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We consider again the fast-slow motions setups in the continuous time $\frac {dX_N(t)}{dt}=N^{1/2} \sig(X_N(t))(\xi(tN))+b(X_N(t)),\, t\in [0,T]$ and the discrete time $X_N((n+1)/N)=X_N(n/N)+N^{-1/2}\sig(X_N(n/N))\xi(n)+N^{-1}b(X_N(n/N)),\, n=0,1,...,[TN]$ where $\sig$ and $b$ are smooth matrix and vector functions, respectively, $\xi$ is a centered vector stationary stochastic process with weak dependence in time and $N$ is a big parameter. We obtain estimates for the almost sure approximations of the process $X_N$ by certain diffusion process $\Sig$. In \cite{FK} and in other recent papers concerning similar setups the results were obtained relying fully on the rough paths theory. Here we derive our probabilistic results as corollaries of quite general deterministic estimates which are obtained with all details provided following somewhat ideology of the rough paths theory but not relying on this theory per se which should allow a more general readership to follow complete arguments., Comment: 24 pages

We obtain strong invariance principles for normalized multiple iterated sums and integrals of the form $\bbS_N^{(\nu)}(t)=N^{-\nu/2}\sum_{0\leq k_1<...

We obtain strong moment invariance principles for normalized multiple iterated sums and integrals of the form $\mathbb{S}^{(\nu)}(t)=N^{-\nu/2}\sum_{0\leq k_1<...0$ in the Prokhorov and the Wasserstein metrics to the distribution of certain stochastic processes $\mathbb{W}_N^{(\nu)}$ constructed recursively starting from $W_N=\mathbb{W}_N^{(1)}$ which is a Brownian motion with covariances. This is done by constructing a coupling between $\mathbb{S}_N^{(1)}$ and $\mathbb{W}_N^{(1)}$, estimating directly the moment variational norm of $\mathbb{S}_N^{(\nu)}-\mathbb{W}_N^{(\nu)}$ for $\nu=1,2$ and extending these estimates to $\nu>2$ relying partially on arguments borrowed from the rough paths theory. In the continuous time we work both under direct weak dependence assumptions and also within the suspension setup which is more appropriate for applications in dynamical systems., Comment: 49 pages

The paper deals with the fast-slow motions setups in the discrete time $X^\epsilon((n+1)\epsilon)=X^\epsilon(n\epsilon)+\epsilon B(X^\epsilon(n\epsilon),\xi(n))$, $n=0,1,...,[T/\epsilon]$ and the continuous time $\frac {dX^\epsilon(t)}{dt}=B(X^\epsilon(t),\xi(t/\epsilon)).\, t\in [0,T]$ where $B$ is a smooth vector function and $\xi$ is a sufficiently fast mixing stationary stochastic process. It is known since 1966 (Khasminskii) that if $\bar X$ is the averaged motion then $G^\epsilon=\epsilon^{-1/2}(X^\epsilon-\bar X)$ weakly converges to a Gaussian process $G$. We will show that for each $\epsilon$ the processes $\xi$ and $G$ can be redefined on a sufficiently rich probability space without changing their distributions so that $E\sup_{0\leq t\leq T}|G^\epsilon(t)-G(t)|^{2M} =O(\epsilon^{\delta})$, $\delta>0$ which gives also $O(\epsilon^{\delta/3})$ Prokhorov distance estimate between the distributions of $G^\epsilon$ and $G$. In the product case $B(x,\xi)=\Sigma(x)\xi$ we obtain almost sure convergence estimates of the form $\sup_{0\leq t\leq T}|G^\epsilon(t)-G(t)|=O(\epsilon^\delta)$ a.s., as well as the functional form of the law of iterated logarithm for $G^\epsilon$. We note that our mixing assumptions are adapted to fast motions generated by important classes of dynamical systems., Comment: arXiv admin note: text overlap with arXiv:2105.01940, arXiv:2111.05390

