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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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Expected value of iid squared conditioned on sum

I would be interested in finding the value of the following expression: $$\mathbb{E}[X_k^2\mid S_N]$$ where $X_k$ are iid random variables with $\mathbb{E}[X_k]=\mu$ and $\operatorname{Var}[X_k]=\...
user3141592's user avatar
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Gambler ruin's: Probability of k consecutive win before j consecutive loss

Assume that a stock has a probability of $p$ to win, a probability of $q$ to lose, and a probability of $(1-p-q)$ to remain every day. What is the probability of $k$ consecutive wins occur before $j$ ...
Zhihao Xu's user avatar
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Proof of Azuma-Hoeffding inequality for Martingales in MAB

Suppose in an MAB problem, $X_t=\mu+\epsilon_t$ represents the reward values at each period (for a given arm). If $\mathcal{M}(t)$ shows the set of periods until $t$ where a specific arm has been ...
Amin's user avatar
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Asymptotic normality of the maximum likelihood estimator with dependent data

In the setup, assume $\left(\mathbb{R}, \mathscr{B}\left(\mathbb{R}\right), P\right)$ is the underlying probability space and suppose that $\left\{\mathcal{F_n}\right\}_{n\in \mathbb{N}}$ is a ...
Yashaswi Mohanty's user avatar
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About the requirement of iids to form a martingale

Most problems about martingales start with an assertion similar to the following one: Suppose that $X_n$ are iid such that $\mathbb{E}[|X_n|]<\infty$ and $\mathbb{E}[X_n]=0$. Then $S_n = \sum_i^n ...
user3141592's user avatar
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71 views

Show that $\xi _{n}^2-n$ is a martingale

Let $\xi_{n}$ be a symmetric random walk, i.e, $\xi_{n}=\eta_{1}+\eta_{2}+\ldots+\eta_{n}$ where $\eta_{1},\ldots$ is a sequence of independent identically distributed random variables such that $P\{\...
A Y's user avatar
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Fundamental theorem of card counting: exchangeability and conditional distributions

I have some elementary queries about the relationship between exchangeability and conditional distributions. For theorem 11.6.4 in the Doctrine of Chances, Ethier S. (2010) it is proved that $\{Z_n\}_{...
microhaus's user avatar
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3 votes
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Distribution Function of Standard Normal is a U.I. Martingale?

I'm a little lost on how to show how $X_{t}=\Phi(\frac{W_{t}}{\sqrt{T-t}})$ $0\leq t\leq T$, where $W_{t}$ is the usual Brownian Motion, is a Uniformly Integrable Martingale? My goal is to try and ...
rockedlauncher's user avatar
3 votes
1 answer
150 views

Uniformly Integrable Martingale

I have $(Y_{n})_{n\in\mathbb{N}}$ as a seq. of positive, independent r.v.'s whose expectation is 1 $\forall n$. I have the canonical filtration $\mathcal{F}_{n}=\sigma(Y_{k},k\leq n)$. I have already ...
rockedlauncher's user avatar
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How to show that this particular card game is a martingale?

An ordinary deck of cards is randomly shuffled and then the cards are exposed one at a time. At some time before all the cards have been exposed you must say “next”, and if the next card exposed is a ...
Win_odd Dhamnekar's user avatar
12 votes
3 answers
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What's the relationship between these two definitions of martingales?

On wikipedia, the definition of a martingale is given as follows: A basic definition of a discrete-time martingale is a discrete-time stochastic process (i.e., a sequence of random variables) $X_1, ...
ExcitedSnail's user avatar
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4 votes
1 answer
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Stopping time with alternating sign random variables

I find myself trying to solve a peculiar stopping time problem. Let $\{X_i\}$ be set of continuous random variables of a stochastic process, each with finite mean value $\mu$ and standard deviation $\...
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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}_{...
Star's user avatar
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Bayes update rule studied as an operator

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? E.g., can we say something about ...
sushant's user avatar
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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{...
Manbearpig's user avatar
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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 ...
Richard Hardy's user avatar
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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 ...
Patrick Star's user avatar
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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_{...
Nonya's user avatar
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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 $q=1-p$....
Not Euler's user avatar
  • 101
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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, ...
user340870's user avatar
1 vote
1 answer
200 views

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 $\...
user10386405's user avatar
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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 ...
Carl's user avatar
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2 votes
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214 views

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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138 views

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. ...
Misius's user avatar
  • 733
1 vote
1 answer
218 views

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, ...
harisf's user avatar
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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 ...
Anil K.'s user avatar
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1 answer
386 views

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 \...
Señora Penn's user avatar
2 votes
0 answers
55 views

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 ...
xxxxxxxx's user avatar
2 votes
1 answer
360 views

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 ...
Garp's user avatar
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4 votes
1 answer
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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\...
Adrian's user avatar
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0 votes
1 answer
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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 ...
nalen's user avatar
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10 votes
2 answers
358 views

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 ...
6 votes
1 answer
512 views

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 $\mathbb{E}[X]=X_0$...
Jan Stuller's user avatar
2 votes
1 answer
254 views

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 ...
xyzt's user avatar
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0 answers
332 views

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 ...
marcusy's user avatar
  • 43
3 votes
1 answer
207 views

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 ...
Caio C.'s user avatar
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2 votes
1 answer
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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\...
Carl's user avatar
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2 votes
2 answers
205 views

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 ...
Theo Broeders's user avatar
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0 answers
176 views

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? ...
llbia's user avatar
  • 361
6 votes
3 answers
2k views

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 $\omega \in \Omega$, $X_n(\omega)\sim \text{Unif}(0,X_{n-1}(\omega))$? ...
D.R's user avatar
  • 123
6 votes
2 answers
4k views

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 + ...
GarlicSTAT's user avatar
3 votes
3 answers
1k views

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 ...
Michael L's user avatar
4 votes
1 answer
145 views

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 ...
Chaos's user avatar
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1 answer
558 views

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?...
mertcan's user avatar
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0 answers
95 views

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 = \...
mertcan's user avatar
  • 105
24 votes
4 answers
2k views

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 ...
ElectronicToothpick's user avatar
2 votes
2 answers
384 views

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 ...
Win_odd Dhamnekar's user avatar
4 votes
2 answers
3k views

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 ...
user3156942's user avatar
2 votes
1 answer
1k views

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 ...
Dio's user avatar
  • 123
6 votes
1 answer
6k views

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 ...
Dan Chaltiel's user avatar
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