23
votes
Accepted
The magic money tree problem
This is a well-known problem. It is called a Kelly bet. The answer, by the way, is 1/3rd. It is equivalent to maximizing the log utility of wealth.
Kelly began with taking time to infinity and ...
- 7,810
17
votes
Accepted
What's the relationship between these two definitions of martingales?
Durrett's definition is the general correct definition of a martingale, while the Wikipedia's definition is at best a "restricted definition". The qualifier "with respect to $\mathcal{...
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12
votes
Accepted
Distribution of $\frac{1}{1+X}$ if $X$ is Lognormal
I will answer a simplified version, so leave the generalization as an exercise. Let $Z$ be a standard normal random variable so $X=e^Z$ is standard lognormal. Since $X>0 $ we have $Y=\frac1{1+X}$ ...
- 82.8k
9
votes
Accepted
Cox PH linearity assumption: reading martingal residual plots
Quoting from Harrell's Regression Modeling Strategies, second edition, page 494:
When correlations among predictors are mild, plots of estimated predictor transformations without adjustment for ...
- 102k
8
votes
Accepted
Prove that a simple random walk is a martingale
\begin{align}
E[X_{t+1} \mid X_1, \ldots, X_t]
&= E[X_t + a_{t+1} \mid X_1, \ldots, X_t]
\\
&= X_t + E[a_{t+1} \mid X_1, \ldots, X_t]
\\
&= X_t
\end{align}
- 2,338
8
votes
ABRACADABRA Problem
The solution is reached thinking about the process as a martingale betting game with some conditions. At every keystroke, a different gambler jumps in the game with a $\small \$1$ bet. If any given ...
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7
votes
What's the relationship between these two definitions of martingales?
I agree with the other two answers here, but just wanted to highlight one reason why the $\sigma$-algebra based definition in Durrett, though more sophisticated, is beneficial. This definition ...
- 4,027
6
votes
The magic money tree problem
I don't think this is much different from the Martingale. In your case, there are no doubling bets, but the winning payout is 3x.
I coded a "living replica" of your tree. I run 10 simulations. In ...
- 169
6
votes
What's the relationship between these two definitions of martingales?
What does the notation $\mathbb E[X\mid X_1,\cdots,X_n]$ mean?
Go to the basics: consider $X\in \mathcal L_1(\Omega, \boldsymbol{\mathfrak B}, \mathbb P)$ and let $\mathcal G\subset\boldsymbol{\...
- 10.3k
5
votes
Accepted
Conditional expectation of random variables defined off of each other
Your first question is key, so let's focus on it. You are concerned about a bivariate random variable $(X_{n-1},X_n)$ with a probability distribution somehow defined by giving $X_{n-1}$ a ...
- 334k
5
votes
The magic money tree problem
I liked the answer given by Dave harris. just though I would come at the problem from a "low risk" perspective, rather than profit maximising
The random walk you are doing, assuming your fraction bet ...
- 25.3k
5
votes
The magic money tree problem
Problem statement
$\mathbf{M_t}$: the amount of money $M_t$ the gambler has at time $t$
$\mathbf{Y_t}$: Let $Y_t = \log_{10}(M_t)$ be the logarithm of $M_t$.
$\mathbf{Y_0}$: Let $Y_0 = 1$ be the ...
- 86.5k
5
votes
Accepted
How to show that this particular card game is a martingale?
$Z_{n-1}$ is the proportion of spades left in the deck before we make the $n^{th}$ draw; therefore, it is equal to the probability that the next card is a spade. Let us assume there are $k$ spades ...
- 41.1k
5
votes
Accepted
Distribution Function of Standard Normal is a U.I. Martingale?
Let $f(t,x) := \Phi\left(\frac{x}{\sqrt{T-t}}\right) = \Phi\circ g(t,x)$. By applying the chain rule we get
$$\frac{\partial f}{\partial x}(t,x) = \frac{1}{\sqrt{T-t}}\phi\left(\frac{x}{\sqrt{T-t}}\...
4
votes
log transform fixed PH in Cox model - how?
If you don't specify the correct linear form for a continuous predictor in a Cox proportional hazards (PH) model, it's quite possible to get this behavior. Tests for PH come after the regression ...
