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Amsterdam October 28, 74 Dear Professor Chung, u The error in the standard deviation of the Poisson distribution is indeed very embarassing my apologies. In spite of the many errors it contains, I was very pleased ax with and would like to thank you for the c~opy~yo sent along with A~nnemarie. I hear you i 1 enjoyed your visit to England. J f written two very readable reports on random sets, one of which Pat Jaicobs had shown me just before I left Stanford. The other report Arrived in the mail today. Cinlar gave a talk on SemiMarkov processes (during which he defined a Chung process, by definition a countable state IP' with standard. transition matrix, all of whose paths are lower semi continuous from the right). In a private talk about Kingman'a I paper he mentioned that he had a simple proof of the formula without SLeplace transforms. This is contained in the second report which Arrived here today. (Discussion paper 104, section 7. ) j .5On one of our walks you said that you might be interested in ~ coming to Holland for a month or two. At the moment the chair of d topology is still vacant from April 1 till September 1 1975. The salary would be DFL 5900 (approx. & 3000) a month, ( or DF1 7800 I 1(approx. $3000 ) if you decide before December 1. ( In order to oiffe you a full Professor's salary, you have to be appointed to the chair and this needs the confirmation of 2 the cabinet minister for education However you would have to pay your ow~n travel expenses and the salary I? will be taxable in the US. You will be expected to give a seminar on any subject you like ( in probability theory). Classes nd i~f he second half of MJay, hence the months of April/viay would suit us very well. I would be very happy to see you argaikn, and we could go for walks along the canals, EC Please give my regards to your wife and Pat JTacobs. Dear Professor Chung, Thank you for your letter. I do not have the Brownian excursion ms at hand, but as far as my recollection goes the argument which you used in Amsterdam to prove the existence of local time for Brownian motion runs like this: 1) Define the maximum process for Brownian motion by M = max {Bs I Ocsit}. Then MB 1 IBl. (Levy's theorem.) 2) The inverse process to M is a Levy process V with stationary independent increments. Its Levy measure is dp~y) = dy//(2nyY3) by the scaling property of Brownian motion. 3) On combining 1) and 2) we see that the lengths of the intervals contiguous to the zero set of Brownian motion correspond to the jumps of the Levy process Y. Hence they form the ordinates of the points of a Poisson point process in [0,oo)X(0,ooi) with mean measure dxdy//(2ny3). 4) Let Nt(E) denote the number of complete excursions of IBI of duration exceeding E in the interval [O,t]. This is equal (in distribution) to the number of jumps of size exceeding E of the Levy process Y before it crosses the level t. 5) Fix s > 0 and let Ms(c) denote the number of jumps of the Levy process of height > c during the time interval (0,s]. Then (*) J(se~/2).MsIE) 4 s a.s. for E 4 0+ by the strong law of large numbers applied to the standard Poisson process Ms(2/xtt2), t,0. Outside some null set SZ0 c S convergence in (*) will hold for all rational s I 0, and hence for all s 2 0. On n\00 the relation (*) will also hold if s depends on w in particular if s denotes the time when the Levy process Y crosses the level t. By 4) this implies J(st/2).Nt(c) , St Bs. for E , 0+ where St is distributed like My . Finally note that Levy's theorem shows that the processes St, t20, and Mt, t,0, have the same distribution . To prove Levy's theorem with the number of excursions replaced by the occupation time of the interval (0,E), E 4 0+, one needs a bound on the second moment of the occupation time of the interval (0,E) for the normalized excursion. We are thinking of spending the summer in Fort Collins. would very much like to visit Stanford again, but am not sure how much travelling we will be doing I hope you and your wife are well We shall certainly make a point of dropping by if we come anywhere close to Stanford . Yours truly, Guus Balkema. Professor K.L. C~hung, Department of Mathematics, Stanford University, Stanford, Ca 94305 May 26, 1966. Dear Professor Chung, I have had a look at the material I have here on Brownian excursions, and think I can manage to write a ten page note on Paul Levy's definition of local time for Brownian motion along the lines of your Amsterdam lectures. We are leaving Ithace next week for a month long camping journey through the States, and hope to arrive at Fort Collins on July 1, so I could let you have it by July 20. The last weeks have been rather busy. With best regards, Guus Balkema. JZo ~ ) 