Distribution of a stochastic process
up vote
0
down vote
favorite
Is it possible to find a distribution of $X(t)$ for a fixed t by looking at a single sample path of X?
I'm kinda lost in the strong assumption, that it is not possible but then I remember that for time series analysis something related to likelihood method implies working with distributions and we have just 1 path there. Is there a way to prove the possibility or impossibility of such thing?
probability-distributions stochastic-processes stochastic-analysis
asked Oct 27 at 18:02
Makina
1,006113
add a comment |
up vote
0
down vote
favorite
Is it possible to find a distribution of $X(t)$ for a fixed t by looking at a single sample path of X?
I'm kinda lost in the strong assumption, that it is not possible but then I remember that for time series analysis something related to likelihood method implies working with distributions and we have just 1 path there. Is there a way to prove the possibility or impossibility of such thing?
probability-distributions stochastic-processes stochastic-analysis
asked Oct 27 at 18:02
Makina
1,006113
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Is it possible to find a distribution of $X(t)$ for a fixed t by looking at a single sample path of X?
I'm kinda lost in the strong assumption, that it is not possible but then I remember that for time series analysis something related to likelihood method implies working with distributions and we have just 1 path there. Is there a way to prove the possibility or impossibility of such thing?
probability-distributions stochastic-processes stochastic-analysis
asked Oct 27 at 18:02
Makina
1,006113
Is it possible to find a distribution of $X(t)$ for a fixed t by looking at a single sample path of X?
I'm kinda lost in the strong assumption, that it is not possible but then I remember that for time series analysis something related to likelihood method implies working with distributions and we have just 1 path there. Is there a way to prove the possibility or impossibility of such thing?
probability-distributions stochastic-processes stochastic-analysis
probability-distributions stochastic-processes stochastic-analysis
asked Oct 27 at 18:02
Makina
1,006113
asked Oct 27 at 18:02
Makina
1,006113
asked Oct 27 at 18:02
Makina
1,006113
asked Oct 27 at 18:02
Makina
1,006113
asked Oct 27 at 18:02
Makina
1,006113
1,006113
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
up vote
0
down vote
The concept of stationarity and ergodicity (see the wikipedia pages: Ergodicity,
Stationary ergodic process) are used to infer several characteristics of a stochastic process by observing only a single sample path of $X$. For example you can compute the moments of the process given a path ${x_k}$ as follows
$$
E[X_t^k] = lim_{Mto infty} frac{1}{M}sum_{t=1}^M x_t^k
$$
In case that the distribution is completely characterized by several or all of the moments, then (in this case) it is in principle possible to characterize the distribution of $X$ using the given path.
answered Nov 17 at 10:03
user144410
9391719
Actually for this question I thought that it would be best to try use simple examples. For example, say, $X_t = at$ where $a$ is equal to 1 or -1 with same probability. In this case by just looking at path $X_t = at$ for $a = 1$ I am technically unable to provide an answer to the question regarding distribution, since I do not know, by just looking at this one path for $a = 1$, if $a$ can take other values and at what probabilities if so.
– Makina
Nov 17 at 16:37
this process is not stationary and it is not ergodic. It is not possible to determine the moments using one sample path. Only in cases where the process is stationary and ergodic that the moments can be found from one sample path. If one is not able to find the moments, then determining the distribution is not possible.
– user144410
Nov 17 at 16:41
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The concept of stationarity and ergodicity (see the wikipedia pages: Ergodicity,
Stationary ergodic process) are used to infer several characteristics of a stochastic process by observing only a single sample path of $X$. For example you can compute the moments of the process given a path ${x_k}$ as follows
$$
E[X_t^k] = lim_{Mto infty} frac{1}{M}sum_{t=1}^M x_t^k
$$
In case that the distribution is completely characterized by several or all of the moments, then (in this case) it is in principle possible to characterize the distribution of $X$ using the given path.
answered Nov 17 at 10:03
user144410
9391719
Actually for this question I thought that it would be best to try use simple examples. For example, say, $X_t = at$ where $a$ is equal to 1 or -1 with same probability. In this case by just looking at path $X_t = at$ for $a = 1$ I am technically unable to provide an answer to the question regarding distribution, since I do not know, by just looking at this one path for $a = 1$, if $a$ can take other values and at what probabilities if so.
– Makina
Nov 17 at 16:37
this process is not stationary and it is not ergodic. It is not possible to determine the moments using one sample path. Only in cases where the process is stationary and ergodic that the moments can be found from one sample path. If one is not able to find the moments, then determining the distribution is not possible.