- 102k
4
votes
Reference Request: Book on Unit Root Theory
Johansen "Likelihood-based inference in cointegrated vector autoregressive models" (1995), Oxford University Press.
This is a pretty technical and theoretical treatment.
Juselius "The ...
4
votes
Accepted
Reference Request: Book on Unit Root Theory
In addition to the references by Richard Hardy, the following may be helpful:
Bierens, Unit Roots, Ch. 29 in "A Companion to Theoretical Econometrics", https://onlinelibrary.wiley.com/doi/10....
4
votes
Bayes update rule studied as an operator
There is a very large literature that touches on the core properties of the Bayesian updating mapping. Some of this literature is statistical and some of it goes further into the domain of philosophy....
- 133k
3
votes
Accepted
What is the meaning of $X \in \mathcal {F}$ in probability space ${\displaystyle (\Omega ,{\mathcal {F}},P)}$?
It's shorthand for saying $X$ is $\mathcal{F}$-measurable (i.e., that $\{\omega \in \Omega : X(\omega) \in B\} \in \mathcal{F}$ for every Borel set $B \subseteq \mathbb{R}$).
- 4,107
3
votes
Prove that a simple random walk is a martingale
Let $\{X_t\}_{t\geq 1}$ be a sequence of independent random variables such that $\Pr\{X_t=1\}=\Pr\{X_t=-1\}=1/2$. Define $\mathscr{F_t}=\sigma(X_1,\dots,X_t)$ and $M_t=X_1+\dots+X_t$. We have (...
- 25.1k
3
votes
log transform fixed PH in Cox model - how?
This paper (Proportionally Difficult: Testing for Nonproportional Hazards in Cox Models) covers this topic nicely
From page 190
The power of the test to detect nonproportionality is dependent on the ...
- 73
3
votes
Is it a valid algorithm to win at the casino roulette?
My level is too low to comment BUT I can post this answer so here it goes.
I see all of you spamming about finite bets and finite money. However, if we do the maths regarding ONLINE roulettes. I ...
3
votes
Accepted
Why white noise process and IID process are considered martingale
A stochastic process $\{X_t\}$ is called a martingale if
$$ \operatorname{E}[X_{t+1} \mid X_{t}, \ldots, X_1\} = X_t $$
That is, the expectation of the future conditional on the past is the present.
...
- 23k
3
votes
Martingales: Why must expected posterior equal prior?
I never heard the term before, but quick googling leads to papers by Kamenica and Gentzkow (2011a, 2011b) on "Bayesian persuasion", who define it as follows (p. 10 in the linked pdf to the 2011a paper)...
- 141k
3
votes
Accepted
Cox PH model: managing continuous variables and linearity assumption
Frank Harrell's recommendations get to the heart of the matter in general, but there may be issues with your dataset that are also giving you problems.
It seems that there must be outliers of ...
- 102k
3
votes
Cox PH model: managing continuous variables and linearity assumption
Don't trust those plots were the data run out, and add confidence bands to the plots.
You are using a multi-step analysis for which model uncertainty will not be recognized in the final model fitting ...
- 98.4k
3
votes
Martingale property & limiting distribution for frequency of last names
All that matters for the analysis are the numbers of Smiths and non-Smiths in the population at any time, say $k.$ Let these be $m$ and $n,$ respectively, so that $X_k=m/(m+n).$
The model supposes ...
- 334k
3
votes
Accepted
Uniformly Integrable Martingale
With all the given conditions, $\{\xi_n\}$ is not uniformly integrable.
First, because $\{\xi_n\}$ is a martingale and $\sup_n E[\xi_n] = 1 < \infty$, by the theorem you cited (i.e., Theorem 35.5 ...
- 21.2k
3
votes
About the requirement of iids to form a martingale
Probably the writer is interested in iid variables, and so not (at the moment) interested in generalizations ... but clearly some can be done. If you keep the independence, but drop identical, and of ...
- 82.8k
3
votes
Expected value of iid squared conditioned on sum
I suspect the only symbolic results you'll be able to find involve those distributions whose sum of iid random variables follow the same family of distributions. That is certainly the case for normal ...
- 4,505
Only top scored, non community-wiki answers of a minimum length are eligible