Questions tagged [martingale]

In probability theory, a martingale is a model of a fair game where knowledge of past events never helps predict the mean of the future winnings and only the current event matters.

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Implications of martingale property

Consider the sequence of random variables $X_1,X_2, X_3,\dots$. Suppose this sequence is a martingale, that is $$ \begin{aligned} &(1) \quad E(|X_t|)<\infty\\ &(2) \quad E(X_t| \mathcal{F}_{...
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Bayes update rule studied as an operator

The bayes rule can be understood as a nonlinear map from the space of probability measures to itself. Are there any reference/books which study it from this perspective? For eg. can we say something ...
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Proving a process is martingale

Consider $\zeta_1,...$ a sequence of independent r.v. with finite expected value and let $S_n=\sum_{i=1}^n\zeta_n$. Fixing n, show that $M_m=\frac{S_{n-m}}{n-m}$ with $0<m<n$ is a martingale ...
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Does the Voter Model converge in $L^1$?

A few questions about the discrete-time voter model, described in the following way. Let $\Lambda = \{1, \dots , L\}^d \subset \mathbb{Z}^d$, and let $(I_n)_{n\in\mathbb{N}}$ and $( N_n)_{n\in\mathbb{...
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Intuition behind martingale residuals and their properties in survival modeling

I am reading Tutz & Schmid "Modeling Discrete Time-to-Event Data" (2016) chapter 4 Evaluation and Model Choice section 4.2 Residuals and Goodness-of-Fit. A martingale residual is defined ...
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Does a martingale difference sequence $X_{t,i}$ imply $E[(\frac{1}{M}\sum_{i=1}^{M}X_{t,i})^2|\mathcal{F}_t]\leq C\times \frac{1}{M}$?

Let $X_{t,i}$ denote a martingale difference sequence in the $i$th time at day $t$, where $i=1,\cdots,M$. If $X_{t,i}$ is independent for $i=1,\cdots,M$, the result is straightforward. And I know that ...
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How to actually apply martingales when conditioning on a random variable (not filtration)?

For a Galton-Watson process, I've shown that $\frac{Z_n}{\mu^n}$ is a martingale i.e. $E[\frac{Z_{n+1}}{\mu^{n+1}}|\mathcal{F}_n]=\frac{Z_n}{\mu^n}$. However, I want to show that, for $n>m$, $$E[Z_{...
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Conditional expectation of this stochastic process?

I'm just beginning to learn about stochastic processes and encountered this very elementary problem that confused me a bit: We toss a coin that lands on Head with probability $p$ and Tail with $q=1-p$....
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Show that the manhattan distance from the origin of an ubiased random walk in $\mathbb{Z}^2$ defines a martingale sequence

Consider the infinite lattice $N \times N$. A pebble starting at the origin walks at random, each time moving equiprobably to one of its four neighbors. Let $X_i$ be the distance from the origin, ...
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Plot the cumulative intensity process survival analysis

Let T be an exponentially distributed survival time with hazard rate $\alpha(t) = 2$ for t > 0. Define $T = min(T,1)$ and $Delta = I(T \le 1)$. Let $\lambda(t)$ be the intensity process and $\...
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What is a Self-Normalized Process?

I came across a vast literature by Victor H. de la Peña (you can find the book here) that concern self-normalized processes, but I am struggling to understand the definition of said processes! What ...
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Random walks and martingales

In class, our professor explained that the martingale process is the in between case of random walk type I (innovations are i.i.d.) and random walk type II (innovations are serially uncorrelated). ...
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Projection operator: why squared norm of the sum of them is equal (or smaller) than the sum of the squared norms?

I am working through the proof of Lemma 2 in this paper (page 25, need it for my own research) and I am stuck at the very first step. Here, I will formulate a bit simplified version of this step. ...
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Additive hazard model: Estimating a martingale residual process

Consider an additive hazard model with a single categorical covariate $$ \alpha(t | x_i) = \beta_0(t) + \beta_1(t) x_i, $$ where e.g., $x_i \in \{0, 1\}$. To assess the goodness-of-fit of this model, ...
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Pulling out constants out conditional expectation

Given $p_t = p_{t-1} + \varepsilon_t$ and $\varepsilon ~ $i.i.d. with$~E(\varepsilon_t) = 0~\forall~t$ I wanna derive the conditional expectation for $p_t$ and I know it is $p_{t-1}$ since it is a ...
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Expectation of sequence of Random Variables

$X_1, X_2, .....X_n$ is a colletion of Random Variables. They are said to be $\textit{multiplicative system}$ if, for any $1 \leq k \leq n$ and for any set of $k$ indices $1 \leq i_1 < i_2 ....i_k \...
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martingale convergence theorem

Let $(X_n)_{n\geq 1}$ be i.i.d. random variables with density $f$ with respect to Lebesgue measure on $\mathbb R$, and $f(x)>0$ for all $x\in \mathbb {R}$. $g$ is another density (with respect to ...
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Are there good examples of martingale processes that are not simple random walks?

