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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:

thanks.

- do a step of lenght 1 in the positive direction
- do a step of lenght 1 in the negative direction
- stay motionless (don't go anywhere, nor the negative nor the positive direction. It's 0-lenght step)

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# I Jump probability of a random walker

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