Dan Asimov <dasimov@earthlink.net> wrote:
E.g., is this asking what the expected number of records will be, before record-keeping began and assuming no knowledge of new records after they are made? Looks that way.
Sorry if I was unclear. There are lots of records, such as hottest recorded July 24th in <your city>. If you look forwards from the beginning of record keeping, there's a 100% chance that the first July 24th will be the hottest ever officially recorded (and the coldest), a 50% chance that the second year's July 24th will be the hottest ever recorded, a 33.3% chance that the third year's July 24th will be the hottest ever recorded, etc. So after N years you'd expect to have set about log(N) such records. (I can never remember whether you're supposed to add or subtract Euler's Constant to get the harmonic sum, so I round that constant to zero. :-) ) If, by contrast, you look backwards from the present, there's always just one hottest July 24th ever recorded, and, assuming no climate change, it's equally likely to be on any of the years since record keeping began. Since there are more days in a year than there are years since record keeping began, it's unusual for the current year not to contain at least one hottest recorded <some date>, and at least one coldest recorded <some other date>. Eventually, of course, weather records will have been kept for eons, and it will be very rare to set a new record. (Assuming, even less plausibly over geological time scales, that there's no climate change. Record cold days will probably be especially unusual after the sun enters its red giant stage.)
Many real-world records, like fastest marathon (https://en.wikipedia.org/wiki/Marathon_world_record_progression#Men), show periods of steady, almost differentiable growth over the years, but also have plenty of what look like discontinuities.
I suspect the discontinuities in marathon records are due to sudden improvements in technique and equipment. Similarly with all other human accomplishments. On the other hand, the literally exponential improvements in the number of known decimal digits of pi have been fairly smooth over the past 70 years or so: 10^1 1400 10^2 1706 306 10^3 1949 243 10^4 1958 9 10^5 1961 3 10^6 1973 12 10^7 1983 10 10^8 1987 4 10^9 1989 2 10^10 1997 8 10^11 1999 2 10^12 2002 3 10^13 2011 9