Harmonic functions, h-transform and large deviations for random walks in random environments in dimensions four and higher

We consider large deviations for nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on $\mathbb{Z}^d$. There exist variational formulae for the quenched and averaged rate functions $I_q$ and $I_a$, obtained by Rosenbluth and Varadhan, respectively. $I_q$ and $I_a$ are not identically equal. However, when $d\geq4$ and the walk satisfies the so-called (T) condition of Sznitman, they have been previously shown to be equal on an open set $\mathcal{A}_{\mathit {eq}}$. For every $\xi\in\mathcal{A}_{\mathit {eq}}$, we prove the existence of a positive solution to a Laplace-like equation involving $\xi$ and the original transition kernel of the walk. We then use this solution to define a new transition kernel via the h-transform technique of Doob. This new kernel corresponds to the unique minimizer of Varadhan's variational formula at $\xi$. It also corresponds to the unique minimizer of Rosenbluth's variational formula, provided that the latter is slightly modified.Comment: Published in at http://dx.doi.org/10.1214/10-AOP556 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Differing averaged and quenched large deviations for random walks in random environments in dimensions two and three

We consider the quenched and the averaged (or annealed) large deviation rate functions $I_q$ and $I_a$ for space-time and (the usual) space-only RWRE on $\mathbb{Z}^d$. By Jensen's inequality, $I_a\leq I_q$. In the space-time case, when $d\geq3+1$, $I_q$ and $I_a$ are known to be equal on an open set containing the typical velocity $\xi_o$. When $d=1+1$, we prove that $I_q$ and $I_a$ are equal only at $\xi_o$. Similarly, when d=2+1, we show that $I_a<I_q$ on a punctured neighborhood of $\xi_o$. In the space-only case, we provide a class of non-nestling walks on $\mathbb{Z}^d$ with d=2 or 3, and prove that $I_q$ and $I_a$ are not identically equal on any open set containing $\xi_o$ whenever the walk is in that class. This is very different from the known results for non-nestling walks on $\mathbb{Z}^d$ with $d\geq4$.Comment: 21 pages. In this revised version, we corrected our computation of the variance of $D(B_1)$ for $d=2+1$ (page 11 of the old version, after (2.31)). We also added details explaining precisely how the space-only case is handled, by mapping the appropriate objects to the space-time setup (see pages 14--17 in the new version). Accepted for publication in Communications in Mathematical Physics

The stochastic encounter-mating model

We propose a new model of permanent monogamous pair formation in zoological populations with multiple types of females and males. According to this model, animals randomly encounter members of the opposite sex at their so-called firing times to form temporary pairs which then become permanent if mating happens. Given the distributions of the firing times and the mating preferences upon encounter, we analyze the contingency table of permanent pair types in three cases: (i) definite mating upon encounter; (ii) Poisson firing times; and (iii) Bernoulli firing times. In the first case, the contingency table has a multiple hypergeometric distribution which implies panmixia. The other two cases generalize the encounter-mating models of Gimelfarb (1988) who gives conditions that he conjectures to be sufficient for panmixia. We formulate adaptations of his conditions and prove that they not only characterize panmixia but also allow us to reduce the model to the first case by changing its underlying parameters. Finally, when there are only two types of females and males, we provide a full characterization of panmixia, homogamy and heterogamy.Comment: 27 pages. We shortened the abstract, added Section 1.1 (Overview), and updated reference

The Rhetorical Algorithm: WikiLeaks and the Elliptical Secrets of Donald J. Trump

Algorithms were a generative force behind many of the leaks and secrets that dominated the 2016 election season. Taking the form of the identity-anonymizing Tor software that protected the identity of leakers, mathematical protocols occupied a prominent place in the secrets generated during the presidential campaign. This essay suggests that the rhetorical trope of ellipsis offers an equally crucial, algorithmic formula for explaining the public production of these secrets and leaks. It then describes the 2016 DNC leak and Donald Trump’s “I love Wikileaks” moment using the trope of ellipsis, which marks a discursive omission or gap in official executive discourse

Averaged large deviations for random walk in a random environment

In his 2003 paper, Varadhan proves the averaged large deviation principle for the mean velocity of a particle taking a nearest-neighbor random walk in a uniformly elliptic i.i.d. environment on $\mathbb{Z}^d$ with $d\geq1$, and gives a variational formula for the corresponding rate function $I_a$. Under Sznitman's transience condition (T), we show that $I_a$ is strictly convex and analytic on a non-empty open set $\mathcal{A}$, and that the true velocity of the particle is an element (resp. in the boundary) of $\mathcal{A}$ when the walk is non-nestling (resp. nestling). We then identify the unique minimizer of Varadhan's variational formula at any velocity in $\mathcal{A}$.Comment: 14 pages. In this revised version, I state and prove all of the results under Sznitman's (T) condition instead of Kalikow's condition. Also, I rewrote many parts of Section 1, streamlined some of the proofs in Section 2, fixed some typos, and improved the wording here and there. Accepted for publication in Annales de l'Institut Henri Poincar

The Predictive Performance of Asymmetric Normal Mixture GARCH in Risk Management: Evidence from Turkey

The purpose of this study is to test predictive performance of Asymmetric Normal Mixture GARCH (NMAGARCH) and other GARCH models based on Kupiec and Christoffersen tests for Turkish equity market. The empirical results show that the NMAGARCH perform better based on %99 CI out-of-sample forecasting Christoffersen test where GARCH with normal and student-t distribution perform better based on %95 Cl out-of-sample forecasting Christoffersen test and Kupiec test. These results show that none of the model including NMAGARCH outperforms other models in all cases as trading position or confidence intervals and the real implications of these results for Value-at-Risk estimation is that volatility model should be chosen according to confidence interval and trading positions. Besides, NMAGARCH increases predictive performance for higher confidence internal as Basel requiresGARCH, Asymmetric Normal Mixture GARCH, Christoffersen Test, Emerging Markets.