Are there non-trivial examples of martingale processes that aren't simple random walks? I'm trying to better understand the difference between martingales and simple random walks. They look pretty ...
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Martingale property & limiting distribution for frequency of last names

Suppose that children always inherit their last names from their father (which implies that no new last names are ever created). Pick a last name of interest (e.g. Smith), and let $X_n \in \left[0, 1\...
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Martingale in Cox Model

Can someone help me to show that $$ \hat{A}(t, \beta_0) = \sum_{i=1}^{n} \int_0^t\frac{1}{\sum_{j}^n Y_j(s) e^{X_i^T \beta_0}} dN_i(s) $$ is a martingale. The setup is the Cox proportional hazard ...
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Reference Request: Book on Unit Root Theory

In trying to do time series analysis, I almost regularly stumble upon unit root and cointegration tests. The design of most these tests is based on a null of unit root (for both linear and non-linear ...
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Distribution of $\frac{1}{1+X}$ if $X$ is Lognormal

Suppose $Z \sim \mathcal{N}(0,1)$. Suppose $X$ is a lognormally distributed random variable, defined as $X:=X_0exp^{(-0.5\sigma^2+\sigma Z)}$, in other words, $X$ is log-normal with $\mathbb{E}[X]=X_0$...
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How can I test if a time-series data satisfies Markov's property and it is a martingale?

My question is about investigating some properties of time-series. How can I test if my time-series data satisfies Markov's property? How can I test if my time-series data is a martingale? I wonder ...
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Carré du champ operator is a quadratic variation

Let $X_t$ be a real valued Markov process (starting at $x$) with generator $L$. Let $\Gamma(f)$ denote Carré du champ operator i.e. $L(f^2) - 2f \cdot L (f)$. As far as I know under suitable ...
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How to show that an m.d.s is not independent?

I have to prove that this Martingale Difference: $x_t = u_t u_{t-1}$ where $u_t \sim^{iid} (0, \sigma^2)$ is not serially independent, but am failing to do such thing. I also have to prove that it's ...
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Are these Bernoulli variables independent?

I was reading a paper in which it was assumed that $\varepsilon_1,\cdots,\varepsilon_n$ conditional on $X$ possess serial (non-linear) dependence, such that \begin{equation} P[\varepsilon_t\geq0\mid\...
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What is the meaning of $X \in \mathcal {F}$ in probability space ${\displaystyle (\Omega ,{\mathcal {F}},P)}$?

I'm studying Martingales. In the text I keep finding $X \in \mathcal {F}$ with random variable $X:\Omega \rightarrow \mathbb{R}$. And $\mathcal {F}$ a $\sigma$-algebra of $\Omega$. What does it ...
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weird outlier in a cox regression model

I'm using normal deviate residual to identify outliers, and I'm confused that my plot seems to suggest that there are unreasonably a lot of outliers...? Has anyone seen something like this? ...
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Conditional expectation of random variables defined off of each other

First of all, when we say that $X_n \sim \text{Unif}(0,X_{n-1})$, what does that mean, rigorously? Does it mean that for every $\omega \in \Omega$, $X_n(\omega)\sim \text{Unif}(0,X_{n-1}(\omega))$? ...
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Prove that a simple random walk is a martingale

Note that $a$ has a mean of 0. My approach: $$X_t=X_{t-1}+a_t$$ $$E[X_{t+1}\mid X_1 + \dots+X_{t-1}]$$ $$=E[X_{t-1}+2a\mid X_1 + \dots+X_{t-1}]$$ $$=E[X_{t-1}\mid X_1 + \dots+X_{t-1}]+E[2a\mid X_1 + ...
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Does a martingale difference sequence (mds) imply strong mixing?