The paper deals with the fast-slow motions setups in the continuous time $\frac {dX^\ve(t)}{dt}=\frac 1\ve\sig(X^\ve(t))\xi(t/\ve^2)+b(X^\ve(t)),\, t\in [0,T]$ and the discrete time $X_N((n+1)/N)=X_N(n/N)+N^{-1/2}\sig(X_N(n/N))\xi(n))+N^{-1}b(X_N(n/N))\xi(n)$, $n=0,1,...,[TN]$ where $\sig$ and $b$ are smooth matrix and vector functions, respectively, $\xi$ is a centered stationary vector stochastic process and $\ve, 1/N$ are small parameters. We derive, first, estimates in the strong invariance principles for sums $S_{N}(t)=N^{-1/2}\sum_{0\leq k< [Nt]}\xi(k)$ and iterated sums $\bbS^{ij}_{N}(t)=N^{-1}\sum_{0\leq k

The paper deals with the fast-slow motions setups in the continuous time $\frac {dX^(t)}{dt}=\frac 1\varepsilon B(X^\varepsilon(t),\xi(t/\varepsilon^2))+b(X^\varepsilon(t),\,\xi(t/\varepsilon^2)),\, t\in [0,T]$ and the discrete time $X^\varepsilon((n+1)\varepsilon^2)=X^\varepsilon(n\varepsilon^2)+\varepsilon B(X^\varepsilon(n\varepsilon^2),\xi(n)) +\varepsilon^2 b(X^\varepsilon(n\varepsilon^2),\xi(n))$, $n=0,1,...,[T/\varepsilon^2]$ where $\Sigma$ and $b$ are smooth vector functions and $\xi$ is a stationary vector stochastic process such that $E\xi(0)=0$ for all $x\in\mathbb{R}^d$. Unlike \cite{Ki20} the assumptions imposed on the process $\xi$ allow applications to a wide class of observables $g$ in the dynamical systems setup so that $\xi$ can be taken in the form $\xi(t)=g(F^t\xi(0))$ or $\xi(n)=g(F^n\xi(0))$ where $F$ is either a flow or a diffeomorphism with some hyperbolicity and $g$ is a vector function. In this paper we show that both $X^\varepsilon$ and a family of diffusions $\Xi^\varepsilon$ can be redefined on a common sufficiently rich probability space so that $E\sup_{0\leq t\leq T}|X^\varepsilon(t)-\Xi^\varepsilon(t)|^{p}\leq C\varepsilon^\delta,\, p\geq 1$ for some $C,\delta>0$ and all $\varepsilon>0$, where all $\Xi^\varepsilon,\, \varepsilon>0$ have the same diffusion coefficients but underlying Brownian motions may change with $\varepsilon$., Comment: 30 pages. arXiv admin note: text overlap with arXiv:2011.07907

We obtain error estimates for strong approximations of a diffusion with a diffusion matrix $\sigma$ and a drift b by the discrete time process defined recursively X_N((n+1)/N) = X_N(n/N)+N^{1/2}\sigma(X_N(n/N))\xi(n+1)+N^{-1}b(XN(n/N)); where \xi(n); n\geq 1 are i.i.d. random vectors, and apply this in order to approximate the fair price of a game option with a diffusion asset price evolution by values of Dynkin's games with payoffs based on the above discrete time processes. This provides an effective tool for computations of fair prices of game options with path dependent payoffs in a multi asset market with diffusion evolution., Comment: arXiv admin note: substantial text overlap with arXiv:2011.07907

