STIT Tessellations via Martingales
|
|
- Esmond Arnold
- 5 years ago
- Views:
Transcription
1 STIT Tessellations via Martingales Christoph Thäle Department of Mathematics, Osnabrück University (Germany) Stochastic Geometry Days, Lille, Christoph Thäle (Uni Osnabrück) STIT Tessellations 1 / 62
2 This is part of joint work with Tomasz Schreiber from Toruń ( ) Christoph Thäle (Uni Osnabrück) STIT Tessellations 2 / 62
3 Overview The model Martingales and STIT tessellations First applications Exact variance of the total surface area Exact variances for other intrinsic volumes Asymptotic expressions Limit theorems Christoph Thäle (Uni Osnabrück) STIT Tessellations 3 / 62
4 Construction and Key Properties We interpret a random tessellation as a special random closed set in R d. STIT tessellations Y (t) formally arise as limits of rescaled iterations of tessellations. In bounded windows W there is an explicit construction of Y (t, W ) in terms of waiting times. There is also an explicit global construction of STIT tessellations Y (t) (Poisson point process on a complicated state space). Important for later considerations: These constructions allow an interpretation of Y (t, W ) (or Y (t)) as a Markov process on (0, ) with values in the space of tessellations. Christoph Thäle (Uni Osnabrück) STIT Tessellations 4 / 62
5 Waiting Time Construction Let W R d be a compact convex polytope and Λ a non-degenerate locally finite hyperplane measure. Assign to W an exponentially distributed lifetime with parameter Λ([W ]). Upon expiry of this life time, a hyperplane is chosen according to Λ([W ]) 1 Λ( [W ]), is introduced in W and splits W into two polyhedral sub-cells W + and W. The construction is now continued recursively and independently in W + and W until some deterministic time threshold t > 0 is reached. Until time t > 0 there was constructed a random tessellation Y (t, W ) inside W with polyhedral cells. Christoph Thäle (Uni Osnabrück) STIT Tessellations 5 / 62
6 Waiting Time Construction Christoph Thäle (Uni Osnabrück) STIT Tessellations 6 / 62
7 Waiting Time Construction Christoph Thäle (Uni Osnabrück) STIT Tessellations 7 / 62
8 Waiting Time Construction Christoph Thäle (Uni Osnabrück) STIT Tessellations 8 / 62
9 Waiting Time Construction Christoph Thäle (Uni Osnabrück) STIT Tessellations 9 / 62
10 Waiting Time Construction Christoph Thäle (Uni Osnabrück) STIT Tessellations 10 / 62
11 Waiting Time Construction Christoph Thäle (Uni Osnabrück) STIT Tessellations 11 / 62
12 Waiting Time Construction Christoph Thäle (Uni Osnabrück) STIT Tessellations 12 / 62
13 Isotropic random STIT Tessellation in 2D Christoph Thäle (Uni Osnabrück) STIT Tessellations 13 / 62
14 Anisotropic random STIT Tessellation in 2D Christoph Thäle (Uni Osnabrück) STIT Tessellations 14 / 62
15 Isotropic random STIT Tessellation in 3D Christoph Thäle (Uni Osnabrück) STIT Tessellations 15 / 62
16 Anisotropic random STIT Tessellation in 3D Christoph Thäle (Uni Osnabrück) STIT Tessellations 16 / 62
17 Extension to R d We cannot start with a single hyperplane at time 0! But, Nagel und Weiß (2005) have shown that Y (t, W ) is spatially consistent, i.e for V W convex, we have Y (t, V ) = D Y (t, W ) V. Thus, by a consistency theorem there exists a random closed set Y (t) in R d with Y (t) W = D Y (t, W ). Christoph Thäle (Uni Osnabrück) STIT Tessellations 17 / 62
