Point Processes, Week 3, Wednesday

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Faculy of Science Poin Processes, Week 3, Wednesday Niels Richard Hansen Deparmen of Mahemaical Sciences Sepember 17, 2014 Slide 1/8

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Poisson processes If N is a homogeneous Poisson process wih rae λ, N N s Poi(λ( s)) for s <, which implies ha P(N N s = 1) = λ( s) + O(( s) 2 ). Slide 2/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Poisson processes If N is a homogeneous Poisson process wih rae λ, N N s Poi(λ( s)) for s <, which implies ha [Big-O! No lile-o] P(N N s = 1) = λ( s) + O(( s) 2 ). Slide 2/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Poisson processes If N is a homogeneous Poisson process wih rae λ, N N s Poi(λ( s)) for s <, which implies ha [Big-O! No lile-o] P(N N s = 1) = λ( s) + O(( s) 2 ). If N is inhomogeneous wih a coninuous rae funcion λ : R + R + hen P(N N s = 1) = λ s ( s) + o(( s)). Slide 2/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Poisson processes If N is a homogeneous Poisson process wih rae λ, N N s Poi(λ( s)) for s <, which implies ha [Big-O! No lile-o] P(N N s = 1) = λ( s) + O(( s) 2 ). If N is inhomogeneous wih a coninuous rae funcion λ : R + R + hen [Here we have lile-o] P(N N s = 1) = λ s ( s) + o(( s)). Slide 2/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Inhomogeneous Poisson processes Theorem If µ is a uni rae homogeneous Poisson random measure on R 2 + and λ : R + R + is a locally inegrable funcion, hen he process N defined by N = µ ({(s, z) (0, ] R + z λ(s)}) is a Poisson process wih rae λ. Slide 3/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Inhomogeneous Poisson processes λ N Slide 4/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Inhomogeneous Poisson processes λ N Slide 4/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Inhomogeneous Poisson processes λ N Slide 4/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Hisories N s s Slide 5/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Hisories N s s N 0:,s s Slide 5/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Hisories N s s N 0:,s s Slide 5/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Hisories N s s N 0:,s s Slide 5/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Examples of hisory dependen inensiies Markov dependence λ = h(n 0:, ) = h(n ). Slide 6/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Examples of hisory dependen inensiies Markov dependence λ = h(n 0:, ) = h(n ). Backward moving averages λ = 0 h( s)dn s. Slide 6/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Examples of hisory dependen inensiies Markov dependence λ = h(n 0:, ) = h(n ). Backward moving averages λ = 0 h( s)dn s. Backward recurren ime dependence λ = H(U 1,..., U k ), U k he ime since k h las even Slide 6/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Inegral equaion Theorem Assume ha λ : R + D lbv R + is measurable and saisfies ha for any couning process N, λ(, N 0: ) is càglàd. If µ is a locally bounded couning measure here exiss a random variable T : Ω (0, ] and a couning process N on [0, T ) ha solves N = 1 (0,] (s)1 [0,λ(s,N0:s )](x)µ(ds, dx). Slide 7/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Soluions of inegral equaion λ N Slide 8/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Soluions of inegral equaion λ N Slide 8/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

d e p a r m e n o f m a h e m a i c a l s c i e n c e s Soluions of inegral equaion λ N Slide 8/8 Niels Richard Hansen Poin Processes, Week 3, Wednesday Sepember 17, 2014

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