What does $\delta_n$ refer to in "distribution of $T_k$", when defining mean recurrence time?

Question

Particularly

$$T_k:= \inf \{ n \geq 1: f_n=k\}$$

then my notes say:

$$\mathbb{P}(f_n=k, f_{n-1} \not=k, ..., f_1 \not=k | f_0=k)= f_{kk}^{(n)}$$

and that the distribution of $T_k$ is

$$\mathbb{P}_{T_k} = \sum_{n=1}^{\infty} f_{kk}^{(n)} \delta_n$$

Is this some "standard" $\delta$?

@kimchilover I think I need an example to get this. E.g. to have some $k$ and then compute the first few terms of the distribution. — mavavilj, May 21 '19 at 00:41

kimchi lover · Answer 1 · 2019-05-21T00:53:40.997

1

$\delta_n$ is a standard notation for the probability distribution that puts all its mass at the point $n$.

So your second displayed formula implies that $P(T_k=n) = f_{kk}^{(n)}$.

edited May 21 '19 at 00:53

answered May 21 '19 at 00:49

kimchi lover

Shouldn't $\delta_n$ signify "value" by definition of probability distribution. Since $f_{kk}^{(n)}$ is the probability of that value, then what's the value? – mavavilj May 21 '19 at 00:51

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