It is known that the slow motion $X^\varepsilon$ in the time-scaled multidimensional averaging setup $\frac {dX^\varepsilon(t)}{dt}=\frac 1\varepsilon B(X^\varepsilon(t),\,\xi(t/\varepsilon^2))+b(X^\varepsilon(t),\,\xi(t/\ve^2)),\, t\in [0,T]$ converges weakly as $\varepsilon\to 0$ to a diffusion process provided $EB(x,\xi(s))\equiv 0$ where $\xi$ is a sufficiently fast mixing stochastic process. In this paper we show that both $X^\varepsilon$ and a family of diffusions $\Xi^\varepsilon$ can be redefined on a common sufficiently rich probability space so that $E\sup_{0\leq t\leq T}|X^\varepsilon(t)-\Xi^\varepsilon(t)|^{2M}\leq C(M)\varepsilon^\del$ for some $C(M),\delta>0$ and all $M\ge 1,\,\varepsilon>0$, where all $\Xi^\varepsilon,\, \varepsilon>0$ have the same diffusion coefficients but underlying Brownian motions may change with $\varepsilon$. This is the first strong approximation result both in the above setup and at all when the limit is a nontrivial multidimensional diffusion. We obtain also a similar result for the corresponding discrete time averaging setup which was not considered before at all. As an application we consider Dynkin's games with path dependent payoffs involving a diffusion and obtain error estimates for computation of values of such games by means of such discrete time approximations which provides a more effective computational tool than the standard discretization of the diffusion itself., Comment: arXiv admin note: text overlap with arXiv:2012.01257

We consider a one dimensional ballistic nearest-neighbor random walk in a random environment. We prove an Erd\H{o}s-R\'enyi strong law for the increments., Comment: 43 pages

We study the strong Borel-Cantelli property both for events and for shifts on sequence spaces considering both a conventional and a nonconventional setups. Namely, under certain conditions on events $\Gamma_1,\Gamma_2,...$ we show that with probability one \[ \left(\sum_{n=1}^N\prod_{i=1}^\ell P(\Gamma_{q_i(n)})\right)^{-1}\sum_{n=1}^N\prod_{i=1}^\ell\mathbb{I}_{\Gamma_{q_i(n)}}\to 1\,\,\mbox{as}\,\, N\to\infty \] where $q_i(n),\, i=1,...,\ell$ are integer valued functions satisfying certain assumptions and $\mathbb{I}_\Gamma$ denotes the indicator of $\Gamma$. When $\ell=1$ (called the conventional setup) this convergence can be established under $\phi$-mixing conditions while when $\ell>1$ (called a nonconventional setup) the stronger $\psi$-mixing condition is required. These results are extended to shifts $T$ of sequence spaces where $\Gamma_{q_i(n)}$ is replaced by $T^{-q_i(n)}C_n^{(i)}$ where $C_n^{(i)},\, i=1,...,\ell,\, n\geq 1$ is a sequence of cylinder sets. As an application we study the asymptotical behavior of maximums of certain logarithmic distance functions and of (multiple) hitting times of shrinking cylinders.

We consider a $\phi$-mixing shift $T$ on a sequence space $\Om$ and study the number $\cN_N$ of returns $\{ T^{q_N(n)}\om\in A^a_n\}$ at times $q_N(n)$ to a cylinder $A^a_n$ constructed by a sequence $a\in\Om$ where $n$ runs either until a fixed integer $N$ or until a time $\tau_N$ of the first return $\{ T^{q_N(n)}\om\in A^b_m\}$ to another cylinder $A^b_m$ constructed by $b\in\Om$. Here $q_N(n)$ are certain functions of $n$ taking on nonnegative integer values when $n$ runs from 0 to $N$ and the dependence on $N$ is the main generalization here which requires certain conditions under which we obtain Poisson distributions limits of $\cN_N$ when counting is until $N$ as $N\to\infty$ and geometric distributions limits when counting is until $\tau_N$ as $N\to\infty$., Comment: arXiv admin note: substantial text overlap with arXiv:1910.01439

