Integrating the path of a random walk
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Suppose I've got a random walk up and down in one dimension. It starts at $S_0=k>0$ and makes a move at every integer time $t$. So:
$$S_t=k+sum_{i=1}^t X_i$$
Each move ($X_i$) has mean 0 and the same distribution (maybe a normal distribution or whatever's easiest to work with). And the walk goes on forever.
I know that the probability that $S$ will return to 0 is arbitrarily close to 1 (it's basically Gambler's Ruin). But I'm interested in how much of the time it spends in the lead as $trightarrow infty$, and how big the lead is.
I want to calculate the integral of $S_t$ from 0 to $infty$. And the probability that this would be positive, the probability that it would be bounded above and below, and the probability that it would be negative. How would I do this?
I initially thought that all 3 probabilities would have to be 0, but then came across the arcsine laws (https://en.wikipedia.org/wiki/Arcsine_laws_(Wiener_process) & http://www2.math.uu.se/~sea/kurser/stokprocmn1/slumpvandring_eng.pdf). These seem to imply that there's a roughly 30% probability that my random walk will spend >80% of the time above 0 in the long run (and just under 30% that it will spend >80% of the time under 0). And that would make the integral positively infinite (negatively infinite if it's under 0 most of the time). Have I interpreted the arcsine laws correctly? And do they imply that the integral is almost always either positively infinite, negatively infinite, or bounded both above and below?
integration random-variables random-walk
asked Dec 14 '18 at 15:14
HW.HW.
61
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add a comment |
$begingroup$
Suppose I've got a random walk up and down in one dimension. It starts at $S_0=k>0$ and makes a move at every integer time $t$. So:
$$S_t=k+sum_{i=1}^t X_i$$
Each move ($X_i$) has mean 0 and the same distribution (maybe a normal distribution or whatever's easiest to work with). And the walk goes on forever.
I know that the probability that $S$ will return to 0 is arbitrarily close to 1 (it's basically Gambler's Ruin). But I'm interested in how much of the time it spends in the lead as $trightarrow infty$, and how big the lead is.
I want to calculate the integral of $S_t$ from 0 to $infty$. And the probability that this would be positive, the probability that it would be bounded above and below, and the probability that it would be negative. How would I do this?
I initially thought that all 3 probabilities would have to be 0, but then came across the arcsine laws (https://en.wikipedia.org/wiki/Arcsine_laws_(Wiener_process) & http://www2.math.uu.se/~sea/kurser/stokprocmn1/slumpvandring_eng.pdf). These seem to imply that there's a roughly 30% probability that my random walk will spend >80% of the time above 0 in the long run (and just under 30% that it will spend >80% of the time under 0). And that would make the integral positively infinite (negatively infinite if it's under 0 most of the time). Have I interpreted the arcsine laws correctly? And do they imply that the integral is almost always either positively infinite, negatively infinite, or bounded both above and below?
integration random-variables random-walk
asked Dec 14 '18 at 15:14
HW.HW.
61
$endgroup$
$begingroup$
Your interpretation of the arcsine law seems reasonable for the amount of time positive (ignoring $k$). As for the integral of $S_t$, you might want to be aware that the integral of a standard Wiener process from $0$ to $t$ is distributed like a normally distributed random variable with mean $0$ and variance $frac{t^3}{3}$ so the probability that it is within given finite bounds tends towards $0$ as $t$ increases without limit
$endgroup$
– Henry
Dec 16 '18 at 18:33
add a comment |
$begingroup$
Suppose I've got a random walk up and down in one dimension. It starts at $S_0=k>0$ and makes a move at every integer time $t$. So:
$$S_t=k+sum_{i=1}^t X_i$$
Each move ($X_i$) has mean 0 and the same distribution (maybe a normal distribution or whatever's easiest to work with). And the walk goes on forever.
I know that the probability that $S$ will return to 0 is arbitrarily close to 1 (it's basically Gambler's Ruin). But I'm interested in how much of the time it spends in the lead as $trightarrow infty$, and how big the lead is.
I want to calculate the integral of $S_t$ from 0 to $infty$. And the probability that this would be positive, the probability that it would be bounded above and below, and the probability that it would be negative. How would I do this?
