"And the time ... in physics class, when we were doing these basic (very basic) labs on probability, and I had a little handheld pachinko machine? With a bunch of balls, and evenly spaced rods, and stalls at the bottom? And you tilt it down, and all the balls roll to the top, and you tilt it back, and they come cascading down, and hit the rods, and either bounce left or right, and in the end you’ve got this lovely little bell curve of balls at the bottom, because law of averages and such most balls bounce left, then right, then left, or some combination thereof, and end up in the middle? And only a few go left-left-left-left, or right-right-right-right, and end up on either end? —Anyway, it’s my turn, so I tilt it down, then back again, and click-clack-click-clack-click, and wouldn’t you know it, I’ve got an almost perfect reverse bell curve. Towering stacks of balls to the left and right, and almost nothing at all in the middle.
"So I go to the teacher running the show and hold it out to him and say, okay, now what, smart guy? (“If it fails to agree, under novel experiments or with refined measuring techniques, it is not said that one should not be happy.”)
"And the teacher looks at the little handheld pachinko machine, cocks an eyebrow, tilts it down, tilts it back, clack-click-clack-click-clack. Perfect bell curve.
"“There,” he says. “Fixed it for you.”
"—And I can’t for the life of me tell you which of those gestures is the argument with the universe, and which the sermon on the way things ought to be, dammit. —And that might just be my problem."

"In this note, we prove a conditionally centered version of the quenched weak invariance principle under the Hannan condition, for stationary processes. In the course, we obtain a (new) construction of the fact that any stationary process may be seen as a functional of a Markov chain."

"Let $(X_t, Y_t)_{tin T}$ be a discrete or continuous-time Markov process with state space $X times R^d$ where $X$ is an arbitrary measurable set. Its transition semigroup is assumed to be additive with respect to the second component, i.e. $(X_t, Y_t)_{tin T}$ is assumed to be a Markov additive process. In particular, this implies that the first component $(X_t)_{tin T}$ is also a Markov process. Markov random walks or additive functionals of a Markov process are special instances of Markov additive processes. In this paper, the process $(Y_t)_{tin T}$ is shown to satisfy the following classical limit theorems: (a) the central limit theorem, (b) the local limit theorem, (c) the one-dimensional Berry-Esseen theorem, (d) the one-dimensional first-order Edgeworth expansion, provided that we have sup{tin(0,1]cap T : E{pi,0}[|Y_t| ^{alpha}] < 1 with the expected order with respect to the independent case (up to some $varepsilon > 0$ for (c) and (d)). For the statements (b) and (d), a Markov nonlattice condition is also assumed as in the independent case. All the results are derived under the assumption that the Markov process $(X_t)_{tin T}$ has an invariant probability distribution $pi$, is stationary and has the $L^2(pi)$-spectral gap property (that is, $(X_t)tin N}$ is $rho$-mixing in the discrete-time case). The case where $(X_t)_{tin T}$ is non-stationary is briefly discussed. As an application, we derive a Berry-Esseen bound for the M-estimators associated with $rho$-mixing Markov chains."

"We study weak convergence of empirical processes of dependent data, indexed by classes of functions. We obtain results that are especially suitable for data arising from dynamical systems and Markov chains, where the Central Limit Theorem for partial sums is commonly derived via the spectral gap technique. Our results apply, e.g. to the empirical process of ergodic torus automorphisms."

"The results of Komlós, Major and Tusnády give optimal Wiener approximation of partial sums of i.i.d. random variables and provide an extremely powerful tool in probability and statistical inference. Recently Wu [Ann. Probab. 35 (2007) 2294–2320] obtained Wiener approximation of a class of dependent stationary processes with finite pth moments, 2 < p ≤ 4, with error term o(n1/p(log n)γ), γ > 0, and Liu and Lin [Stochastic Process. Appl. 119 (2009) 249–280] removed the logarithmic factor, reaching the Komlós–Major–Tusnády bound o(n1/p). No similar results exist for p > 4, and in fact, no existing method for dependent approximation yields an a.s. rate better than o(n1/4). In this paper we show that allowing a second Wiener component in the approximation, we can get rates near to o(n1/p) for arbitrary p > 2. This extends the scope of applications of the results essentially, as we illustrate it by proving new limit theorems for increments of stochastic processes and statistical tests for short term (epidemic) changes in stationary processes. Our method works under a general weak dependence condition covering wide classes of linear and nonlinear time series models and classical dynamical systems."

"The concentration inequality approach for normal approximation by Stein's method is generalized to the multivariate setting. This approach is used to prove a multivariate normal approximation theorem for standardized sums of independent random vectors with an error bound of the order $k^{1/2}gamma$, where $k$ is the dimension of the random vectors and $gamma$ is the sum of absolute third moments of the random vectors."