For a $\psi$-mixing process $\xi_0,\xi_1,\xi_2,...$ we consider the number $\mathcal{N}_N$ of multiple returns $\{\xi_{q_{i,N}(n)}\in\Gamma_N,\, i=1,...,\ell\}$ to a set $\Gamma_N$ for $n$ until either a fixed number $N$ or until the moment $\tau_N$ when another multiple return $\{\xi_{q_{i,N}(n)}\in\Delta_N,\, i=1,...,\ell\}$ takes place for the first time where $\Gamma_N\cap\Delta_N=\emptyset$ and $q_{i,N},\, i=1,...,\ell$ are certain functions of $n$ taking on nonnegative integer values when $n$ runs from 0 to $N$. The dependence of $q_{i,N}(n)$'s on both $n$ and $N$ is the main novelty of the paper. Under some restrictions on the functions $q_{i,N}$ we obtain Poisson distributions limits of $\mathcal{N}_N$ when counting is until $N$ as $N\to\infty$ and geometric distributions limits when counting is until $\tau_N$ as $N\to\infty$. We obtain also similar results in the dynamical systems setup considering a $\psi$-mixing shift $T$ on a sequence space $\Omega$ and studying the number of multiple returns $\{ T^{q_{i,N}(n)}\omega\in A^a_n,\, i=1,...,\ell\}$ until the first occurrence of another multiple return $\{ T^{q_{i,N}(n)}\omega\in A^b_m,\, i=1,...,\ell\}$ where $A^a_n,\, A_m^b$ are cylinder sets of length $n$ and $m$ constructed by sequences $a,b\in\Omega$, respectively, and chosen so that their probabilities have the same order.

We consider a $\phi$-mixing shift $T$ on a sequence space $\Omega$ and study the number of returns $\{ T^k\omega\in U\}$ to a union $U$ of cylinders of length $n$ until the first return $\{ T^k\omega\in V\}$ to another union $V$ of cylinder sets of length $m$. It turns out that if probabilities of the sets $U$ and $V$ are small and of the same order then the above number of returns has approximately geometric distribution. Under appropriate conditions, we extend this result for some dynamical systems to geometric balls and Young towers with integrable tails. This work is motivated by a number of papers on asymptotical behavior of numbers of returns to shrinking sets, as well as by the papers on open systems studying their behavior until an exit through a "hole".

We prove existence of a self-financing strategy which minimizes shortfall for game options in discrete time

Let $\xi_1,\xi_2,...$ be independent identically distributed random variables and $F:\bbR^\ell\to SL_d(\bbR)$ be a Borel measurable matrix-valued function. Set $X_n=F(\xi_{q_1(n)},\xi_{q_2(n)},...,\xi_{q_\ell(n)})$ where $0\leq q_1

For a $\psi$-mixing stationary process $\xi_0,\xi_1,\xi_2,...$ we consider the number $\mathcal N_N$ of multiple recurrencies $\{\xi_{q_i(n)}\in\Gamma_N,\, i=1,...,\ell\}$ to a set $\Gamma_N$ for $n$ until the moment $\tau_N$ (which we call a hazard) when another multiple recurrence $\{\xi_{q_i(n)}\in\Delta_N,\, i=1,...,\ell\}$ takes place for the first time where $\Gamma_N\cap\Delta_N= \emptyset$ and $q_i(n)

The paper is primarily concerned with the asymptotic behavior as $N\to\infty$ of averages of nonconventional arrays having the form $N^{-1}\sum_{n=1}^N\prod_{j=1}^\ell T^{P_j(n,N)}f_j$ where $f_j$'s are bounded measurable functions, $T$ is an invertible measure preserving transformation and $P_j$'s are polynomials of $n$ and $N$ taking on integer values on integers. It turns out that when $T$ is weakly mixing and $P_j(n,N)=p_jn+q_jN$ are linear or, more generally, have the form $P_j(n,N)=P_j(n)+Q_j(N)$ for some integer valued polynomials $P_j$ and $Q_j$ then the above averages converge in $L^2$ but for general polynomials $P_j$ the $L^2$ convergence can be ensured even in the case $\ell=1$ only when $T$ is strongly mixing. Studying also weakly mixing and compact extensions and relying on Furstenberg's structure theorem we derive an extension of Szemer\' edi's theorem saying that for any subset of integers $\Lambda$ with positive upper density there exists a subset $\mathcal N_\Lambda$ of positive integers having uniformly bounded gaps such that for $N\in\mathcal N_\Lambda$ and at least $\varepsilon N,\,\varepsilon>0$ of $n$'s all numbers $p_jn+q_jN,\, j=1,...,\ell$ belong to $\Lambda$. We obtain also a version of these results for several commuting transformations which yields a corresponding extension of the multidimensional Szemer\' edi theorem., Comment: 28 pages

We obtain a functional Erd\H os-R\' enyi law of large numbers for "nonconventional" sums of the form $\Sig_n=\sum_{m=1}^nF(X_m,X_{2m},...,X_{\ell m})$ where $X_1,X_2,...$ is a sequence of exponentially fast $\psi$-mixing random vectors and $F$ is a Borel vector function extendin in several directions our previous result concerning i.i.d. random variables $X_1,X_2,...$.