18 Parameter and special cases The law of Y (t) is characterized by t and Λ. In general, Y (t) is neither stationary nor isotropic. If Λ is translation-invariant, then Y (t) is stationary and thus a STIT tessellation. If Λ is the the isometry-invariant measure Λ iso, then Y (t) is stationary and isotropic. Christoph Thäle (Uni Osnabrück) STIT Tessellations 18 / 62
19 Generator of Y (t, W ) The random process Y (t, W ) is a pure jump Markov process on the space of tessellations in W and its generator L := L Λ;W is given by: LF (Y ) = [F (Y {f }) F (Y )] Λ(dH), [W ] }{{} f Cells(Y H) add-one cost with F measurable on the space of tessellation in W for which the integral is well defined. Christoph Thäle (Uni Osnabrück) STIT Tessellations 19 / 62
20 A derived Martingale From the standard theory (Dynkin s formula) it follows that the following real-valued process is a martingale with respect to the natural filtration of Y (t, W ): t F (Y (t, W )) LF (Y (s, W ))ds 0 for F D(L). Christoph Thäle (Uni Osnabrück) STIT Tessellations 20 / 62
21 A more special case Denote by MaxFacets(Y ) the set of maximal facets of the tessellation Y. We regard now the special function Σ φ (Y ) := φ(f ) f MaxFacets(Y ) with φ measurable and bounded on the space of (d 1)-polytopes in W. Applications of the general result from the last slide shows that t Σ φ (Y (t, W )) φ(f )Λ(dH)ds 0 [W ] is a martingale. f Cells(Y H) Christoph Thäle (Uni Osnabrück) STIT Tessellations 21 / 62
22 First Consequences Consider a STIT tessellation Y (t, W ) = Y (tλ, W ) and the measures M Y (t,w ) := δ c, M Y (t,w ) := EM Y (t,w ) c Cells(Y (t,w )) and likewise M PHT(tΛ,W ) and M PHT(tΛ,W ). Christoph Thäle (Uni Osnabrück) STIT Tessellations 22 / 62
23 First Consequences Consider a STIT tessellation Y (t, W ) = Y (tλ, W ) and the measures M Y (t,w ) := c Cells(Y (t,w )) δ c, M Y (t,w ) := EM Y (t,w ) and likewise M PHT(tΛ,W ) and M PHT(tΛ,W ). Define further F Y (t,w ) k := δ f, F Y (t,w ) k := EF Y (t,w ) k, k = 1,..., d 1, and F PHT(tΛ,W ) k := for k = 1,..., d 1,. f MaxFaces k (Y (t,w )) f Faces k (PHT(tΛ,W )) δ f, F PHT(tΛ,W ) k := EF PHT(tΛ,W ) k Christoph Thäle (Uni Osnabrück) STIT Tessellations 23 / 62
24 First Consequences Theorem: It holds and F Y (t,w ) k for k = 1,..., d 1. M Y (t,w ) = M PHT(tΛ,W ) t = (d k)2 d k s FPHT(sΛ,W ) k ds Christoph Thäle (Uni Osnabrück) STIT Tessellations 24 / 62
25 First Consequences Theorem: It holds and F Y (t,w ) k for k = 1,..., d 1. M Y (t,w ) = M PHT(tΛ,W ) t = (d k)2 d k s FPHT(sΛ,W ) k This means that STITs and PHTs have the same typical cell distribution, but the spatial arrangement of the cells is different in both tessellation models. ds Christoph Thäle (Uni Osnabrück) STIT Tessellations 25 / 62
26 First Consequences Theorem: It holds and F Y (t,w ) k for k = 1,..., d 1. Idea of the proof: M Y (t,w ) = M PHT(tΛ,W ) t = (d k)2 d k s FPHT(sΛ,W ) k (1) uses a uniqueness theorem for the solution of a certain operator differential equation on a space of measures together with a derived martingale. (2) uses a derived martingale, part (1), Slivnyak s theory for Poisson point process, scaling properties and adjacency relationships for Poisson hyperplane tessellations. ds Christoph Thäle (Uni Osnabrück) STIT Tessellations 26 / 62
27 Application: Typical k-dimensional maximal faces Let us consider the isotropic case, i.e. Λ = Λ iso and Y (t) = Y (tλ iso ). Fix a measurable and translation-invariant ϕ k : MaxFaces k R, ϕ 0. Define the ϕ k -density of Y (t): ϕ k (Y (t)) = lim r 1 r d Vol d (W ) E Apply now the theory from above. f MaxFaces k (Y (t,rw )) ϕ k (f ). Christoph Thäle (Uni Osnabrück) STIT Tessellations 27 / 62