– user144410
Nov 17 at 16:41
add a comment |
up vote
0
down vote
The concept of stationarity and ergodicity (see the wikipedia pages: Ergodicity,
Stationary ergodic process) are used to infer several characteristics of a stochastic process by observing only a single sample path of $X$. For example you can compute the moments of the process given a path ${x_k}$ as follows
$$
E[X_t^k] = lim_{Mto infty} frac{1}{M}sum_{t=1}^M x_t^k
$$
In case that the distribution is completely characterized by several or all of the moments, then (in this case) it is in principle possible to characterize the distribution of $X$ using the given path.
answered Nov 17 at 10:03
user144410
9391719
Actually for this question I thought that it would be best to try use simple examples. For example, say, $X_t = at$ where $a$ is equal to 1 or -1 with same probability. In this case by just looking at path $X_t = at$ for $a = 1$ I am technically unable to provide an answer to the question regarding distribution, since I do not know, by just looking at this one path for $a = 1$, if $a$ can take other values and at what probabilities if so.
– Makina
Nov 17 at 16:37
this process is not stationary and it is not ergodic. It is not possible to determine the moments using one sample path. Only in cases where the process is stationary and ergodic that the moments can be found from one sample path. If one is not able to find the moments, then determining the distribution is not possible.
– user144410
Nov 17 at 16:41
add a comment |
up vote
0
down vote
up vote
0
down vote
The concept of stationarity and ergodicity (see the wikipedia pages: Ergodicity,
Stationary ergodic process) are used to infer several characteristics of a stochastic process by observing only a single sample path of $X$. For example you can compute the moments of the process given a path ${x_k}$ as follows
$$
E[X_t^k] = lim_{Mto infty} frac{1}{M}sum_{t=1}^M x_t^k
$$
In case that the distribution is completely characterized by several or all of the moments, then (in this case) it is in principle possible to characterize the distribution of $X$ using the given path.
answered Nov 17 at 10:03
user144410
9391719
The concept of stationarity and ergodicity (see the wikipedia pages: Ergodicity,
Stationary ergodic process) are used to infer several characteristics of a stochastic process by observing only a single sample path of $X$. For example you can compute the moments of the process given a path ${x_k}$ as follows
$$
E[X_t^k] = lim_{Mto infty} frac{1}{M}sum_{t=1}^M x_t^k
$$
In case that the distribution is completely characterized by several or all of the moments, then (in this case) it is in principle possible to characterize the distribution of $X$ using the given path.
answered Nov 17 at 10:03
user144410
9391719
edited Nov 17 at 18:59
answered Nov 17 at 10:03
user144410
9391719
answered Nov 17 at 10:03
user144410
9391719
answered Nov 17 at 10:03
user144410
9391719
9391719
Actually for this question I thought that it would be best to try use simple examples. For example, say, $X_t = at$ where $a$ is equal to 1 or -1 with same probability. In this case by just looking at path $X_t = at$ for $a = 1$ I am technically unable to provide an answer to the question regarding distribution, since I do not know, by just looking at this one path for $a = 1$, if $a$ can take other values and at what probabilities if so.
– Makina
Nov 17 at 16:37
this process is not stationary and it is not ergodic. It is not possible to determine the moments using one sample path. Only in cases where the process is stationary and ergodic that the moments can be found from one sample path. If one is not able to find the moments, then determining the distribution is not possible.
– user144410
Nov 17 at 16:41
add a comment |
Actually for this question I thought that it would be best to try use simple examples. For example, say, $X_t = at$ where $a$ is equal to 1 or -1 with same probability. In this case by just looking at path $X_t = at$ for $a = 1$ I am technically unable to provide an answer to the question regarding distribution, since I do not know, by just looking at this one path for $a = 1$, if $a$ can take other values and at what probabilities if so.
– Makina
Nov 17 at 16:37
this process is not stationary and it is not ergodic. It is not possible to determine the moments using one sample path. Only in cases where the process is stationary and ergodic that the moments can be found from one sample path. If one is not able to find the moments, then determining the distribution is not possible.
– user144410
Nov 17 at 16:41
Actually for this question I thought that it would be best to try use simple examples. For example, say, $X_t = at$ where $a$ is equal to 1 or -1 with same probability. In this case by just looking at path $X_t = at$ for $a = 1$ I am technically unable to provide an answer to the question regarding distribution, since I do not know, by just looking at this one path for $a = 1$, if $a$ can take other values and at what probabilities if so.
– Makina
Nov 17 at 16:37
Actually for this question I thought that it would be best to try use simple examples. For example, say, $X_t = at$ where $a$ is equal to 1 or -1 with same probability. In this case by just looking at path $X_t = at$ for $a = 1$ I am technically unable to provide an answer to the question regarding distribution, since I do not know, by just looking at this one path for $a = 1$, if $a$ can take other values and at what probabilities if so.
– Makina
Nov 17 at 16:37
this process is not stationary and it is not ergodic. It is not possible to determine the moments using one sample path. Only in cases where the process is stationary and ergodic that the moments can be found from one sample path. If one is not able to find the moments, then determining the distribution is not possible.
– user144410
Nov 17 at 16:41
this process is not stationary and it is not ergodic. It is not possible to determine the moments using one sample path. Only in cases where the process is stationary and ergodic that the moments can be found from one sample path. If one is not able to find the moments, then determining the distribution is not possible.
– user144410
Nov 17 at 16:41
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2973707%2fdistribution-of-a-stochastic-process%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
