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18

I think that this question is best approached through careful modelling, rather than pure mathematics. Here's the approach I took. I don't claim that this is the perfect approach by any means, but it's a start. Spoiler: My simulations give a rate of approximately once every 0.66 years, for a population of 7 billion people who share US mortality statistics. ...

8

The Gerontology Research Group keeps records I brought this up on the GRG and Louis Epstein posted the table "CHRONOLOGICAL OLDEST LIVING LISTED PERSONS (Since 1955)". I extracted the final column, death dates, and formatted it and extracted the intervals between the death dates of each person, reasoning that if the Oldest Person In The World who died ...

4

Super-simple answer, depending only on some crude properties of mortality rates: It appears that at very advanced ages, mortality rate varies only slowly with age and is on the order of 50% per year. (If this is wrong, everything else in this answer is wrong. On the other hand, if it's right then it's all we need.) It will basically always be the case that ...

2

These are the basics of the exponential distributions, probably explained on the WP page. If $T$ is exponential with parameter $a$, then $F_T(t)=1-\mathrm e^{-at}$ for every $t\geqslant0$ hence $a=F_T'(0^+)$ and $F_T(t)=1-\mathrm e^{-F_T'(0^+)t}$. Furthermore, for every $t\geqslant0$, $P[t\leqslant T\leqslant t+s\mid T\geqslant t]=1-\mathrm e^{-as}\sim ... 2 To summarize a discussion in the comments, the OP uses as a textbook Introduction to Probability, by Grinstead and Snell. The authors explain how to find the stationary distribution(s) of a Markov chain on a finite number of states from its transition matrix. Section 11.3 Ergodic Markov chains gives the theory in Theorem 11.8, and explains the practice ... 2 An irreducible Markov chain is recurrent if and only if every non-negative, superharmonic function$f$is constant. Here is a typical non-negative, superharmonic function$f$: select a state, say$0$and define$f(x)=\mathbb{P}_x(T<\infty)$where$T:=\inf(n\geq 0: X_n=0)$is the hitting time of$0$. We want to figure out whether this function is ... 2 Your initial$P^1$matrix is has first row$[p,1-p]$and second row the reverse of that. Your goal matrix for$P^n$also has its entries in the same form, with first row say$[a_n,b_n]$and second row the reverse of that. So an approach would be to multiply the matrix for$P^n$by the matrix$P$, and its top row will be $$[p a_n + (1-p)b_n, (1-p)a_n +p ... 1 Not sure one can get explicit formulas for E[X_t] but anyway, your function f is not rich enough to capture the dynamics of the process. The canonical way to go is to consider u(t,s)=E[s^{X_t}] for every t\geqslant0 and, say, every s in (0,1). Then, pending some errors in computations done too quickly, the function u solves an ... 1 First, form the transition matrix corresponding to the random walk with 0 and 3 as absorbing states:$$P=\begin{bmatrix} 1&0&0&0\\2/5&0&3/5&0\\0&2/5&0&3/5\\0&0&0&1 \end{bmatrix}.$$Then, rearrange to have the absorbing states first:$$\tilde{P}=\left[ \begin{array}{c|c} I & 0 \\ \hline S & Q ... 1 Super-simpler answer: "How often will the oldest living person die?" is just a paraphrase of "How long, typically, will the oldest person (on earth/in country/in any given population) live?" According to one source (http://life-span.findthedata.org/q/101/166/What-is-the-life-expectancy-of-a-100-year-old-man), life expectancy for a man is 2.07 years, for a ... 1 1. Let$s=(2-p-q)^{-1}$, then$\pi_0=(1-q)s$,$\pi_1=(1-p)s$, defines a stationary distribution$\pi$. If the initial distribution is$\pi$, at each step the distribution is$\pi$hence the probability that a jump occurs is $$r=(1-p)\pi_0+(1-q)\pi_1=2(1-p)(1-q)s.$$ In particular, the mean number of jumps during$T$periods is exactly$rT$. 2. By a ... 1 A second order Markov chain is a random process$(X_n)_n$on an alphabet$A$, whose distribution is specified by its transition probabilities$Q(x\mid y,z)=P[X_n=x\mid X_{n-1}=y,X_{n-2}=z]$, for every$(x,y,z)$in$A\times A\times A$(and by an initial distribution on$A\times A$). A stationary distribution of$(X_n)$is a probability measure$\pi$on ... 1 Concerning references, I've seen the problem of first passage time distributions treated in two books, unfortunately both are in German: Nollau, V.: Semi-Markovsche Prozesse, H. Deutsch (1981). Störmer, H.: Semi-Markoff-Prozesse mit endlich vielen Zuständen, Springer Lecture Notes in Operations Research and Math. Systems, Vol. 34 (1970). As the titles ... 1$T_i$is the expected number of steps that we'll take to get from$i$to$1$. So if you think about any one of the paths from$i$to$1\$, we can break it into "the first step" and "the rest of the steps". Clearly the total number steps in the path is "the first step" plus "the rest of the steps". So the first step in the path takes us to any one of the ...

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