اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA Maximal Inequality for Supermartingales
104
0
0.0
(
0
)
تأليف
Bruce Hajek
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
A tight upper bound is given on the distribution of the maximum of a supermartingale. Specifically, it is shown that if $Y$ is a semimartingale with initial value zero and quadratic variation process $[Y,Y]$ such that $Y + [Y,Y]$ is a supermartingale, then the probability the maximum of $Y$ is greater than or equal to a positive constant $a$ is less than or equal to $1/(1+a).$ The proof makes use of the semimartingale calculus and is inspired by dynamic programming.
قيم البحث
اقرأ أيضاً
In this note besides two abstra
In this note, we present a version of Hoeffdings inequality in a continuous-time setting, where the data stream comes from a uniformly ergodic diffusion process. Similar to the well-studied case of Hoeffdings inequality for discrete-time uniformly er
godic Markov chain, the proof relies on techniques ranging from martingale theory to classical Hoeffdings lemma as well as the notion of deviation kernel of diffusion process. We present two examples to illustrate our results. In the first example we consider large deviation probability on the occupation time of the Jacobi diffusion, a popular process used in modelling of exchange rates in mathematical finance, while in the second example we look at the exponential functional of a finite interval analogue of the Ornstein-Uhlenbeck process introduced by Kessler and S{o}rensen (1999).
Consider a parabolic stochastic PDE of the form $partial_t u=frac{1}{2}Delta u + sigma(u)eta$, where $u=u(t,,x)$ for $tge0$ and $xinmathbb{R}^d$, $sigma:mathbb{R}tomathbb{R}$ is Lipschitz continuous and non random, and $eta$ is a centered Gaussian no
ise that is white in time and colored in space, with a possibly-signed homogeneous spatial correlation function $f$. If, in addition, $u(0)equiv1$, then we prove that, under a mild decay condition on $f$, the process $xmapsto u(t,,x)$ is stationary and ergodic at all times $t>0$. It has been argued that, when coupled with moment estimates, spatial ergodicity of $u$ teaches us about the intermittent nature of the solution to such SPDEs cite{BertiniCancrini1995,KhCBMS}. Our results provide rigorous justification of of such discussions. The proof rests on novel facts about functions of positive type, and on strong localization bounds for comparison of SPDEs.
As an extension of a central limit theorem established by Svante Janson, we prove a Berry-Esseen inequality for a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables.
An easy consequence of Kantorovich-Rubinstein duality is the following: if $f:[0,1]^d rightarrow infty$ is Lipschitz and $left{x_1, dots, x_N right} subset [0,1]^d$, then $$ left| int_{[0,1]^d} f(x) dx - frac{1}{N} sum_{k=1}^{N}{f(x_k)} right| leq le
ft| abla f right|_{L^{infty}} cdot W_1left( frac{1}{N} sum_{k=1}^{N}{delta_{x_k}} , dxright),$$ where $W_1$ denotes the $1-$Wasserstein (or Earth Movers) Distance. We prove another such inequality with a smaller norm on $ abla f$ and a larger Wasserstein distance. Our inequality is sharp when the points are very regular, i.e. $W_{infty} sim N^{-1/d}$. This prompts the question whether these two inequalities are specific instances of an entire underlying family of estimates capturing a duality between transport distance and function space.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