We extend the Erd\H os-R\' enyi law of large numbers to the averaging setup both in discrete and continuous time cases. We consider both stochastic processes and dynamical systems as fast motions whenever they are fast mixing and satisfy large deviations estimates. In the continuous time case we consider flows with large deviations estimates which allow a suspension representation and it turns out that fast mixing of corresponding base transformation suffices for our results.

We obtain decay rates of probabilities of tails of polynomials in several independent random variables with heavy tails and derive stable limit theorems for nonconventional sums of such polynomials

We obtain rge Erd\" os-R\' enyi type law of large numbers for "nonconventional" sums of the form $S_n=\sum^n_{m=1}F(X_m,X_{2m},...,X_{\ell m})$ where $X_1,X_2,...$ is a sequence of i.i.d. random variables and $F$ is a bounded Borel function. The proof relies on nonconventional large deviations obtained in [8].

We obtain first decay rates of probabilities of tails of multivariate polynomials built on independent random variables with heavy tails. Then we derive stable limit theorems for nonconventional sums of the form $\sum_{Nt\geq n\geq 1}F(X(q_1(n)),...,X(q_\ell(n)))$ where $F$ is a polynomial, $1\leq q_1(n)<\cdots

We obtain Berry-Esseen type estimates for "nonconventional" expressions of the form $\xi_N=\frac 1{\sqrt N}\sum_{n=1}^N(F(X(q_1(n)),...,X(q_\ell(n)))-\bar F)$

We study partial hedging for game options in markets with transaction costs bounded from below. More precisely, we assume that the investor's transaction costs for each trade are the maximum between proportional transaction costs and a fixed transaction costs. We prove that in the continuous time Black--Scholes (BS) model, there exists a trading strategy which minimizes the shortfall risk. Furthermore, we use binomial models in order to provide numerical schemes for the calculation of the shortfall risk and the corresponding optimal portfolio in the BS model.

We extend local limit theorem type results to nonconventional sums of the form $S_N=\sum_{n=1}^NF(\xi_n,\xi_{2n},...,\xi_{\ell n})$., Comment: 23 pages

We show that for all $\psi$-mixing shifts distributions of the numbers of multiple recurrencies to shrinking cylindrical neighborhoods of all points are close either to Poisson or to compound Poisson distributions. We also describe completely the asymptotic behavior of these distributions., Comment: slightly revised version

We obtain a strong invariance principle for nonconventional sums and applying this result we derive for them a version of the law of iterated logarithm, as well as an almost sure central limit theorem. Among motivations for such results are their applications to multiple recurrence for stochastic processes and dynamical systems.

We start briefly surveying research on optimal stopping games since their introduction by E.B.Dynkin more than 40 years ago. Recent renewed interest to dynkin's games is due, in particular, to the study of Israeli (game) options introduced in 2000. We discuss the work on these options and related derivative securities for the last decade. Among various results on game options we consider error estimates for their discrete approximations, swing game options, game options in markets with transaction costs and other questions., Comment: survey article

We construct algorithms for computation of prices and superhedging strategies for game options in general discrete markets both from the seller and the buyer points of view.

We obtain large deviations theorems for nonconventional sums with underlying process being a Markov process satisfying the Doeblin condition or a dynamical system such as subshift of finite type or hyperbolic or expanding transformation.

The classical Poisson theorem says that if $\xi_1,\xi_2,...$ are i.i.d. 0--1 Bernoulli random variables taking on 1 with probability $p_n\equiv \la/n$ then the sum $S_n=\sum_{i=1}^n\xi_i$ is asymptotically in $n$ Poisson distributed with the parameter $\la$. It turns out that this result can be extended to sums of the form $S_n=\sum_{i=1}^n\xi_{q_1(i)}... \xi_{q_\ell(i)}$ where now $p_n\equiv(\la/n)^{1/\ell}$ and $1\leq q_1(i) <...