"Classical Edgeworth expansions provide asymptotic correction terms to the Central Limit Theorem (CLT) up to an order that depends on the number of moments available. In this paper, we provide subsequent correction terms beyond those given by a standard Edgeworth expansion in the general case of regularly varying distributions with diverging moments (beyond the second). The subsequent terms can be expressed in a simple closed form in terms of certain special functions (Dawson’s integral and parabolic cylinder functions), and there are qualitative differences depending on whether the number of moments available is even, odd, or not an integer, and whether the distributions are symmetric or not. If the increments have an even number of moments, then additional logarithmic corrections must also be incorporated in the expansion parameter. An interesting feature of our correction terms for the CLT is that they become dominant outside the central region and blend naturally with known large-deviation asymptotics when these are applied formally to the spatial scales of the CLT."

"Let $(U_n(t))_{tinR^d}$ be the empirical process associated to an $R^d$-valued stationary process $(X_i)_{ige 0}$. We give general conditions, which only involve processes $(f(X_i))_{ige 0}$ for a restricted class of functions $f$, under which weak convergence of $(U_n(t))_{tinR^d}$ can be proved. This is particularly useful when dealing with data arising from dynamical systems or functional of Markov chains. This result improves those of [DDV09] and [DD11], where the technique was first introduced, and provides new applications."

"The Shannon-McMillan-Breiman theorem asserts that the sample entropy of a stationary and ergodic stochastic process converges to the entropy rate of the same process almost surely. In this paper, we focus our attention on the convergence behavior of the sample entropy of a hidden Markov chain. Under certain positivity assumption, we prove that a central limit theorem (CLT) with some Berry-Esseen bound for the sample entropy of a hidden Markov chain, and we use this CLT to establish a law of iterated logarithm (LIL) for the sample entropy."

"We establish a central limit theorem and an invariance principle for stationary random fields. In particular, we extend the Maxwell--Woodroofe condition on stationary processes to the multiparameter setting. Our result is obtained via an $m$-dependent approximation method. As applications, we improve known results on the invariance principles for orthomartingales and functionals of linear random fields."

From a quick skim, looks reasonably teachable.

"A Central Limit Theorem is proved for linear random fields when sums are taken over finite disjoint union of rectangles. The approach does not rely upon the use of Beveridge Nelson decomposition and the conditions needed are similar to those given by Ibragimov for linear processes. When specializing this result to the case when sums are being taken over rectangles, an analogue of Ibragimov result is obtained with a lot of uniformity."

"This study discusses the history of the central limit theorem and related probabilistic limit theorems from about 1810 through 1950. In this context the book also describes the historical development of analytical probability theory and its tools, such as characteristic functions or moments. The central limit theorem was originally deduced by Laplace as a statement about approximations for the distributions of sums of independent random variables within the framework of classical probability, which focused upon specific problems and applications."

"We show that the conditional central limit theorem can take place for a stationary process defined on a nonergodic dynamical system while this last does not satisfy the central limit theorem for any ergodic component. There exists an ergodic Markov chain such that the conditional central limit theorem is satisfied for an invariant measure but fails to hold for almost all starting points."

"We consider a partially hyperbolic set $K$ on a Riemannian manifold $M$ whose tangent space splits as $T_K M=E^{cu}\oplus E^{s}$, for which the centre-unstable direction $E^{cu}$ expands non-uniformly on some local unstable disk. We show that under these assumptions $f$ induces a Gibbs-Markov structure. Moreover, the decay of the return time function can be controlled in terms of the time typical points need to achieve some uniform expanding behavior in the centre-unstable direction. As an application of the main result we obtain certain rates for decay of correlations, large deviations, an almost sure invariance principle and the validity of the Central Limit Theorem."

Sounds suspiciously like they're rediscovering the connection between random walks and stable distributions.

Shorter Prof. Dr. Peter Grassberger; Tsallis is teh suxx0r!

"The asymptotic normality of U-statistics has so far been proved for iid data and under various mixing conditions such as absolute regularity, but not for strong mixing. We use a coupling technique introduced in 1983 by Bradley to prove a new generalized covariance inequality similar to Yoshihara's. It follows from the Hoeffding-decomposition and this inequality that U-statistics of strongly mixing observations converge to a normal limit if the kernel of the U-statistic fulfills some moment and continuity conditions.
The validity of the bootstrap for U-statistics has until now only been established in the case of iid data (see Bickel and Freedman). For mixing data, Politis and Romano proposed the circular block bootstrap, which leads to a consistent estimation of the sample mean's distribution. We extend these results to U-statistics of weakly dependent data and prove a CLT for the circular block bootstrap version of U-statistics under absolute regularity and strong mixing. We also calculate a rate of convergence for the bootstrap variance estimator of a U-statistic and give some simulation results."

Monday's seminar speaker.

Here "pet" = polynomial ergodic theorem, "split" = sum-product limit theorem

Central limit theorems and exponential growth