Jump probability of a random walker

In summary: Now in what you've done above you have the numerator correct, but you have not divided by the total number of possibilities, which would be ##P(A)##.
  • #1
grquanti
17
0
Hello everybody.
I have a Markowian homogeneous random walk. Given the transition matrix of the chain, I know that

##P[ X(t) = i | X(t-1) = j ] ≡ P_{j→i}=T_{ij}##

where ##T## is the transition matrix and ##X(t)## is the position of the walker at time ##t##.
Given this formula, I think the following two formulas hold:

##P[ X(t) ≠ j | X(t-1) = i ] = ∑_{k≠j} P_{i→k} = ∑_{k≠j} T_{ki}##

and

##P[X(t) = j | X(t-1) ≠ j ] = ∑_{i≠j} P_{i→j} P_{i}(t-1) = ∑_{i≠j} T_{ji} P_{i}(t-1)##

First of all: it is right?
However the most importat question is: what can I say about

##P[X(t) ≠ j | X(t-1) ≠ j ]## ?

I think my question is quite general, however I let you note that: in my particular case, in a single time step the walker can:
  • do a step of length 1 in the positive direction
  • do a step of length 1 in the negative direction
  • stay motionless (don't go anywhere, nor the negative nor the positive direction. It's 0-lenght step)
thanks.
 
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  • #2
grquanti said:
First of all: it is right?
That second one is not correct.
Taking a sum works for the first case because the events ##X(t)=k## for ##k\neq j## are disjoint for different values of ##k##, so we can just add the probabilities. In the second case it is the conditional event ##X(t-1)\neq j## that has multiple parts and there is no theorem we can use about adding probabilities there.

To get a handle on what you need to do, try to write ##P[A\ |\ B\cup C]## in terms of ##P[A|B],P[ B ],P[A|C]## and ##P[C]## where ##B## and ##C## are disjoint events. You could then think of ##A## as the event ##X(t)=j## and ##B## and ##C## as the events ##X(t-1)=j-1,\ X(t-1)=j+1##.
 
  • #3
@andrewkirk: I don't understand the problem you have with the second expression. The initial probabilities are taken into account.
1->3 and 2->3 are disjoint events, for example, and you can sum their probabilities.

grquanti said:
However the most importat question is: what can I say about

##P[X(t) ≠ j | X(t-1) ≠ j ]## ?
Write down a 3x3 matrix (or larger) for T and see which transitions contribute. You can express it as double sum, but you can find a more compact expression.
 
  • #4
mfb said:
I don't understand the problem you have with the second expression. The initial probabilities are taken into account.
1->3 and 2->3 are disjoint events, for example, and you can sum their probabilities.
Certainly it is true that ##X(t-1)=1\wedge X(t)=3## and ##X(t-1)=2\wedge X(t)=3## are disjoint events. But the sum of their probabilities is ##Pr(X(t)=3\wedge (X(t-1)\in\{1,2\})\ )## not ##Pr(X(t)=3\ |\ X(t-1)\in\{1,2\}\ )##. To get the latter we need to divide by ##Pr(X(t-1)\in\{1,2\})##.
 
  • #5
mfb said:
You can express it as double sum, but you can find a more compact expression.
I think you are suggesting me this:

##P[ X(t) ≠ j | X(t-1) ≠ j ] = ∑_{i≠j}^{k≠j} P_{i→k}P_{i}(t) = ∑_{i≠j}^{k≠j} T_{ki}P_{i}(t)##

It seems reasonable to me, but am I ensured that this summation in not greater than 1?

andrewkirk said:
try to write ##P[A\ |\ B\cup C]## in terms of ##P[A|B],P[ B ],P[A|C]## and ##P[C]## where ##B## and ##C## are disjoint events.

I would say

##P( A | B ∪ C ) = \frac{P(A | B )P(B) + P( A | C )P(C)}{P(B) + P(C)}##

Is it right?
 
  • #6
grquanti said:
I would say

##P( A | B ∪ C ) = \frac{P(A | B )P(B) + P( A | C )P(C)}{P(B) + P(C)}##

Is it right?
Yes.

Now in what you've done above you have the numerator correct, but you have not divided by the denominator, as in this example you just did.
The denominator to divide by is
$$\sum_{k\neq j}P_k(t-1)$$
 
  • #7
@andrewkirk: Ah you are right, I missed the part about the conditional probability.
grquanti said:
It seems reasonable to me, but am I ensured that this summation in not greater than 1?
You know something about sums of Tij.
 
  • #8
mfb said:
You know something about sums of Tij.

Yes, I know that

##∑_i T_{ij}=1 ∀ j##

but I have to do (at least, I think I have to do)

##∑_{i≠j}^{k≠j} T_{ki}P_i##

For sure, the summation over a single index is not greater than one, but here I have to do the summation over two index...
 
  • #9
You can write it as explicit double sum, then it is easier to see which summation you can do first (as inequality if you just want to compare to 1). Afterwards you can use a similar equation for Pi to get rid of the remaining sum.
 
  • #10
Yes, you mean

##∑_{i≠j}^{k≠j} T_{ki}P_i = ∑_{i≠j} P_i ∑_{k≠j}T_{ki} ≤ ∑_{i}P_i = 1 ##

Thank you!
 
  • #11
$$∑_{i≠j}^{k≠j} T_{ki}P_i = ∑_{i≠j} P_i ∑_{k≠j}T_{ki} ≤ ∑_{i\neq j}P_i \leq 1$$
 
  • #12
##∑_{i≠j}^{k≠j}T_{ki}P_i = ∑_{i≠j}P_i∑_{k≠j}T_{ki} ≤ ∑_{i≠j}P_i ≤ ∑_i P_i = 1##
 

Related to Jump probability of a random walker

1. What is a random walker?

A random walker is a mathematical model used to represent the movement of a particle or object in a random or unpredictable manner. It is often used in physics, chemistry, biology, and other fields to study diffusion processes and to describe the behavior of complex systems.

2. What is jump probability in the context of a random walker?

Jump probability refers to the likelihood that a random walker will move from its current location to a neighboring point in a given time step. It is a measure of the randomness or unpredictability of the walker's movement and is often represented by the symbol p.

3. How is jump probability calculated in a random walker model?

Jump probability is typically calculated by dividing the number of possible jump locations by the total number of locations the walker can occupy. This can vary depending on the specific model being used, but in general, the more possible jump locations, the higher the jump probability.

4. Why is the study of jump probability important in random walker models?

Understanding jump probability is crucial in random walker models because it affects the overall behavior and dynamics of the system. It can provide insights into the diffusion rate, the likelihood of the walker reaching a certain location, and the overall stability or randomness of the system.

5. How does jump probability relate to other concepts in random walker models?

Jump probability is closely related to other concepts such as step size, step frequency, and step direction in random walker models. These factors all play a role in determining the overall jump probability and therefore have a significant impact on the behavior of the system.

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