* Measure Theory-Basic Notions * Measure Theory-Key Results * Processes, Distributions, and Independence * Random Sequences, Series, and Averages * Characteristic Functions and Classical Limit Theorems * Conditioning and Disintegration * Martingales and Optional Times * Markov Processes and Discrete-Time Chains * Random Walks and Renewal Theory * Stationary Processes and Ergodic Theory * Special Notions of Symmetry and Invariance * Poisson and Pure Jump-Type Markov Processes * Gaussian Processes and Brownian Motion * Skorohod Embedding and Invariance Principles * Independent Increments and Infinite Divisibility * Convergence of Random Processes, Measures, and Sets * Stochastic Integrals and Quadratic Variation * Continuous Martingales and Brownian Motion * Feller Processes and Semigroups * Ergodic Properties of Markov Processes * Stochastic Differential Equations and Martingale Problems * Local Time, Excursions, and Additive Functionals * One-Dimensional SDEs and Diffusions * Connections with PDEs and Potential Theory * Predictability, Compensation, and Excessive Functions * Semimartingales and General Stochastic Integration * Large Deviations * Appendix 1: Advanced Measure Theory * Appendix 2: Some Special Spaces * Historical and Bibliographical Notes * Bibliography * Indices

http://cyber.sibsutis.ru:82/Monarev/docs/nauka/PROBABILITY/Kallenberg%20O.%20Foundations%20of%20Modern%20Probability%20(Springer,1997)(ISBN%200387949577)(535s).pdf

Foundations of modern probability

(2002). Modelling Extremal Events for Insurance and Finance. Journal of the American Statistical Association: Vol. 97, No. 457, pp. 360-360.

Modelling Extremal Events for Insurance and Finance

We extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence. We do not attempt a reduction to i.i.d. values. The jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of $l$ consecutive data once. In the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero. Under appropriate conditions consistency is obtained if $l = l(n) \rightarrow \infty$ and $l(n)/n \rightarrow 0$. General statistics are approximated by an arithmetic mean. In regular cases this approximation determines the asymptotic behavior. Bootstrap replicates are constructed by selecting blocks of length $l$ randomly with replacement among the blocks of observations. The procedures are illustrated by using the sunspot numbers and some simulated data.

https://projecteuclid.org/download/pdf_1/euclid.aos/1176347265

The Jackknife and the Bootstrap for General Stationary Observations

Let $(X, Y)$ be a pair of random variables such that $X$ is $\mathbb{R}^d$-valued and $Y$ is $\mathbb{R}^{d'}$-valued. Given a random sample $(X_1, Y_1), \cdots, (X_n, Y_n)$ from the distribution of $(X, Y)$, the conditional distribution $P^Y(\bullet \mid X)$ of $Y$ given $X$ can be estimated nonparametrically by $\hat{P}_n^Y(A \mid X) = \sum^n_1 W_{ni}(X)I_A(Y_i)$, where the weight function $W_n$ is of the form $W_{ni}(X) = W_{ni}(X, X_1, \cdots, X_n), 1 \leqq i \leqq n$. The weight function $W_n$ is called a probability weight function if it is nonnegative and $\sum^n_1 W_{ni}(X) = 1$. Associated with $\hat{P}_n^Y(\bullet \mid X)$ in a natural way are nonparametric estimators of conditional expectations, variances, covariances, standard deviations, correlations and quantiles and nonparametric approximate Bayes rules in prediction and multiple classification problems. Consistency of a sequence $\{W_n\}$ of weight functions is defined and sufficient conditions for consistency are obtained. When applied to sequences of probability weight functions, these conditions are both necessary and sufficient. Consistent sequences of probability weight functions defined in terms of nearest neighbors are constructed. The results are applied to verify the consistency of the estimators of the various quantities discussed above and the consistency in Bayes risk of the approximate Bayes rules.

