freezing, which is a variant of the classical model of random recursive trees introduced in a companion paper. Here vertices are allowed to freeze, and arriving vertices cannot be attached to already frozen ones. We identify a phase transition when the number of non-frozen vertices roughly evolves as the total number of vertices to a given power. In particular, we observe a critical regime where the scaling limit is a random compact real tree, closely related to a time non-homogenous Kingman coalescent process identified by Aldous. Interestingly, in this critical regime, a condensation phenomenon can occur.
... It is known [10] that in this case n −2 (x 2 1 + . . . + x 2 n ) converges in distribution to the appropriate 1/2-stable law. ...
A notion of the limit spectral measure of a metric triple (i.e., a metric measure space) is defined. If the metric is square integrable, then the limit spectral measure is deterministic and coinsides with the spectrum of the integral operator in $L^2(\mu)$ with kernel $\rho$. We construct an example in which there is no deterministic spectral measure.
... where the innovations ε i are independent and identical distributed (i.i.d.) random variables belonging to the domain of attraction of an α-stable law with α ∈ (1, 2], E ε 1 = 0 and E ε 2 1 = ∞, a i = i −1 ℓ(i) with ℓ being a slowly varying function at infinity. For the innovation ε 1 , by Theorems 2.6.1 and 2.6.2 in [4], there exist nonnegative constants σ 1 and σ 2 such that P(ε 1 ≤ −x) = (σ 1 + o(1))x −α h(x) and P(ε 1 > x) = (σ 2 + o(1))x −α h(x) (1.2) as x tends to infinity, where σ 1 , σ 2 ≥ 0, σ 1 + σ 2 > 0 and h(x) is a positive slowly varying function at infinity. For α ∈ (1,2], it is well known that (1.2) is equivalent to ln Ee ιuε 1 = −(σ + o(1))|u| α H(|u| −1 )(1 − ιDsgn (u)) (1. 3) as u tends to 0, where ι = √ −1, σ = (σ 1 + σ 2 )Γ(|α − 1|) cos( πα 2 ), D = σ 1 −σ 2 σ 1 +σ 2 tan( πα 2 ) and (see, e.g., [1]). ...
Let $X=\{X_n: n\in\mathbb{N}\}$ be a linear process in which the coefficients are of the form $a_i=i^{-1}\ell(i)$ with $\ell$ being a slowly varying function at the infinity and the innovations are independent and identically distributed random variables belonging to the domain of attraction of an $\alpha$-stable law with $\alpha\in (1, 2]$. We will establish the asymptotic behavior of the partial sum process \[ \bigg\{\sum\limits_{n=1}^{[Nt]} X_n: t\geq 0\bigg\} \] as $N$ tends to infinity, where $[t]$ is the integer part of the non-negative number $t$.
Motivated by metastability in the zero-range process, we consider i.i.d.\ random variables with values in $\N_0$ and Weibull-like (stretched exponential) law $\mathbb P(X_i =k) = c \exp( - k^\alpha)$, $\alpha \in (0,1)$. We condition on large values of the sum $S_n= \mu n + s n^\gamma$ and prove large deviation principles for the rescaled maximum $M_n /n^\gamma$ and for the reversed order statistics. The scale is $n^\gamma$ with $\gamma = 1/(2-\alpha)$; on that scale, the big-jump principle for heavy-tailed variables and a naive normal approximation for moderate deviations yield bounds of the same order $n^{\gamma \alpha} = n^{2\gamma-1}$, the speed of the large deviation principles. The rate function for $M_n/n^\gamma$ is non-convex and solves a recursive equation similar to a Bellman equation.
We consider local level and local linear estimators for estimation and inference in time-varying parameter (TVP) regressions with general stationary covariates. The latter estimator also yields estimates for parameter derivatives that are utilized for the development of time invariance tests for the regression coefficients. Our theoretical framework is general enough to allow for a wide range of stationary regressors, including stationary long memory. We demonstrate that neglecting time variation in the regression parameters has a range of adverse effects in inference, in particular, when regressors exhibit long-range dependence. For instance, parametric tests diverge under the null hypothesis when the memory order is strictly positive. The finite sample performance of the methods developed is investigated with the aid of a simulation experiment. The proposed methods are employed for exploring the predictability of SP500 returns by realized variance. We find evidence of time variability in the intercept as well as episodic predictability when realized variance is utilized as a predictor in TVP specifications.
