## István Berkes

### Alfréd Rényi Institute of Mathematics, Hungarian Academy of SciencesGraz University of Technology

#### Strong approximation, Skorohod representation, St. Petersburg paradox: some forgotten results of Paul Lévy

In this lecture, we discuss some remarks in Paul Lévy's paper "Propriétés asymptotiques des sommes de variables aléatoires indépendantes ou enchainées", J. Math. Pures Appl. 14 (1935), 347-402. Lévy's remarks have apparently been forgotten in the shadow of the celebrated results of his paper, even though they contain important results and ideas of probability theory discovered decades later. Among others, Lévy proves a remarkable limit theorem for i.i.d. sums with irregularly varying tails, containing the striking limit theorem for St. Petersburg sums used to clarify the St. Petersburg paradox 50 years later. The proof uses a coupling argument, similar to Skorohod representation, and provides a strong (pointwise) approximation result, 30 years before Strassen's theory. Lévy was also the first to use the quantile transform, the basic tool leading to the deep approximation results in the years 1970-1990 and utilized it, among others, to prove a pointwise decomposition of stable variables, another celebrated result from the 1980's. In addition to discussing Lévy's arguments, we also formulate some new results using his ideas.

## Siegfried Hörmann

### Université libre de Bruxelles

#### Dimension reduction for functional data

Data in many fields of science are sampled from processes that can most naturally be described as functional. Examples include growth curves, temperature curves, curves of financial transaction data and patterns of pollution data. Functional data analysis (FDA) is concerned with the statistical analysis of such data.
An important tool in many empirical and theoretical problems related to FDA is the functional principal analysis (FPCA) which allows to represent or approximate curves in low dimension. It is certainly the most common approach to obtain dimension reduction for functional data. In fact, it achieves in some sense optimal dimension reduction if data are independent. However, it is all but uncommon that functional data are serially correlated. A typical example is if the observations are segments from a continuous time process (e.g. days). Then, although cross-sectionally uncorrelated for a fixed observation, the classical FPC-score vectors have non-diagonal cross-correlations. This means that we cannot analyze them componentwise (like in the i.i.d. case), but we need to consider them as vector time series which are less easy to handle and interpret. In this talk we will present a time series version of functional PCA. The idea is to transform the (possibly infinite dimensional) functional time series, into a vector time series of low dimension (in practice 3 or 4, say), where the individual component processes are mutually uncorrelated, and explain a bigger part of the dynamics and variability of the original process. We demonstrate the method on real data and discuss potential applications.

The talk is based on joint work with Łukasz Kidziński (Stanford) and Marc Hallin (ULB).

## Thomas Mikosch

### University of Copenhagen

#### The eigenstructure of sample covariance matrices for high-dimensional heavy-tailed stochastic volatility models

This is joint work with Johannes Heiny (Aarhus University).

We are interested in the asymptotic behavior of the eigenvalues of the sample covariance matrix ${\mathbf X}{\mathbf X}'$ where the data matrix ${\mathbf X}={\mathbf X}_n=(X_{it})_{i=1,\ldots,p;t=1,\ldots,n}$ has the structure of a stochastic volatility model, i.e., $X_{it}=\sigma_{it}Z_{it}$ and the stationary volatility field $(\sigma_{it})$ is independent of the iid field $(Z_{it})$. We assume that $Z_{it}$ has regularly varying tails with index $\alpha\in (0,4)$ and that $\sigma_{it}$ has lighter tails. We are interested in the case when the dimension $p=p_n\to\infty$. Then the eigenvalues of ${\mathbf X}{\mathbf X}'$ are essentially determined by the diagonal elements ${\mathbf X}_{ii}$, $i=1,\ldots,n$. We consider limit theory of Poisson-type for these eigenvalues and also discuss the structure of the corresponding eigenvectors.

## Bálint Tóth

### Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences University of Bristol

#### Central limit theorem for random walk in doubly stochastic random environment

It is proved that an infrared bound on the correlations of the local drift field, called the $H_{-1}$-condition, is sufficient for the central limit theorem with diffusive scaling to hold for the random walks in doubly stochastic random environments. This result settles a notorious issue in the context of random walks in random environments. The talk will be based partly on joint work with Gady Kozma (Weizmann Institute).

### Ulyanovsk State University

#### The Mittag-Leﬄer functions, related distributions, processes and limit theorems

The Mittag-Leﬄer (M-L)functions can be considered as a generalization of the exponential function whose role in theoretical physics can hardly be overestimated. Nevertheless, only the last two decades or so have been marked by increased attention to these functions [1]. This review gives a short description of their main properties, containing both known results related to fractional diﬀerential equations and original results concerning the properties of the M-L probability distribution such as its geometric stability, various mixture representations and relations with the Linnik distributions. Limit theorems for statistics constructed from samples with random sizes (including continuous-time random walks and sums of a random number of independent random variables with ﬁnite variances) are presented establishing the convergence of the distributions of such statistics to these distributions. The applications of the M-L functions to random processes are considered including fractional Poisson process, time-fractional Boltzmann process, bifractional generalization of the Brownian diﬀusion process and some others. It is shown that long-time asymptotics of such processes generates fractional stable distributions and fractional Poisson limit distributions. Some examples of application of these models to description of processes in cosmic scales are demonstrated: cosmic ray transport, magnetic ﬁeld turbulence, pulsar signal propagation, image processing in astronomy. This work is partially supported by Russian Foundation for Basic Research (projects 15-07-04040 and 16-01-00556).

[1] Gorenﬂo, R, Kilbas, A., Mainardi, F., Rogosin, S., Mittag-Leﬄer Functions, Related Topics and Applications. Springer, 2014.

## Victor Korolev

### Moscow State University

#### GG-mixed Poisson distributions

A wide and flexible family of discrete distributions is considered, the so-called GG-mixed Poisson distributions that are mixed Poisson distributions in which the mixing laws belong to the class of generalized gamma distributions (GG-distributions). The latter contains practically all most popular absolutely continuous distributions concentrated on the non-negative half-line. The GG-mixed Poisson distributions seem to be very promising in the statistical description of many real phenomena being almost universal models for the description of statistical regularities in discrete data. Analytic properties of GG-mixed Poisson distributions are studied. A GG-distribution is proved to be a mixed exponential distribution if and only if the shape and exponent power parameters are no greater than one. The mixing distribution is written out explicitly as a scale mixture of strictly stable laws concentrated on the nonnegative halfline. As a corollary, the representation is obtained for the GG-mixed Poisson distribution as a mixed geometric distribution. The corresponding scheme of Bernoulli trials with random probability of success is considered. Within this scheme, a random analog of the Poisson theorem is proved establishing the convergence of mixed binomial distributions to mixed Poisson laws. Limit theorems are proved for random sums of independent random variables in which the number of summands has the GG-mixed Poisson distribution and the summands have both light- and heavy-tailed distributions. The class of limit laws is wide enough and includes the so-called generalized variance gamma distributions. Various representations for the limit laws are obtained in terms of mixtures of Mittag-Leffler, Linnik or Laplace distributions. Limit theorems are proved establishing the convergence of the distributions of statistics constructed from samples with random sizes obeying the GG-mixed Poisson distribution to generalized variance gamma distributions. Some applications of GG-mixed Poisson distributions in meteorology are discussed.