We consider "nonconventional" averaging setup in the form $\frac {dX^\epsilon(t)}{dt}=\epsilon B\big(X^\epsilon(t),\xi(q_1(t)), \xi(q_2(t)),...,\xi(q_\ell(t))\big)$ where $\xi(t),t\geq 0$ is either a stochastic process or a dynamical system (i.e. then $\xi(t)=F^tx$) with sufficiently fast mixing while $q_j(t)=\al_jt,\,\al_1<\al_2<...<\al_k$ and $q_j,\, j=k+1,...,\ell$ grow faster than linearly. We show that the properly normalized error term in the "nonconventional" averaging principle is asymptotically Gaussian.

We prove a nonconventional invariance principle (functional central limit theorem) for random fields., Comment: corrected version

This paper deals with strong invariance principles (known also as strong approximation theorems) for sums of the form $\sum_{n=1}^{[Nt]}F\big(X(n),X(2n),...,X(kn), X(q_{k+1}(n)),X(q_{k+2}(n)),..., X(q_\ell(n))\big)$, Comment: 17 pages

We provide conditions which yield a strong law of large numbers for expressions of the form $1/N\sum_{n=1}^{N}F\big(X(q_1(n)),..., X(q_\ell(n))\big)$ where $X(n),n\geq 0$'s is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, $F$ is a continuous function with polinomial growth and certain regularity properties and $q_i,i>m$ are positive functions taking on integer values on integers with some growth conditions. Applying these results we study certain multifractal formalism type questions concerning Hausdorff dimensions of some sets of numbers with prescribed asymptotic frequencies of combinations of digits at places $q_1(n),...,q_\ell(n)$., Comment: 18 pages

We obtain functional central limit theorems for both discrete time expressions of the form $1/\sqrt{N}\sum_{n=1}^{[Nt]}(F(X(q_1(n)),\ldots, X(q_{\ell}(n)))-\bar{F})$ and similar expressions in the continuous time where the sum is replaced by an integral. Here $X(n),n\geq0$ is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, $F$ is a continuous function with polynomial growth and certain regularity properties, $\bar{F}=\int F\,d(\mu\times\cdots\times\mu)$, $\mu$ is the distribution of $X(0)$ and $q_i(n)=in$ for $i\le k\leq\ell$ while for $i>k$ they are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when $q_i$'s are polynomials of increasing degrees. These results decisively generalize [Probab. Theory Related Fields 148 (2010) 71-106], whose method was only applicable to the case $k=2$ under substantially more restrictive moment and mixing conditions and which could not be extended to convergence of processes and to the corresponding continuous time case. As in [Probab. Theory Related Fields 148 (2010) 71-106], our results hold true when $X_i(n)=T^nf_i$, where $T$ is a mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well as in the case when $X_i(n)=f_i({\Upsilon }_n)$, where ${\Upsilon }_n$ is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure. Moreover, our relaxed mixing conditions yield applications to other types of dynamical systems and Markov processes, for instance, where a spectral gap can be established. The continuous time version holds true when, for instance, $X_i(t)=f_i(\xi_t)$, where $\xi_t$ is a nondegenerate continuous time Markov chain with a finite state space or a nondegenerate diffusion on a compact manifold. A partial motivation for such limit theorems is due to a series of papers dealing with nonconventional ergodic averages., Comment: Published in at http://dx.doi.org/10.1214/12-AOP796 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We show that prices and shortfall risks of game (Israeli) barrier options in a sequence of binomial approximations of the Black--Scholes (BS) market converge to the corresponding quantities for similar game barrier options in the BS market with path dependent payoffs and the speed of convergence is estimated, as well. The results are new also for usual American style options and they are interesting from the computational point of view, as well, since in binomial markets these quantities can be obtained via dynamical programming algorithms. The paper continues the study of [11]and [7] but requires substantial additional arguments in view of pecularities of barrier options which, in particular, destroy the regularity of payoffs needed in the above papers.