I initially thought that all 3 probabilities would have to be 0, but then came across the arcsine laws (https://en.wikipedia.org/wiki/Arcsine_laws_(Wiener_process) & http://www2.math.uu.se/~sea/kurser/stokprocmn1/slumpvandring_eng.pdf). These seem to imply that there's a roughly 30% probability that my random walk will spend >80% of the time above 0 in the long run (and just under 30% that it will spend >80% of the time under 0). And that would make the integral positively infinite (negatively infinite if it's under 0 most of the time). Have I interpreted the arcsine laws correctly? And do they imply that the integral is almost always either positively infinite, negatively infinite, or bounded both above and below?
integration random-variables random-walk
asked Dec 14 '18 at 15:14
HW.HW.
61
$endgroup$
Suppose I've got a random walk up and down in one dimension. It starts at $S_0=k>0$ and makes a move at every integer time $t$. So:
$$S_t=k+sum_{i=1}^t X_i$$
Each move ($X_i$) has mean 0 and the same distribution (maybe a normal distribution or whatever's easiest to work with). And the walk goes on forever.
I know that the probability that $S$ will return to 0 is arbitrarily close to 1 (it's basically Gambler's Ruin). But I'm interested in how much of the time it spends in the lead as $trightarrow infty$, and how big the lead is.
I want to calculate the integral of $S_t$ from 0 to $infty$. And the probability that this would be positive, the probability that it would be bounded above and below, and the probability that it would be negative. How would I do this?
I initially thought that all 3 probabilities would have to be 0, but then came across the arcsine laws (https://en.wikipedia.org/wiki/Arcsine_laws_(Wiener_process) & http://www2.math.uu.se/~sea/kurser/stokprocmn1/slumpvandring_eng.pdf). These seem to imply that there's a roughly 30% probability that my random walk will spend >80% of the time above 0 in the long run (and just under 30% that it will spend >80% of the time under 0). And that would make the integral positively infinite (negatively infinite if it's under 0 most of the time). Have I interpreted the arcsine laws correctly? And do they imply that the integral is almost always either positively infinite, negatively infinite, or bounded both above and below?
integration random-variables random-walk
integration random-variables random-walk
asked Dec 14 '18 at 15:14
HW.HW.
61
asked Dec 14 '18 at 15:14
HW.HW.
61
asked Dec 14 '18 at 15:14
HW.HW.
61
asked Dec 14 '18 at 15:14
HW.HW.
61
asked Dec 14 '18 at 15:14
HW.HW.
61
61
$begingroup$
Your interpretation of the arcsine law seems reasonable for the amount of time positive (ignoring $k$). As for the integral of $S_t$, you might want to be aware that the integral of a standard Wiener process from $0$ to $t$ is distributed like a normally distributed random variable with mean $0$ and variance $frac{t^3}{3}$ so the probability that it is within given finite bounds tends towards $0$ as $t$ increases without limit
$endgroup$
– Henry
Dec 16 '18 at 18:33
add a comment |
$begingroup$
Your interpretation of the arcsine law seems reasonable for the amount of time positive (ignoring $k$). As for the integral of $S_t$, you might want to be aware that the integral of a standard Wiener process from $0$ to $t$ is distributed like a normally distributed random variable with mean $0$ and variance $frac{t^3}{3}$ so the probability that it is within given finite bounds tends towards $0$ as $t$ increases without limit
$endgroup$
– Henry
Dec 16 '18 at 18:33
$begingroup$
Your interpretation of the arcsine law seems reasonable for the amount of time positive (ignoring $k$). As for the integral of $S_t$, you might want to be aware that the integral of a standard Wiener process from $0$ to $t$ is distributed like a normally distributed random variable with mean $0$ and variance $frac{t^3}{3}$ so the probability that it is within given finite bounds tends towards $0$ as $t$ increases without limit
$endgroup$
– Henry
Dec 16 '18 at 18:33
$begingroup$
Your interpretation of the arcsine law seems reasonable for the amount of time positive (ignoring $k$). As for the integral of $S_t$, you might want to be aware that the integral of a standard Wiener process from $0$ to $t$ is distributed like a normally distributed random variable with mean $0$ and variance $frac{t^3}{3}$ so the probability that it is within given finite bounds tends towards $0$ as $t$ increases without limit
$endgroup$
– Henry
Dec 16 '18 at 18:33
add a comment |
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$begingroup$
Your interpretation of the arcsine law seems reasonable for the amount of time positive (ignoring $k$). As for the integral of $S_t$, you might want to be aware that the integral of a standard Wiener process from $0$ to $t$ is distributed like a normally distributed random variable with mean $0$ and variance $frac{t^3}{3}$ so the probability that it is within given finite bounds tends towards $0$ as $t$ increases without limit
$endgroup$
– Henry
Dec 16 '18 at 18:33