/pdf/consistent-nonparametric-regression-1i3ntzz647.pdf

Consistent Nonparametric Regression

Exchangeability and related topics

Strong approximations in probability and statistics

Simple Symmetric Random Walk in ℤ1: Introduction of Part I Distributions Recurrence and the Zero-One Law From the Strong Law of Large Numbers to the Law of Iterated Logarithm Levy Classes Wiener Process and Invariance Principle Increments Strassen Type Theorems Distribution of the Local Time Local Time and Invariance Principle Strong Theorems of the Local Time Excursions Frequently and Rarely Visited Sites An Embedding Theorem A Few Further Results Summary of Part I Simple Symmetric Random Walk in ℤd: The Recurrence Theorem Wiener Process and Invariance Principle The Law of Iterated Logarithm Local Time The Range Heavy Points and Heavy Balls Crossing and Self-crossing Large Covered Balls Long Excursions Speed of Escape A Few Further Problems Random Walk in Random Environment: Introduction of Part III In the First Six Days After the Sixth Day What Can a Physicist Say About the Local Time ξ(0,n)? On the Favourite Value of the RWIRE A Few Further Problems Random Walks in Graphs: Introduction of Part IV Random Walk in Comb Random Walk in a Comb and in a Brush with Crossings Random Walk on a Spider Random Walk in Half-Plane-Half-Comb.

Random Walk in Random and Non-Random Environments

Let $q_n(y), 0 < y < 1,$ be a quantile process based on a sequence of i.i.d. rv with distribution function $F$ and density function $f.$ Given some regularity conditions on $F$ the distance of $q_n(y)$ and the uniform quantile process $u_n(y),$ respectively defined in terms of the order statistics $X_{k:n}$ and $U_{k:n} = F(X_{k:n}),$ is computed with rates. As a consequence we have an extension of Kiefer's result on the distance between the empirical and quantile processes, a law of iterated logarithm for $q_n(y)$ and, using similar results for the uniform quantile process $u_n(y),$ it is also shown that $q_n(y)$ can be approximated by a sequence of Brownian bridges as well as by a Kiefer process.

/pdf/strong-approximations-of-the-quantile-process-3bml7wum6l.pdf

Strong Approximations of the Quantile Process

Let $\beta_T = (2a_T\lbrack\log(T/a_T) + \log\log T\rbrack)^{-\frac{1}{2}}, 0 0$, the Erdos-Renyi law of large numbers for the Wiener process. A number of other results for the Wiener process also follow via choosing $a_T$ appropriately. Connections with strong invariance principles and the P. Levy modulus of continuity for $W(t)$ are also established.

https://projecteuclid.org/download/pdf_1/euclid.aop/1176994994

How Big are the Increments of a Wiener Process

A new method is developed to produce strong laws of invariance principle without making use of the Skorohod representation. As an example, it will be proved that 
$${{\mathop {\lim }\limits_{n \to \infty } \left( {S_n - W(n)} \right)} \mathord{\left/ {\vphantom {{\mathop {\lim }\limits_{n \to \infty } \left( {S_n - W(n)} \right)} {n^{{1 \mathord{\left/ {\vphantom {1 {6 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {6 + \varepsilon }}} }}} \right. \kern-\nulldelimiterspace} {n^{{1 \mathord{\left/ {\vphantom {1 {6 + \varepsilon }}} \right. \kern-\nulldelimiterspace} {6 + \varepsilon }}} }} = 0$$

 with probability 1, for any g3>0, where S

 n
 =X
1 + ... +X

 n
 , X

 i
 is a sequence of i.i.d.r.v.'s with P(X

 i
 <t)=F(t), and F(t) is a distribution function obeying (i), (ii) and W(n) is a suitable Wiener-process. Strassen in [1], proved (under weaker conditions): 
$$S_n - W\left( n \right) = O\left( {\sqrt[4]{{n{\text{ log log }}n}}\sqrt {{\text{log }}n} {\text{ }}} \right)$$

 with probability one. He conjectured that if 
$$S_n - W\left( n \right) = o\left( {\sqrt[4]{{n{\text{ log log }}n}}\sqrt {{\text{log }}n} {\text{ }}} \right)$$

 then F(x) = Φ(x) where Φ(.) is the unit normal distribution function. (See also [2], [6] and [7].) Our result above is a negative answer to this question.

Pál Révész

Papers

Strong approximations in probability and statistics

Random Walk in Random and Non-Random Environments

Strong Approximations of the Quantile Process

How Big are the Increments of a Wiener Process

A new method to prove strassen type laws of invariance principle. 1