28 Application: Typical k-dimensional maximal faces We have ϕ k (Y (t)) = (d k)2 d k 1 t For ϕ k 1 this yields 0 1 s ϕ k(pht(s))ds. t ( ) ( ) N k = (d k)2 d k 1 1 d d κd 1 κ d s d ds 0 s k dκ d = (d k)2 d k 1 κ ( ) ( ) d d d κd 1 t d. d k dκ d Christoph Thäle (Uni Osnabrück) STIT Tessellations 28 / 62
29 Application: Typical k-dimensional maximal faces Now the same procedure for ϕ k (f ) := 1[ ](f c(f )) with some center function c, f MaxFaces k (Y (t)). We get N k Q Y (t) k = (d k)2 d k 1 t 0 1 s NPHT(s) k Q PHT(s) k ds (the distribution of the naked polytopes). Inserting the values for N k and N PHT(s) k t Q Y (t) k = 0 ds d 1 t d gives Q PHT(s) k ds. The mixing distribution is a beta-distribution on (0, t) with parameters d and 1. Christoph Thäle (Uni Osnabrück) STIT Tessellations 29 / 62
30 Application: Typical maximal segment Let p (d) l (x) be the length density of the typical maximal Segment in R d. Then p (d) l (x) = t 0 λ 1 se λ 1sx dsd 1 t d ds = Γ ( ) d 2 λ 1 = ( πγ d+1 ). 2 d (λ 1 t) d x d+1 γ(d + 1, λ 1tx) with 1 ( d = 2: t 2 x 3 π 2 (π 2 + 2πtx + 2t 2 x 2 )e 2 tx) π, mean π t, no higher moments exist. 3 ( d = 3: t 3 x 4 48 ( tx + 6t 2 x 2 + t 3 x 3 )e 1 tx) 2, mean 3 t, variance 24, no higher moments exist. t 2 Christoph Thäle (Uni Osnabrück) STIT Tessellations 30 / 62
31 Application: Typical k-dimensional maximal faces For the mean j-th intrinsic volume of the typical k-dimensional maximal face, 0 j k, we have d EV j (I k ) = (d j)κ j ( k j ) ( 2 πγ ( ) d+1 2 Γ ( ) d 2 ) j 1 t j. Christoph Thäle (Uni Osnabrück) STIT Tessellations 31 / 62
32 Next goal The next goal is to use the martingale approach to study second-order properties of the STIT tessellations, such as variances and covariances. Christoph Thäle (Uni Osnabrück) STIT Tessellations 32 / 62
33 Another derived Martingale - Notation Let φ be bounded and measurable on the space of (d 1)-polytopes. Define Σ φ (Y (t, W )) := φ(f ) and A φ (Y (t, W )) := f MaxFacets(Y (t,w )) [W ] f Cells(Y (t,w ) H) φ(f )Λ(dH) and put Σ φ (Y (t, W )) := Σ φ (Y (t, W )) EΣ φ (Y (t, W )), Ā φ (Y (t, W )) := A φ (t, W ) EA φ (t, W ). Christoph Thäle (Uni Osnabrück) STIT Tessellations 33 / 62
34 Another derived Martingale Let φ be bounded and measurable on the space of (d 1)-polytopes. Define and Σ φ (Y (t, W )) := A φ (Y (t, W )) := A second-order martingale The random process Σ 2 φ (Y (t, W )) t 0 f MaxFacets(Y (t,w )) [W ] f Cells(Y (t,w ) H) φ(f ) φ(f )Λ(dH). t A φ 2(Y (s, W ))ds 2 Ā φ (Y (s, W )) Σ φ (Y (s, W ))ds 0 is a martingale wrt. filtration induced by (Y (t, W )) t>0. Christoph Thäle (Uni Osnabrück) STIT Tessellations 34 / 62
35 Variance of the Total Surface Area A second-order martingale The random process Σ 2 φ (Y (t, W )) t 0 t A φ 2(Y (s, W ))ds 2 Ā φ (Y (s, W )) Σ φ (Y (s, W ))ds 0 is a martingale wrt. filtration induced by (Y (t, W )) t>0. Take φ(f ) := Vol d 1 (f ). Then, A Vold 1 (Y (, W )) is constant, thus Ā Vold 1 0. Taking expectations in the above formula, we get Var Vol d 1 (Y (t, W )) = t It remains to calculate EA Vol 2 d 1 (Y (s, W )). 0 EA Vol 2 d 1 (Y (s, W ))ds. Christoph Thäle (Uni Osnabrück) STIT Tessellations 35 / 62
