In this thesis we investigate large deviation and path properties of a few models within the Kardar-Parisi-Zhang (KPZ) universality class. The KPZ equation is the central object in the KPZ universality class. It is a stochastic PDE describing various objects in statistical mechanics such as random interface growth, directed polymers, interacting particle systems. In the first project we study one point upper tail large deviations of the KPZ equation 𝜢(t,x) started from narrow wedge initial data. We obtain precise expression of the upper tail LDP in the long time regime for the KPZ equation. We then extend our techniques and methods to obtain upper tail LDP for the asymmetric exclusion process model, which is a prelimit of the KPZ equation. In the next direction, we investigate temporal path properties of the KPZ equation. We show that the upper and lower law of iterated logarithms for the rescaled KPZ temporal process occurs at a scale (log log 𝑡)²/³ and (log log 𝑡)¹/³ respectively. We also compute the exact Hausdorff dimension of the upper level sets of the solution, i.e., the set of times when the rescaled solution exceeds 𝛼(log log 𝑡)²/³. This has relevance from the point of view of fractal geometry of the KPZ equation. We next study superdiffusivity and localization features of the (1+1)-dimensional continuum directed random polymer whose free energy is given by the KPZ equation. We show that for a point-to-point polymer of length 𝑡 and any 𝑝 ⋲ (0,1), the point on the path which is 𝑝𝑡 distance away from the origin stays within a 𝑂(1) stochastic window around a random point 𝙈_𝑝,𝑡 that depends on the environment. This provides an affirmative case of the folklore `favorite region' conjecture. Furthermore, the quenched density of the point when centered around 𝙈_𝑝,𝑡 converges in law to an explicit random density function as 𝑡 → ∞ without any scaling. The limiting random density is proportional to 𝑒^{-𝓡(x)} where 𝓡(x) is a two-sided 3D Bessel process with diffusion coefficient 2. Our proof techniques also allow us to prove properties of the KPZ equation such as ergodicity and limiting Bessel behaviors around the maximum. In a follow up project, we show that the annealed law of polymer of length 𝑡, upon 𝑡²/³ superdiffusive scaling, is tight (as 𝑡 → ∞) in the space of 𝐶([0,1]) valued random variables. On the other hand, as 𝑡 → 0, under diffusive scaling, we show that the annealed law of the polymer converges to Brownian bridge. In the final part of this thesis, we focus on an integrable discrete half-space variant of the CDRP, called half-space log-gamma polymer.We consider the point-to-point log-gamma polymer of length 2𝑁 in a half-space with i.i.d.Gamma⁻¹(2𝛳) distributed bulk weights and i.i.d. Gamma⁻¹(𝛼+𝛳) distributed boundary weights for 𝛳 > 0 and 𝛼 > -𝛳. We establish the KPZ exponents (1/3 fluctuation and 2/3 transversal) for this model when 𝛼 ≥ 0. In particular, in this regime, we show that after appropriate centering, the free energy process with spatial coordinate scaled by 𝑁²/³ and fluctuations scaled by 𝑁¹/³ is tight. The primary technical contribution of our work is to construct the half-space log-gamma Gibbsian line ensemble and develop a toolbox for extracting tightness and absolute continuity results from minimal information about the top curve of such half-space line ensembles. This is the first study of half-space line ensembles. The 𝛼 ≥ 0 regime correspond to a polymer measure which is not pinned at the boundary. In a companion work, we investigate the 𝛼 < 0 setting. We show that in this case, the endpoint of the point-to-line polymer stays within 𝑂(1) window of the diagonal. We also show that the limiting quenched endpoint distribution of the polymer around the diagonal is given by a random probability mass function proportional to the exponential of a random walk with log-gamma type increments.

