Palm Versions and Extra Heads
In this talk we shall consider Palm versions in the transparent context of doubly infinite coin tosses: a coin (possibly first randomly chosen from different biased coins) is flipped independently at each integer location. The general theory for stationary random measures will be indicated along the way with Poisson processes and Brownian motion as examples.
Although Palm versions are an important tool in both applied and theoretical probability, it is not too well-known that that there are in fact two Palm versions, with rather different interpretations. For lack of better terms we shall call them standard and modified. The standard version is obtained by conditioning on there being a head at the origin. The modified version is obtained by shifting the origin from 0 to a typical head, a head chosen uniformly at random (stretching the imagination) among all the (infinitely many) heads (choose first among n heads and then send n to infinity).
After considering these two versions in some detail, we turn to the related question of actually finding a typical head in tosses of a fair coin. This would be an Extra Head in the sense that if you remove it you would be left with a two-sided sequence of independent tosses of a fair coin, as if nothing had happened.
In part based on joint work with Guenter Last and Peter Moerters. The Extra Heads are due to Tom Liggett.