36 Calculation of EA Vol 2 d 1 (Y (s, W )) EA Vol 2 (Y (s, W )) d 1 = E = E = [W ] f Cells(Y (s,w ) H) [W ] [W ] W H W H W H W H Vol 2 d 1 (f )Λ(dH) 1[x, y in the same cell of Y (s, W ) H]dxdyΛ(dH) e sλ([xy]) dxdyλ(dh), since STIT tessellations have Poisson typical cells and Y (s) H is also a STIT tessellation. Christoph Thäle (Uni Osnabrück) STIT Tessellations 36 / 62
37 Variance of the Total Surface Area Putting together and Var Vol d 1 (Y (t, W )) = EA Vol 2 (Y (s, W )) = d 1 [W ] t 0 EA Vol 2 d 1 (Y (s, W ))ds W H W H e sλ([xy]) dxdyλ(dh), we get by integration Var Vol d 1 (Y (t, W )) = [W ] W H W H 1 e tλ([xy]) dxdyλ(dh). Λ([xy]) Christoph Thäle (Uni Osnabrück) STIT Tessellations 37 / 62
38 Variance of the Total Surface Area Assume that Λ = Λ iso. Then, the affine Blaschke-Petkantschin formula implies g(x, y)dxdyλ iso (dh) Take now [W ] W H W H = (d 1)κ d 1 dκ d W W g(x, y) x y dxdy. g(x, y) = 1 exp( tλ iso([xy])) Λ iso ([xy]) = 1 e 2κ d 1 dκ d 2κ d 1 t x y dκ d x y. Christoph Thäle (Uni Osnabrück) STIT Tessellations 38 / 62
39 Variance of the Total Surface Area Theorem (Schreiber and T. 2010) For a stationary and isotropic STIT tessellation Y (t) we have Var(Vol d 1 (Y (t, W ))) = d 1 1 e 2κ d 1 t x y dκ d 2 W W x y 2 dxdy = d(d 1)κ ( ) d γ 2 W (r)r d 3 1 e 2κ d 1 tr dκ d dr. 0 Christoph Thäle (Uni Osnabrück) STIT Tessellations 39 / 62
40 Variance of the Total Surface Area Theorem (Schreiber and T. 2010) For a stationary and isotropic STIT tessellation Y (t) we have Var(Vol d 1 (Y (t, W ))) = d(d 1)κ ( ) d γ 2 W (r)r d 3 1 e 2κ d 1 tr dκ d dr. Corollary (Weiß, Ohser, Nagel, d = 2 and Schreiber and T. in general) The pair-correlation function g d (r) of the random surface measure of a stationary and isotropic Y (t) is given by g d (r) = 1 + d 1 ( ) 2t 2 r 2 1 e 2κ d 1 tr dκ d. 0 Christoph Thäle (Uni Osnabrück) STIT Tessellations 40 / 62
41 Variances for Intrinsic Volumes We consider only the stationary and isotropic case Λ = Λ iso. Our aim is to calculate Var Σ Vj (Y (t, W )), where, recall, Σ Vj (Y (t, W )) = V j (f ). f MaxFacets(Y (t,w )) Christoph Thäle (Uni Osnabrück) STIT Tessellations 41 / 62
42 Variances for Intrinsic Volumes We consider only the stationary and isotropic case Λ = Λ iso. Our aim is to calculate Var Σ Vj (Y (t, W )), where, recall, Σ Vj (Y (t, W )) = V j (f ). f MaxFacets(Y (t,w )) However, we start with F j (Y (t, W )) := V j (c). c Cells(Y (t,w )) If f splits c into c + and c, we have F j (Y (t, W ) {f }) F j (Y (t, W )) = V j (c + )+V j (c ) V j (c) = V j (f ), thus F j (Y (t, W )) = Σ Vj (Y (t, W )) + V j (W ). Christoph Thäle (Uni Osnabrück) STIT Tessellations 42 / 62
43 Variances for Intrinsic Volumes Recall, that A Vj (Y (t, W )) = [W ] f Cells(Y (t,w ) H) V j (f )Λ iso (dh). Crofton s formula and F j (Y (t, W )) = Σ Vj (Y (t, W )) + V j (W ) implies now A Vj (Y (t, W )) = V j (c H)Λ iso (dh) = c Cells(Y (t,w )) c Cells(Y (t,w )) [W ] γ j+1 V j+1 (c) = γ j+1 F j+1 (Y (t, W )) = γ j+1 (Σ Vj+1 (Y (t, W )) + V j+1 (W )), thus Ā Vj (Y (t, W )) = γ j+1 Σ Vj+1 (Y (t,w )). Christoph Thäle (Uni Osnabrück) STIT Tessellations 43 / 62
44 Another Martingale Denote Σ φ;t := Σ φ (Y (t, W )). By considering once again the time-augmented process (Y (t, W ), t) and Dynkin s formula one shows that t t Σ Vi ;t Σ Vj ;t A Vi V j ;sds [ĀV i ;s Σ Vj ;s + ĀV j ;s Σ Vi ;s]ds 0 0 is a martingale wrt. filtration induced by (Y (t, W )) t>0. Christoph Thäle (Uni Osnabrück) STIT Tessellations 44 / 62