For a Tychonoff space X, we consider the space Pτ (X) of probability τ-smooth measures on X, and exp X of all closed nonempty subsets of X. In the paper we establish that for every positive integer n and m-bounded space X the spaces Pτ, n(X) and expn X are also
m-bounded spaces.
We consider the partial-sum process \({\sum }_{k=1}^{\left[nt\right]}{X}_{k}^{\left(n\right)},\) where \(\left\{{X}_{k}^{\left(n\right)}={\sum }_{j=0}^{\infty }{\alpha }_{j}^{\left(n\right)}{\xi }_{k-j}\left(b\left(n\right)\right), k\in {\mathbb{Z}}\right\},\) n ≥ 1, is a series of linear processes with tapered filter \({\alpha }_{j}^{\left(n\right)}={\alpha }_{j} {1}_{\left\{0\le j\le\lambda\left(n\right)\right\}}\) and heavy-tailed tapered innovations ξj(b(n)), j ∈ Z. Both tapering parameters b(n) and ⋋ (n) grow to ∞ as n→∞. The limit behavior of the partial-sum process (in the sense of convergence of finite-dimensional distributions) depends on the growth of these two tapering parameters and dependence properties of a linear process with nontapered filter ai, i ≥ 0, and nontapered innovations. We consider the cases where b(n) grows relatively slowly (soft tapering) and rapidly (hard tapering) and all three cases of growth of ⋋(n) (strong, weak, and moderate tapering).
This paper proposes a spatial k-nearest neighbor method for nonparametric prediction of real-valued spatial data and supervised classification for categorical spatial data. The proposed method is based on a double nearest neighbor rule which combines two kernels to control the distances between observations and locations. It uses a random bandwidth in order to more appropriately fit the distributions of the covariates. The almost complete convergence with rate of the proposed predictor is established and the almost sure convergence of the supervised classification rule was deduced. Finite sample properties are given for two applications of the k-nearest neighbor prediction and classification rule to the soil and the fisheries datasets.
We describe a construction of random meromorphic functions with prescribed simple poles with unit residues at a given stationary point process. We characterize those stationary processes with finite second moment for which, after subtracting the mean, the random function becomes stationary. These random meromorphic functions can be viewed as random analogues of the Weierstrass zeta function from the theory of elliptic functions, or equivalently as electric fields generated by an infinite random distribution of point charges.
We consider the hard-edge scaling of the Mittag-Leffler ensemble confined to a fixed disk inside the droplet. Our primary emphasis is on fluctuations of rotationally-invariant additive statistics that depend on the radius and thus give rise to radius-dependent stochastic processes. For the statistics originating from bounded measurable functions, we establish a central limit theorem in the appropriate functional space. By assuming further regularity, we are able to extend the result to a vector functional limit theorem that additionally includes the first hitting "time" of the radius-dependent statistic. The proof of the first theorem involves an approximation by exponential random variables alongside a coupling technique. The proof of the second result rests heavily on Skorohod's almost sure representation theorem and builds upon a result of Galen Shorack (1973).
Motivated by applications to COVID dynamics, we describe a model of a branching process in a random environment $\{Z_n\}$ whose characteristics change when crossing upper and lower thresholds. This introduces a cyclical path behavior involving periods of increase and decrease leading to supercritical and subcritical regimes. Even though the process is not Markov, we identify subsequences at random time points $\{(\tau_j, \nu_j)\}$ —specifically the values of the process at crossing times, viz. $\{(Z_{\tau_j}, Z_{\nu_j})\}$ —along which the process retains the Markov structure. Under mild moment and regularity conditions, we establish that the subsequences possess a regenerative structure and prove that the limiting normal distributions of the growth rates of the process in supercritical and subcritical regimes decouple. For this reason, we establish limit theorems concerning the length of supercritical and subcritical regimes and the proportion of time the process spends in these regimes. As a byproduct of our analysis, we explicitly identify the limiting variances in terms of the functionals of the offspring distribution, threshold distribution, and environmental sequences.