We show that the shortfall risk of binomial approximations of game (Israeli) options converges to the shortfall risk in the corresponding Black--Scholes market considering Lipschitz continuous path-dependent payoffs for both discrete- and continuous-time cases. These results are new also for usual American style options. The paper continues and extends the study of Kifer [Ann. Appl. Probab. 16 (2006) 984--1033] where estimates for binomial approximations of prices of game options were obtained. Our arguments rely, in particular, on strong invariance principle type approximations via the Skorokhod embedding, estimates from Kifer [Ann. Appl. Probab. 16 (2006) 984--1033] and the existence of optimal shortfall hedging in the discrete time established by Dolinsky and Kifer [Stochastics 79 (2007) 169--195]., Comment: Published in at http://dx.doi.org/10.1214/07-AAP503 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Various forms of the polynomial ergodic theorem (PET) which attracted substantial attention in ergodic theory study the limits of expressions having the form $1/N\sum_{n=1}^NT^{q_1(n)}f_1... T^{q_\ell (n)}f_\ell$ where $T$ is a weakly mixing measure preserving transformation, $f_i$'s are bounded measurable functions and $q_i$'s are polynomials taking on integer values on the integers. Motivated partially by these results we obtain a central limit theorem for expressions of the form $1/\sqrt{N}\sum_{n=1}^N (X_1(q_1(n))X_2(q_2(n))... X_\ell(q_\ell(n))-a_1a_2... a_\ell)$ (sum-product limit theorem--SPLIT) where $X_i$'s are fast $\alpha$-mixing bounded stationary processes, $a_j=EX_j(0)$ and $q_i$'s are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when $q_i$'s are polynomials of growing degrees. This result can be applied to the case when $X_i(n)=T^nf_i$ where $T$ is a mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well, as to the case when $X_i(n)=f_i(\xi_n)$ where $\xi_n$ is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.

The Annals of Applied Probability 16 (2006) 984--1033 [URL: http://projecteuclid.org/euclid.aoap/1151592257], Comment: Published in at http://dx.doi.org/10.1214/07-AAP479 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The work treats systems combining slow and fast motions depending on each other where fast motions are perturbations of families of either dynamical systems or Markov processes with freezed slow variable. In the first case we consider hyperbolic dynamical systems and in the second case we deal with random evolutions which are combinations of diffusions and continuous time Markov chains. We study first large deviations of the slow motion from the averaged one and then use these results together with some Markov property type arguments in order to describe very long time behavior of the slow motion such as its transitions between attractors of the averaged system., Comment: 132 pages, 4 figures, to appear in Memoirs of AMS

Correction for Error estimates for binomial approximations of game options [math.PR/0607123], Comment: Published at http://dx.doi.org/10.1214/105051606000000808 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We justify and give error estimates for binomial approximations of game (Israeli) options in the Black--Scholes market with Lipschitz continuous path dependent payoffs which are new also for usual American style options. We show also that rational (optimal) exercise times and hedging self-financing portfolios of binomial approximations yield for game options in the Black--Scholes market ``nearly'' rational exercise times and ``nearly'' hedging self-financing portfolios with small average shortfalls and initial capitals close to fair prices of the options. The estimates rely on strong invariance principle type approximations via the Skorokhod embedding., Comment: Published at http://dx.doi.org/10.1214/105051606000000088 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The usual random walk on a group (homogeneous both in time and in space) is determined by a probability measure on the group. In a random walk with random transition probabilities this single measure is replaced with a stationary sequence of measures, so that the resulting (random) Markov chains are still space homogeneous, but no longer time homogeneous. We study various notions of measure theoretical boundaries associated with this model and establish an analogue of the Poisson formula for (random) bounded harmonic functions. Under natural conditions on transition probabilities we identify these boundaries for several classes of groups with hyperbolic properties and prove the boundary triviality (i.e., the absence of non-constant random bounded harmonic functions) for groups of subexponential growth, in particular, for nilpotent groups.