45 Another Martingale Denote Σ φ;t := Σ φ (Y (t, W )). By considering once again the time-augmented process (Y (t, W ), t) and Dynkin s formula one shows that t t Σ Vi ;t Σ Vj ;t A Vi V j ;sds [Ā Vi ;s Σ Vj ;s + Ā Vj ;s Σ Vi ;s]ds 0 0 is a martingale wrt. filtration induced by (Y (t, W )) t>0. Using that Ā Vj (Y (t, W )) = γ j+1 Σ Vj+1 (Y (t,w )), also t Σ Vi ;t Σ Vj ;t A Vi V j ;sds 0 is a martingale as well. t 0 [γ i+1 ΣVi+1 ;s Σ Vj ;s + γ j+1 ΣVi ;s Σ Vj+1 ;s]ds Christoph Thäle (Uni Osnabrück) STIT Tessellations 45 / 62
46 The Recursion Formula Using that ĀV j (Y (t, W )) = γ j+1 ΣVj+1 (Y (t,w )), also t Σ Vi ;t Σ Vj ;t A Vi V j ;sds 0 is a martingale as well. t Taking expectation, we get the recursion formula + t 0 Cov(Σ Vi ;t, Σ Vj ;t) = 0 [γ i+1 ΣVi+1 ;s Σ Vj ;s + γ j+1 ΣVi ;s Σ Vj+1 ;s]ds t 0 EA Vi V j ;sds [γ i+1 Cov(Σ Vi+1 ;s, Σ Vj ;s) + γ j+1 Cov(Σ Vi ;s, Σ Vj+1 ;s)]ds. The recursion terminates, since Σ Vd ;t 0, which allows an explicit expression for all Cov(V i;t, V j;t ). Recall that Var Σ Vd 1 ;t = t 0 EA V 2 d 1 ;sds. Christoph Thäle (Uni Osnabrück) STIT Tessellations 46 / 62
47 Exact Expression I n (f ; t) := = t s (n 1)! sn 1 0 t Theorem (Schreiber and T. 2010) 0 f (s n )ds n ds n 1 ds 1 (t s) n 1 f (s)ds. The covariance between Σ Vd 1 k ;t and Σ Vd 1 l ;t for k, l {0,..., d 1} of a stationary and isotropic random STIT tessellation Y (t, W ) is given by ( k i=m+1 Cov(Σ Vd 1 k ;t, Σ Vd 1 l ;t) = γ d i ) l j=n+1 γ d j k m=0 n=0 l ( ) k + l m n k m I k+l m n+1 (EA Vd 1 m V d 1 n ;( ); t). Christoph Thäle (Uni Osnabrück) STIT Tessellations 47 / 62
48 Asyptotic Expressions: The Problem Take a convex body W R d and consider the sequence W R := R W as R. How does Var Σ Vi (Y (t, W R )) behave, as R? Which terms dominate the expression? How is the formula influenced by the geometry of W? The answers for d = 2 and d 3 are different! For simplicity we consider only d 3. Christoph Thäle (Uni Osnabrück) STIT Tessellations 48 / 62
49 Asymptotics Proposition For k, l, m, n {0,..., d 1}, with t fixed we have with p := k + l m n + 1 I p (EA W R V d 1 m V d 1 n ;( ) ; t) = O(R 2(d 1) m n ), if m + n d 3, O(R d log R), if m + n = d 2, O(R 2d 1 m n ), if m + n d 1. Christoph Thäle (Uni Osnabrück) STIT Tessellations 49 / 62
50 Asymptotic picture Proposition For k, l, m, n {0,..., d 1}, with t fixed we have with p := k + l m n + 1 I p (EA W R V d 1 m V d 1 n ;( ) ; t) = Corollary O(R 2(d 1) m n ), if m + n d 3, O(R d log R), if m + n = d 2, O(R 2d 1 m n ), if m + n d 1. The asymptotic covariance Cov(Σ Vd 1 k ;t, Σ Vd 1 l ;t) is dominated by the term n = m = 0, i.e. by I k+l+1 (EA W R ; t) = I k+l (Var Σ W Vol 2 R V d 1 d 1 ). Christoph Thäle (Uni Osnabrück) STIT Tessellations 50 / 62
51 Influence of W The geometry of W is for d 3 reflected by either one of the (non-additive) functionals dxdy E 2 (W ) = x y 2 or since I d 1 (W ) = dκ d 2 E 2 (W ) = W L W Vol d 1 1 (W L)dL, 2 (d 1)(d 2) I d 1(W ). For d = 2 only the area Vol 2 (W ) matters, but not the precise shape of W. But here some logarithm enters the discussion... Christoph Thäle (Uni Osnabrück) STIT Tessellations 51 / 62
52 The Asymptotic result Theorem (Schreiber and T. 2010) Asymptotically, as R, we have for k, l {0,..., d 1} ( k ) 1 γ d i d 2 i=1 Cov(Σ W R V d 1 k ;t, ΣW R V d 1 l ;t ) = l j=1 γ d j tk+l k!l! I d 1(W )R 2(d 1) + O(R 2d 3 ) and for k = l ( k ) 2 Var(Σ W R V d 1 k ;t ) = 1 t 2k γ d i d 2 (k!) 2 I d 1(W )R 2(d 1) + O(R 2d 3 ). i=1 Christoph Thäle (Uni Osnabrück) STIT Tessellations 52 / 62
