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Convergence of an implicitly defined sequence of random variables

Let $\{X_n\}_{n\ge 1}$ be a sequence of independent identically distributed Poisson random variables with mean $\lambda^*$. Consider a sequence of random variables $\{\hat{\lambda}_{n}\}_{n\ge 1}$ ...
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Martingales in both discrete and continuous setting

I am wondering, polynomials like $S_n^4-6n S_n^2+3n^2+2n$ for $$S_n=\sum_{i=1}^n{X_i}$$ where $$\mathbb{P}(X_i=1)=\mathbb{P}(X_i=-1)=\frac{1}{2}$$ is a martingale (under the conventional filtration). ...
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Brownian motion, martingales, Markov Chains - Rosetta Stone

What are the most fundamental/useful/interesting ways in which the concepts of Brownian motion, martingales and markov chains are related? I'm a graduate student doing a crash course in ...
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Savings property: A transformation which turns nonnegative martingales into uniformly integrable ones

Background I work in a subfield of computability theory called algorithmic randomness. We have been using martingales as long as probability theory (going back to work of von Mises). However, since ...
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Suppose $D \subset \mathbb{R}^d$ is a domain and $f: \overline{D} \to \mathbb{R}$ is a continuous function, $C^2$ in $D$, satisfying$$f(x) = 0\text{ for }x\in \partial D,$$$${1\over2} \Delta f(x) = -1 ... 2answers 281 views Examples of a continuous martingale with E[\sup\limits_{0\leq s\leq t} |M_s|]=\infty? A local martingale is a martingale iff it is in the class DL. The condition: for every t\in[0,\infty)$$E[\sup\limits_{0\leq s\leq t} |M_s|]<\infty\tag1$$guarantees a local martingale M is ... 1answer 123 views concentration inequality for d-dimensional martingale Are any concentration inequality available for d-dimensional martingale. It is easy to find such inequality using the inequalities for single dimension, but that will contain the dimension d in ... 1answer 225 views Stochastic integration by parts to obtain Kailath Segall identity for iterated stochastic integrals? If (M_t)_{t \geq 0} is a continuous local martingale, one can define the iterated integrals I_0=1, I_1(t)=M_t and for n \geq 2$$I_{n}(t) = \int_0^t I_{n-1} (s) \mathrm{d} M_s.$$By noting ... 1answer 239 views explicit characterization of the stochastic integrand Let V be a cadlag positive supermartingale with the following decomposition:$$V_t=V_0+\int_0^tH_sdX_s-K_t$$where X is a cadlag local martingale and K is an adapted increasing process with ... 1answer 389 views a L^1 convergence for backward martingale I have a question which may be naive, but I can not find the related result in the classical reference such as "Foundations of Modern Probability" and "Probability"(Billingsley). So if someone knows ... 0answers 50 views Existence of martingales given some constraint on laws Let X=(X)_{0\le t\le 1} be a continuous martingale starting at 0, then denote by \mu and \nu the probability laws of \int_0^1X_t \mathrm{d}t and X_1. Then it is easy to see that the couple ... 0answers 230 views Some constants in Martingale Stein inequality Dear all, the following is a special case of Stein inequalities for martingales. \textbf{Theorem} Let (\Omega, \mathbb{P}) be a (standard) probability space equipped with a filtration of ... 4answers 812 views Men in a bar - stoch. processes Hello everyone, I'm trying to solve a applied stochastic process problem and even the example is beautiful, I don't know how to approach it. Here the problem: 10 men want to get out of a bar, they do ... 1answer 274 views Proof of Pinelis (1992) - Banach space inequalities I am reading Pinelis "An approach to inequalities for the distributions of infinite -dimensional martingales" and cannot follow his proof of Theorem 3: Let (f_n) be a martingale in a separable ... 1answer 461 views Stochastic integrals as honest martingales — comparison criterion We have a given positive martingale \rho_t, with the dynamics:$$\textrm{d} \rho_t = \lambda_t \rho_t \textrm{d} W_t$$where W_t is a standard Brownian motion. Now we have a "dumped" process p_t: ... 1answer 155 views Does martingale convergence hold for arbitrary time? Let \{\mathcal B_i:i\in I\} be a family of \sigma-algebras (over the same set \Omega) which are totally ordered by inclusion, in the sense that for any i,j\in I either \mathcal B_i\subset\... 1answer 592 views Stochastic integrals as honest martingales — exponential damping We have a given positive martingale ρt, with the dynamics:$$\textrm{d}\rho_t = \lambda_t \rho_t \textrm{d}W_t$$where W_t is a standard Brownian motion. Now we have an "exponentially dampened" ... 2answers 945 views Is the truncated Brownian motion of the class DL? Let W be a standard Brownian motion under given probability space. For a given constant a, W^a is a truncated Brownian motion by stopping time T^a = \inf(t>0:W(t) = a). That is, W^a(t) = ... 1answer 153 views Conditional Form of Rosenthal's Inequality Rosenthal's Inequality as stated in the book "Martingale Limit Theory and Its Application" by Hall and Heyde states the following: If \{S_i, \mathcal{F}_i, 1\leq i \leq n\} is a martingale and 2\... 1answer 604 views Lower-semicomputable supermartingales with bounded increments I'm interested in whether Levin and Solomonoff's results on "universal semimeasures" can be extended to other settings. One case that especially interests me is finding "universal" strategies in the ... 1answer 139 views Poisson kernel, expectation, an absolute value comes in See here. Let d = 2, and consider the domain D = \mathbb{H}, the upper half-plane. Let W_t = (X_t, Y_t). We see that for any \theta \in \mathbb{R} and any t \ge 0, we have$$E^{(x, y)}\...
I am trying to prove that a certain sequence of Markov chains $x^N_k$ converges towards a diffusion process. The invariant measure of $x^N$ is $\pi^N$ and the Markov chain $x^N$ is started in ...
Let $(\Omega, \mathcal{F}, P)$ be a probability space, on which $\mathcal{F}_t$ is filtration satisfying general conditions. $W_{t}=\left(W_{t}^{1},W_{t}^{2}\right)^{T}$ is a two dimensional Brownian ...