Cette thèse est dévolue à l'étude de certaines propriétés asymptotiques du processus autorégressif d'ordre p. Ce dernier qualifie communément une suite aléatoire $(Y_{n})$ définie sur $\dN$ ou $\dZ$ et entièrement décrite par une combinaison linéaire de ses $p$ valeurs passées, perturbée par un bruit blanc $(\veps_{n})$. Tout au long de ce mémoire, nous traitons deux problématiques majeures de l'étude de tels processus : l'\textit{autocorrélation résiduelle} et la \textit{stationnarité}. Nous proposons en guise d'introduction un survol nécessaire des propriétés usuelles du processus autorégressif. Les deux chapitres suivants sont consacrés aux conséquences inférentielles induites par la présence d'une autorégression significative dans la perturbation $(\veps_{n})$ pour $p=1$ tout d'abord, puis pour une valeur quelconque de $p$, dans un cadre de stabilité. Ces résultats nous permettent d'apposer un regard nouveau et plus rigoureux sur certaines procédures statistiques bien connues sous la dénomination de \textit{test de Durbin-Watson} et de \textit{H-test}. Dans ce contexte de bruit autocorrélé, nous complétons cette étude par un ensemble de principes de déviations modérées liées à nos estimateurs. Nous abordons ensuite un équivalent en temps continu du processus autorégressif. Ce dernier est décrit par une équation différentielle stochastique et sa solution est plus connue sous le nom de \textit{processus d'Ornstein-Uhlenbeck}. Lorsque le processus d'Ornstein-Uhlenbeck est lui-même engendré par une diffusion similaire, cela nous permet de traiter la problématique de l'autocorrélation résiduelle dans le processus à temps continu. Nous inférons dès lors quelques propriétés statistiques de tels modèles, gardant pour objectif le parallèle avec le cas discret étudié dans les chapitres précédents. Enfin, le dernier chapitre est entièrement dévolu à la problématique de la stationnarité. Nous nous plaçons dans le cadre très général où le processus autorégressif possède une tendance polynomiale d'ordre $r$ tout en étant engendré par une marche aléatoire intégrée d'ordre $d$. Les résultats de convergence que nous obtenons dans un contexte d'instabilité généralisent le \textit{test de Leybourne et McCabe} et certains aspects du \textit{test KPSS}. De nombreux graphes obtenus en simulations viennent conforter les résultats que nous établissons tout au long de notre étude. / This thesis is devoted to the study of some asymptotic properties of the $p-$th order \textit{autoregressive process}. The latter usually designates a random sequence $(Y_{n})$ defined on $\dN$ or $\dZ$ and completely described by a linear combination of its $p$ last values and a white noise $(\veps_{n})$. All through this manuscript, one is concerned with two main issues related to the study of such processes: \textit{serial correlation} and \textit{stationarity}. We intend, by way of introduction, to give a necessary overview of the usual properties of the autoregressive process. The two following chapters are dedicated to inferential consequences coming from the presence of a significative autoregression in the disturbance $(\veps_{n})$ for $p=1$ on the one hand, and then for any $p$, in the stable framework. These results enable us to give a new light on some statistical procedures such as the \textit{Durbin-Watson test} and the \textit{H-test}. In this autocorrelated noise framework, we complete the study by a set of moderate deviation principles on our estimates. Then, we tackle a continuous-time equivalent of the autoregressive process. The latter is described by a stochastic differential equation and its solution is the well-known \textit{Ornstein-Uhlenbeck process}. In the case where the Ornstein-Uhlenbeck process is itself driven by an Ornstein-Uhlenbeck process, one deals with the serial correlation issue for the continuous-time process. Hence, we infer some statistical properties of such models, keeping the parallel with the discrete-time framework studied in the previous chapters as an objective. Finally, the last chapter is entirely devoted to the stationarity issue. We consider the general autoregressive process with a polynomial trend of order $r$ driven by a random walk of order $d$. The convergence results in the unstable framework generalize the \textit{Leybourne and McCabe test} and some angles of the \textit{KPSS test}. Many graphs obtained by simulations come to strengthen the results established all along the study.