53 Limit theorems We restrict ourself from now on to the case of the total surface area, but limit theorems for other functionals are also available. We approach the problem in two different (but closely related) settings: We can consider the surface area constructed by facets born after an initial time instant s 0 > 0. It is also natural to ask for the surface area constructed by all facets. Interestingly both approaches lead to results of very different qualitative nature! Christoph Thäle (Uni Osnabrück) STIT Tessellations 53 / 62
54 First limit theorem Theorem (Schreiber and T. 2010) For each s 0 > 0, the random variable 1 R d/2 [(Vol d 1(Y (1, W R )) EVol d 1 (Y (1, W R ))) (Vol d 1 (Y (s 0, W R )) EVol d 1 (Y (s 0, W R )))] converges, as R, in law to N (0, V W (Vol d 1, Λ) 1 s 0 s 1 d ds), a normal distribution with mean 0 and variance V W (Vol d 1, Λ) 1 s 0 s 1 d ds, where V W (Vol d 1, Λ) is explicitly known. Christoph Thäle (Uni Osnabrück) STIT Tessellations 54 / 62
55 First limit theorem Theorem (Schreiber and T. 2010) For each s 0 > 0, the random variable 1 R d/2 [(Vol d 1(Y (1, W R )) EVol d 1 (Y (1, W R ))) (Vol d 1 (Y (s 0, W R )) EVol d 1 (Y (s 0, W R )))] converges, as R, in law to N (0, V W (Vol d 1, Λ) 1 s 0 s 1 d ds), a normal distribution with mean 0 and variance V W (Vol d 1, Λ) 1 s 0 s 1 d ds, where V W (Vol d 1, Λ) is explicitly known. Idea of the proof: Use STIT scaling and ergodicity to calculate the asymptotic variance. Check the assumptions of the martingale CLT for cadlag martingales. Christoph Thäle (Uni Osnabrück) STIT Tessellations 55 / 62
56 Planar CLT Theorem (Schreiber and T. 2010) We have for the stationary and isotropic STIT tessellation Y (1) in the plane 1 R log R [Vol 1(Y (1, W R )) EVol 1 (Y (1, W R ))] = N (0, π Vol 2 (W )), where = means convergence in law. Christoph Thäle (Uni Osnabrück) STIT Tessellations 56 / 62
57 Planar CLT Theorem (Schreiber and T. 2010) We have for the stationary and isotropic STIT tessellation Y (1) in the plane 1 R log R [Vol 1(Y (1, W R )) EVol 1 (Y (1, W R ))] = N (0, π Vol 2 (W )), where = means convergence in law. Idea of the proof: Use a space-time scaling to construct an asymptotically equivalent martingale with good properties. Check again the assumptions of the martingale CLT for cadlag martingales. Christoph Thäle (Uni Osnabrück) STIT Tessellations 57 / 62
58 Higher dimensions Theorem (Schreiber and T. 2010) For d > 2, Λ = d + i=1 δ re i +ei dr and W = [0, 1] d, R 2(d 1) [Vol d 1 (1, W R ) EVol d 1 (1, W R )] converges, as R, to a non-gaussian square-integrable random variable Ξ(W ) with explicitly known variance. Christoph Thäle (Uni Osnabrück) STIT Tessellations 58 / 62
59 Higher dimensions Theorem (Schreiber and T. 2010) For d > 2, Λ = d + i=1 δ re i +ei dr and W = [0, 1] d, R 2(d 1) [Vol d 1 (1, W R ) EVol d 1 (1, W R )] converges, as R, to a non-gaussian square-integrable random variable Ξ(W ) with explicitly known variance. Idea of the proof: Show by a conditioning argument that Ξ(W ) satisfies P(Ξ(W ) > ξ) = exp( Θ(ξ log ξ)), whereas a Gaussian random variable has tails of order exp( Θ(ξ 2 )). Christoph Thäle (Uni Osnabrück) STIT Tessellations 59 / 62
60 Why this difference? Overall variance is of order R 2(d 1) for d > 2 and of order R 2 log R for d = 2. The variance contribution after the initial time instant s 0 > 0 is of order R d for all d. Thus, for d = 2 the initial segments contribute a negligible amount to the total variance. But, in higher dimensions d > 2, the big bang phase is crucial and dominates the scenery. Christoph Thäle (Uni Osnabrück) STIT Tessellations 60 / 62
