Tail asymptotics for the supercritical Galton-Watson process in the heavy-tailed case

As well known, for a supercritical Galton-Watson process $Z_n$ whose offspring distribution has mean $m>1$, the ratio $W_n:=Z_n/m^n$ has a.s. limit, say $W$. We study tail behaviour of the distributions of $W_n$ and $W$ in the case where $Z_1$ has heavy-tailed distribution, that is, \E e^{\lambda Z_1}=\infty for every $\lambda>0$. We show how different types of distributions of $Z_1$ lead to different asymptotic behaviour of the tail of $W_n$ and $W$. We describe the most likely way how large values of the process occur

Tail behaviour of stationary distribution for Markov chains with asymptotically zero drift

We consider a Markov chain on $R^+$ with asymptotically zero drift and finite second moments of jumps which is positive recurrent. A power-like asymptotic behaviour of the invariant tail distribution is proven; such a heavy-tailed invariant measure happens even if the jumps of the chain are bounded. Our analysis is based on test functions technique and on construction of a harmonic function.Comment: 27 page

At the Edge of Criticality: Markov Chains with Asymptotically Zero Drift

In Chapter 2 we introduce a classification of Markov chains with asymptotically zero drift, which relies on relations between first and second moments of jumps. We construct an abstract Lyapunov functions which looks similar to functions which characterise the behaviour of diffusions with similar drift and diffusion coefficient. Chapter 3 is devoted to the limiting behaviour of transient chains. Here we prove converges to $\Gamma$ and normal distribution which generalises papers by Lamperti, Kersting and Klebaner. We also determine the asymptotic behaviour of the cumulative renewal function. In Chapter 4 we introduce a general strategy of change of measure for Markov chains with asymptotically zero drift. This is the most important ingredient in our approach to recurrent chains. Chapter 5 is devoted to the study of the limiting behaviour of recurrent chains with the drift proportional to $1/x$. We derive asymptotics for a stationary measure and determine the tail behaviour of recurrence times. All these asymptotics are of power type. In Chapter 6 we show that if the drift is of order $x^{-\beta}$ then moments of all orders $k\le [1/\beta]+1$ are important for the behaviour of stationary distributions and pre-limiting tails. Here we obtain Weibull-like asymptotics. In Chapter 7 we apply our results to different processes, e.g. critical and near-critical branching processes, risk processes with reserve-dependent premium rate, random walks conditioned to stay positive and reflected random walks. In Chapter 8 we consider asymptotically homogeneous in space Markov chains for which we derive exponential tail asymptotics

On lower limits and equivalences for distribution tails of randomly stopped sums

For a distribution $F^{*\tau}$ of a random sum $S_{\tau}=\xi_1+...+\xi_{\tau}$ of i.i.d. random variables with a common distribution $F$ on the half-line $[0,\infty)$, we study the limits of the ratios of tails $\bar{F^{*\tau}}(x)/\bar{F}(x)$ as $x\to\infty$ (here, $\tau$ is a counting random variable which does not depend on $\{\xi_n\}_{n\ge1}$). We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ111 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Visualizing the strain evolution during the indentation of colloidal glasses

We use an analogue of nanoindentation on a colloidal glass to elucidate the incipient plastic deformation of glasses. By tracking the motion of the individual particles in three dimensions, we visualize the strain field and glass structure during the emerging deformation. At the onset of flow, we observe a power-law distribution of strain indicating strongly correlated deformation, and reflecting a critical state of the glass. At later stages, the strain acquires a Gaussian distribution, indicating that plastic events become uncorrelated. Investigation of the glass structure using both static and dynamic measures shows a weak correlation between the structure and the emerging strain distribution. These results indicate that the onset of plasticity is governed by strong power-law correlations of strain, weakly biased by the heterogeneous glass structure.Comment: 13 pages, 8 figure

PROBABILITY METHODS FOR ASSESSING FINANCIAL RISKS FOR ENTERPRISES

The paper reviews methods for risk assessment which could be employed in the financial management of enterprises. The set of methods proposed renders it possible to assess the impact which different risk factors upon the efficiency of implemented projects and the effect of antirisk measures on the financial performance of companies and thus identify the most efficient measures according to the criterion selected for project evaluation

Asymptotics of randomly stopped sums in the presence of heavy tails

We study conditions under which $P(S_\tau>x)\sim P(M_\tau>x)\sim E\tau P(\xi_1>x)$ as $x\to\infty$, where $S_\tau$ is a sum $\xi_1+...+\xi_\tau$ of random size $\tau$ and $M_\tau$ is a maximum of partial sums $M_\tau=\max_{n\le\tau}S_n$. Here $\xi_n$, $n=1$, 2, ..., are independent identically distributed random variables whose common distribution is assumed to be subexponential. We consider mostly the case where $\tau$ is independent of the summands; also, in a particular situation, we deal with a stopping time. Also we consider the case where $E\xi>0$ and where the tail of $\tau$ is comparable with or heavier than that of $\xi$, and obtain the asymptotics $P(S_\tau>x) \sim E\tau P(\xi_1>x)+P(\tau>x/E\xi)$ as $x\to\infty$. This case is of a primary interest in the branching processes. In addition, we obtain new uniform (in all $x$ and $n$) upper bounds for the ratio $P(S_n>x)/P(\xi_1>x)$ which substantially improve Kesten's bound in the subclass ${\mathcal S}^*$ of subexponential distributions.Comment: 22 page

Renewal Theory for Transient Markov Chains with Asymptotically Zero Drift

We solve the problem of asymptotic behaviour of the renewal measure (Green function) generated by a transient Lamperti's Markov chain $X_n$ in $\R$, that is, when the drift of the chain tends to zero at infinity. Under this setting, the average time spent by $X_n$ in the interval $(x,x+1]$ is roughly speaking the reciprocal of the drift and tends to infinity as $x$ grows. For the first time we present a general approach relying on a diffusion approximation to prove renewal theorems for Markov chains. We apply a martingale type technique and show that the asymptotic behaviour of the renewal measure heavily depends on the rate at which the drift vanishes. The two main cases are distinguished, either the drift of the chain decreases as $1/x$ or much slower than that, say as $1/x^\alpha$ for some $\alpha\in(0,1)$. The intuition behind how the renewal measure behaves in these two cases is totally different. While in the first case $X_n^2/n$ converges weakly to a $\Gamma$-distribution and there is no law of large numbers available, in the second case a strong law of large numbers holds true for $X_n^{1+\alpha}/n$ and further normal approximation is available