I read this from an econometrics paper " The typical hypothesis which is imposed in the time series literature is that the $u_t$'s are either independent and identically distributed (i.i.d.) or a ...
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Help to understand martingale example from Billingsley

I am studying Billingsley's section 35 about martingales and I have some difficulties understanding one of the examples. This is the example 35.4. Suppose we have a measurable space $(\Omega,\mathcal ...
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Martingale Difference Sequence

I saw that in the link on page 3 it is said $Y_t = e_t\cdot e_{t-1}$ is martingale difference sequence and dependent where $e_t$ is i.i.d with $N(0,\sigma^2)$ Could you provide me with the proof of it?...
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Martingale Difference Sequence CLT

Could you provide me with the proof of the following: $$n^{-1/2} \cdot\sum_t a_{t-j}e_t$$ converges to normal distribution as $n$ goes infinity by martingale difference sequence CLT where $a_t = \...
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The magic money tree problem

I thought of this problem in the shower, it was inspired by investment strategies. Let's say there was a magic money tree. Every day, you can offer an amount of money to the money tree and it will ...
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How to check these sequences generated by i.i.d random variables are martingales?

Let $\{Y_n\}_{n\geq 1}$ be a sequence of independent, identically distributed random variables. $P(Y_i=1)=P(Y_i=-1)=\frac12$ Set $S_0=0$ and $S_n=Y_1+...+Y_n$ if $n\geq 1$ I want to check if the ...
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log transform fixed PH in Cox model - how?

I have survival data to which I am fitting a Cox model with a continuous predictor. The cumulative martingale residual method (supremum test) of Lin, Wei and Ying suggested that both proportional ...
2 votes
1 answer
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When is the posterior distribution equal to the prior?

So I have heard that if the prior distribution is in the subexponential class, applying Bayes rule does not change the belief. I have been trying to find an example of this but I am unable to do so. I ...
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Cox PH linearity assumption: reading martingal residual plots

According to a lot of ressources about Cox PH model, continuous numeric variables should be tested for linearity assymption by plotting the Martingale residuals. In R, you can use ...
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Cox PH model: managing continuous variables and linearity assumption

In an epidemiological study, I'm using martingale plot to assess the linearity of continuous variables. Here are the Martingale Residuals (from Null Model) using R's ...
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How to approach Basketball "Beat the Pro" drill with Markov Chain

Suppose, similarly to Gambler's ruin problem a Basketball player is doing "Beat the Pro" drill. That is, for every shot made, he scores one point, and for those missed, two points are to be deducted. ...
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Azuma's inequality Use

For binary vectors, x = (x1,x2,...,xn) and y=(y1,y2,...,yn), the Hammond distance between them is defined by d(x,y) = |x1 - y1| + |x2 - y2| + ... + |xn - yn|. Let A be a finite set of such vectors ...
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Write expectation of brownian motion conditional on filtration as an integral?

Let $W_t$ be a Brownian motion, so $W_t=z_t \sqrt{t}$ where $z_t \in N(0,1)$ and the pdf of $z$ is $f(z)=\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}$. So $$E(W_t)=\int_{-\infty}^{\infty} W_t f(z) dz =\...
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Martingale Difference & Conditional Heteroskedasticity

I was reading this passage in a book and I am confused about the last line. As far as I understand, the white noise process does not allow for conditional heteroskedasticity due to its i.i.d. property....
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Is a random walk necessarily a martingale?

I read in the following notes (http://www2.econ.iastate.edu/classes/econ672/Falk/_notes/lecture_4_martingales.pdf, p.2) that "a random walk is a martingale." Although it seems logical with the used ...
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Upper bound for randomly weighted sum of independent random variables

I have a sequence of independent random variables {$\epsilon_j$} with mean 0. I also have another sequence of Bernoulli random variables $\delta_1, \delta_2,\dots$ which are dependent on the previous ...
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1_event set notation

I do not understand the notation as given here: The source for the image is page 17 in this paper after equation 18. I understand the first subset notation but what does 1_\e_t mean?
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Why white noise process and IID process are considered martingale [closed]

Can anyone explain to me why A white noise process (εt) and An IID process (εt) are martingales?
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$M_t$ Local Martingale if and only if satisfies $\frac{\partial f}{\partial t}+\frac{1}{2}\frac{\partial^2 f}{\partial x^2}=0$ proof?

I was reading the wikipedia article on local martingales and it states the following theorem Let $W_t$ be standard Brownian motion and let $f:[0,\infty)\times\mathbb{R}\rightarrow\mathbb{R}$ such ...
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Martingale Difference implies white noise

A white noise sequence needs to have 0 mean, uncorrelated $\epsilon_t$ and $\epsilon_{t+k}$ and constant variance. A martingale difference sequence has 0 mean and uncorrelated $\epsilon_t$ and $\...
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