We establish a conditional limit theorem for occupation times of infinite ergodic transformations under a tied-down condition, that is, the condition that the orbit returns to a reference set with finite measure at the final observation time. The class of limit distributions is the generalization of the uniform distribution which was discovered by M. Barlow, J. Pitman and M. Yor in [S\'eminaire de Probabilit\'es XXIII. Lecture Notes in Mathematics, volume 1372 (1989), 294--314]. For the proof we utilize operator renewal theory. Our result can be applied to intermittent maps with two or more indifferent fixed points.
We explore the class of probability distributions on the real line whose Laplace transform admits a strong upper bound of subgaussian type. Using Hadamard's factorization theorem, we extend the class $\mathfrak L$ of Newman and propose new sufficient conditions for this property in terms of location of zeros of the associated characteristic functions in the complex plane. The second part of this note deals with Laplace transforms of strictly subgaussian distributions with periodic components. This subclass contains interesting examples, for which the central limit theorem with respect to the R\'enyi entropy divergence of infinite order holds.
The local limit theorem may fail for additive functionals whose range can be reduced by subtracting a center-tight functional. In this chapter we study the structure of such functionals, and calculate the smallest possible algebraic range which can be obtained this way.
We prove quenched local limit theorems for Markov chains in random environments, with stationary ergodic noise processes.
We analyze the variance of \(S_N=f_1(X_1,X_2)+\cdots +f_N(X_N,X_{N+1})\), and characterize the additive functionals for which \(\mathrm {Var}(S_N)\not \to \infty \). Then we prove Dobrushin’s theorem: If \(\mathrm {Var}(S_N)\to \infty \), then \(S_N\) satisfies the central limit theorem.
This chapter presents the main objects of our study. We define Markov arrays and additive functionals, discuss the uniform ellipticity condition, and introduce the structure constants.
We prove the local limit theorem in the regimes of moderate deviations and large deviations. In these cases the asymptotic behavior of \({\mathbb {P}}(S_N-z_N\in (a,b))\) is determined by the Legendre transforms of the log-moment generating functions.
We find the asymptotic behavior of \({\mathbb {P}}(S_N-z_N\in (a,b))\), assuming that \({(z_N-\mathbb E(S_N))}/{\sqrt {{\mathrm {Var}}(S_N)}}\) converges to a finite limit, and subject to the irreducibility condition: The algebraic range cannot be reduced by a center-tight modification.
We prove the local limit theorem for \({\mathbb {P}}(S_{N}-z_{N}\in (a,b))\) when \(\frac {z_{N}-\mathbb E(S_{N})}{\sqrt {\text{Var}(S_{N})}}\) converges to a finite limit and \(\mathsf {f}\) is reducible. In the reducible case, the asymptotic behavior of \({\mathbb {P}}(S_{N}-z_{N}\in (a,b))\) is not universal, and it depends on \(f_{n}(X_{n},X_{n+1})\). The dependence is strong for small intervals, and weak for large intervals.
In this chapter we consider several special cases where our general results take stronger form. These include sums of independent random variables, homogeneous or asymptotically homogeneous additive functionals, and equicontinuous additive functionals.
We establish sharp concentration inequalities for sums of dependent random matrices. Our results concern two models. First, a model where summands are generated by a $\psi$-mixing Markov chain. Second, a model where summands are expressed as deterministic matrices multiplied by scalar random variables. In both models, the leading-order term is provided by free probability theory. This leading-order term is often asymptotically sharp and, in particular, does not suffer from the logarithmic dimensional dependence which is present in previous results such as the matrix Khintchine inequality. A key challenge in the proof is that techniques based on classical cumulants, which can be used in a setting with independent summands, fail to produce efficient estimates in the Markovian model. Our approach is instead based on Boolean cumulants and a change-of-measure argument. We discuss applications concerning community detection in Markov chains, random matrices with heavy-tailed entries, and the analysis of random graphs with dependent edges.