We provide a probabilistic proof of various Stein's factors for Poisson approximation in terms of the Wasserstein distance.

We prove the max-martingale conjecture of Obłój and Yor. We show that for a continuous local martingale $(N_t:t\ge 0)$ and a function $H:R\times R_{+}\to R$, $H(N_t,{sup}_{s\le t}{N}_s)$ is a local martingale if and only if there exists a locally integrable function $f$ such that $H(x,y)={\int }_0^{y}f\left(s\right)ds-f\left(y\right)(x-y)+H(0,0)$. This readily implies, via Lévy's equivalence theorem, an analogous result with the maximum process replaced by the local time at $0$.

Peter offers to play exactly one St Petersburg game with each of $n\ge 2$ players, Paul${}_1$, ${}_n$, Paul${pp}_n=(p_{1,n},...,p_{n,n})$, whose conceivable pooling strategies are described by all possible probability distributions ${pp}_n$. Comparing infinite expectations, we characterize among all $V_{{pp}_n}={\sum }_{k=1}^n{p}_{k,n}{X}_k$ those admissible strategies for which the pooled winnings, each distributed as ${}_1$, yield a finite added value for each and every one of Paul${}_n$, $X_1,...,X_n$, Paul$S_n=X_1+\dots +X_n$ in comparison with their individual winnings ${pp}_n$, even though their total winnings $H\left({pp}_n\right)$ is the same. We show that the added value of an admissible ${pp}_n^{*}$ is just its entropy $n\ge 2$, and we determine the best admissible strategy ${pp}_n$. Moreover, for every $S_{{pp}_n}=V_{{pp}_n}-H\left({pp}_n\right)$ and $S_{{pp}_n}$ we construct semistable approximations to $n\to \mathrm{\infty }$. We show in particular that $max\{p_{1,n},\dots ,p_{n,n}\}\to 0$ has a proper semistable asymptotic distribution as ${pp}_n$ along the entire sequence of natural numbers whenever $S_n/n$ for a sequence $S_{{pp}_n^{*}}$ of admissible strategies, which is in sharp contrast to Peter offers to play exactly one St Petersburg game with each of $n\ge 2$ players, Paul${}_1$, ..., Paul${}_n$, whose conceivable pooling strategies are described by all possible probability distributions ${pp}_n=(p_{1,n},...,p_{n,n})$. Comparing infinite expectations, we characterize among all ${pp}_n$ those admissible strategies for which the pooled winnings, each distributed as $V_{{pp}_n}={\sum }_{k=1}^n{p}_{k,n}{X}_k$, yield a finite added value for each and every one of Paul${}_1$, ..., Paul${}_n$ in comparison with their individual winnings $X_1,...,X_n$, even though their total winnings $S_n=X_1+\dots +X_n$ is the same. We show that the added value of an admissible ${pp}_n$ is just its entropy $H\left({pp}_n\right)$, and we determine the best admissible strategy ${pp}_n^{*}$. Moreover, for every $n\ge 2$ and ${pp}_n$ we construct semistable approximations to $S_{{pp}_n}=V_{{pp}_n}-H\left({pp}_n\right)$. We show in particular that $S_{{pp}_n}$ has a proper semistable asymptotic distribution as $n\to \mathrm{\infty }$ along the entire sequence of natural numbers whenever $max\{p_{1,n},\dots ,p_{n,n}\}\to 0$ for a sequence ${pp}_n$ of admissible strategies, which is in sharp contrast to $S_n/n$, and the rate of convergence is very fast for $S_{{pp}_n^{*}}$. , and the rate of convergence is very fast for $n\ge 2$.

In a finite mixture of location--scale distributions the maximum likelihood estimator does not exist because of the unboundedness of the likelihood function when the scale parameter of some mixture component approaches zero. In order to study the strong consistency of the maximum likelihood estimator, we consider the case where the scale parameters of the component distributions are restricted from below by $c_n$, where $\left\{c_n\right\}$ is a sequence of positive real numbers which tend to zero as the sample size $n$ increases. We prove that under mild regularity conditions the maximum likelihood estimator is strongly consistent if the scale parameters are restricted from below by $c_n=exp(-n^d)$, $0<d<1$.

With the availability of high-frequency financial data, nonparametric estimation of the volatility of an asset return process becomes feasible. A major problem is how to estimate the volatility consistently and efficiently, when the observed asset returns contain error or noise, for example, in the form of microstructure noise. The issue of consistency has been addressed in the recent literature. However, the resulting estimator is not efficient. In work by Zhang, Myland and Aït-Sahalia, the best estimator converges to the true volatility only at the rate of n^-1/6. In this paper, we propose an estimator, the multi-scale realized volatility (MSRV), which converges to the true volatility at the rate of n^-1/4, which is the best attainable. We show a central limit theorem for the MSRV estimator, which permits intervals to be set for the true integrated volatility on the basis of the MSRV.

We study the binary classification problem with hinge loss. We consider classifiers that are linear combinations of base functions. Instead of an ${\ell }_2$ penalty, which is used by the support vector machine, we put an ${\ell }_1$ penalty on the coefficients. Under certain conditions on the base functions, hinge loss with this complexity penalty is shown to lead to an oracle inequality involving both model complexity and margin.

We present an algorithm for exact simulation of a class of Itô's diffusions. We demonstrate that when the algorithm is applicable, it is also straightforward to simulate diffusions conditioned to hit specific values at predetermined time instances. We also describe a method that exploits the properties of the algorithm to carry out inference on discretely observed diffusions without resorting to any kind of approximation apart from the Monte Carlo error.

Starting from the moving average (MA) integral representation of fractional Brownian motion (FBM), the class of fractional Lévy processes (FLPs) is introduced by replacing the Brownian motion by a general Lévy process with zero mean, finite variance and no Brownian component. We present different methods of constructing FLPs and study second-order and sample path properties. FLPs have the same second-order structure as FBM and, depending on the Lévy measure, they are not always semimartingales. We consider integrals with respect to FLPs and MA processes with the long memory property. In particular, we show that the Lévy-driven MA process with fractionally integrated kernel coincides with the MA process with the corresponding (not fractionally integrated) kernel and driven by the corresponding FLP.

Acknowledgement of refereeing work BJ vol. 12

Annual index BJ vol. 12