61 Thank you for your attention! Christoph Thäle (Uni Osnabrück) STIT Tessellations 61 / 62
Normal approximation of geometric Poisson functionals
Institut für Stochastik Karlsruher Institut für Technologie Normal approximation of geometric Poisson functionals (Karlsruhe) joint work with Daniel Hug, Giovanni Peccati, Matthias Schulte presented at
More informationarxiv: v1 [math.mg] 27 Jan 2016
On the monotonicity of the moments of volumes of random simplices arxiv:67295v [mathmg] 27 Jan 26 Benjamin Reichenwallner and Matthias Reitzner University of Salzburg and University of Osnabrueck Abstract
More informationConvex hull of a random point set
Convex hull of a random point set Pierre Calka Journées nationales 26 GdR Informatique Mathématique Villetaneuse, 20 January 26 Outline Random polytopes: an overview Main results: variance asymptotics
More informationPoisson-Voronoi graph on a Riemannian manifold
Poisson-Voronoi graph on a Riemannian manifold Pierre Calka October 3, 2018 Random graphs and its applications for networks Saint-Étienne Poisson-Voronoi tessellation in R n P λ homogeneous Poisson point
More informationGravitational allocation to Poisson points
Gravitational allocation to Poisson points Sourav Chatterjee joint work with Ron Peled Yuval Peres Dan Romik Allocation rules Let Ξ be a discrete subset of R d. An allocation (of Lebesgue measure to Ξ)
More informationAsymptotic theory for statistics of the Poisson-Voronoi approximation
Submitted to Bernoulli arxiv: arxiv:1503.08963 Asymptotic theory for statistics of the Poisson-Voronoi approximation CHRISTOPH THÄLE1 and J. E. YUKICH 2 1 Faculty of Mathematics, Ruhr University Bochum,
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationIntroduction The Poissonian City Variance and efficiency Flows Conclusion References. The Poissonian City. Wilfrid Kendall.
The Poissonian City Wilfrid Kendall w.s.kendall@warwick.ac.uk Mathematics of Phase Transitions Past, Present, Future 13 November 2009 A problem in frustrated optimization Consider N cities x (N) = {x 1,...,
More informationTheorem 2.1 (Caratheodory). A (countably additive) probability measure on a field has an extension. n=1
Chapter 2 Probability measures 1. Existence Theorem 2.1 (Caratheodory). A (countably additive) probability measure on a field has an extension to the generated σ-field Proof of Theorem 2.1. Let F 0 be
More informationSome Aspects of Universal Portfolio
1 Some Aspects of Universal Portfolio Tomoyuki Ichiba (UC Santa Barbara) joint work with Marcel Brod (ETH Zurich) Conference on Stochastic Asymptotics & Applications Sixth Western Conference on Mathematical
More informationTopics in Stochastic Geometry. Lecture 4 The Boolean model
Institut für Stochastik Karlsruher Institut für Technologie Topics in Stochastic Geometry Lecture 4 The Boolean model Lectures presented at the Department of Mathematical Sciences University of Bath May
More informationEstimation of arrival and service rates for M/M/c queue system
Estimation of arrival and service rates for M/M/c queue system Katarína Starinská starinskak@gmail.com Charles University Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics
More information{σ x >t}p x. (σ x >t)=e at.
3.11. EXERCISES 121 3.11 Exercises Exercise 3.1 Consider the Ornstein Uhlenbeck process in example 3.1.7(B). Show that the defined process is a Markov process which converges in distribution to an N(0,σ
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationAsymptotics for posterior hazards
Asymptotics for posterior hazards Pierpaolo De Blasi University of Turin 10th August 2007, BNR Workshop, Isaac Newton Intitute, Cambridge, UK Joint work with Giovanni Peccati (Université Paris VI) and
More informationLARGE DEVIATION PROBABILITIES FOR SUMS OF HEAVY-TAILED DEPENDENT RANDOM VECTORS*
LARGE EVIATION PROBABILITIES FOR SUMS OF HEAVY-TAILE EPENENT RANOM VECTORS* Adam Jakubowski Alexander V. Nagaev Alexander Zaigraev Nicholas Copernicus University Faculty of Mathematics and Computer Science
More informationFunctional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals
Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals Noèlia Viles Cuadros BCAM- Basque Center of Applied Mathematics with Prof. Enrico
More informationStability of the two queue system
Stability of the two queue system Iain M. MacPhee and Lisa J. Müller University of Durham Department of Mathematical Science Durham, DH1 3LE, UK (e-mail: i.m.macphee@durham.ac.uk, l.j.muller@durham.ac.uk)
More informationRandom Locations of Periodic Stationary Processes
Random Locations of Periodic Stationary Processes Yi Shen Department of Statistics and Actuarial Science University of Waterloo Joint work with Jie Shen and Ruodu Wang May 3, 2016 The Fields Institute
More informationMarch 1, Florida State University. Concentration Inequalities: Martingale. Approach and Entropy Method. Lizhe Sun and Boning Yang.
Florida State University March 1, 2018 Framework 1. (Lizhe) Basic inequalities Chernoff bounding Review for STA 6448 2. (Lizhe) Discrete-time martingales inequalities via martingale approach 3. (Boning)
More informationBRANCHING PROCESSES 1. GALTON-WATSON PROCESSES
BRANCHING PROCESSES 1. GALTON-WATSON PROCESSES Galton-Watson processes were introduced by Francis Galton in 1889 as a simple mathematical model for the propagation of family names. They were reinvented
More informationLecture 2. We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales.
Lecture 2 1 Martingales We now introduce some fundamental tools in martingale theory, which are useful in controlling the fluctuation of martingales. 1.1 Doob s inequality We have the following maximal
More informationLogarithmic scaling of planar random walk s local times
Logarithmic scaling of planar random walk s local times Péter Nándori * and Zeyu Shen ** * Department of Mathematics, University of Maryland ** Courant Institute, New York University October 9, 2015 Abstract
More informationFast-slow systems with chaotic noise
Fast-slow systems with chaotic noise David Kelly Ian Melbourne Courant Institute New York University New York NY www.dtbkelly.com May 12, 215 Averaging and homogenization workshop, Luminy. Fast-slow systems
More informationP (A G) dp G P (A G)
First homework assignment. Due at 12:15 on 22 September 2016. Homework 1. We roll two dices. X is the result of one of them and Z the sum of the results. Find E [X Z. Homework 2. Let X be a r.v.. Assume
More informationApproximately Gaussian marginals and the hyperplane conjecture
Approximately Gaussian marginals and the hyperplane conjecture Tel-Aviv University Conference on Asymptotic Geometric Analysis, Euler Institute, St. Petersburg, July 2010 Lecture based on a joint work
More informationThe concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.
The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes
More informationExponential Distribution and Poisson Process
Exponential Distribution and Poisson Process Stochastic Processes - Lecture Notes Fatih Cavdur to accompany Introduction to Probability Models by Sheldon M. Ross Fall 215 Outline Introduction Exponential
More informationComparison Method in Random Matrix Theory
Comparison Method in Random Matrix Theory Jun Yin UW-Madison Valparaíso, Chile, July - 2015 Joint work with A. Knowles. 1 Some random matrices Wigner Matrix: H is N N square matrix, H : H ij = H ji, EH
More information. Find E(V ) and var(v ).
Math 6382/6383: Probability Models and Mathematical Statistics Sample Preliminary Exam Questions 1. A person tosses a fair coin until she obtains 2 heads in a row. She then tosses a fair die the same number
More informationIntroduction to Spatial Data and Models
Introduction to Spatial Data and Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Biostatistics, School of Public Health, University of Minnesota, Minneapolis, Minnesota, U.S.A. 2 Department of Forestry
More informationGravitational allocation to Poisson points
Gravitational allocation to Poisson points Ron Peled joint work with Sourav Chatterjee Yuval Peres Dan Romik Allocation rules Let Ξ be a discrete subset of R d. Allocation rules Let Ξ be a discrete subset
More informationDifferent scaling regimes for geometric Poisson functionals
Different scaling regimes for geometric Poisson functionals Raphaël Lachièze-Rey, Univ. Paris 5 René Descartes Joint work with Giovanni Peccati, University of Luxembourg isson Point Processes: Malliavin
More informationFiltrations, Markov Processes and Martingales. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition
Filtrations, Markov Processes and Martingales Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition David pplebaum Probability and Statistics Department,
More information9 Brownian Motion: Construction
9 Brownian Motion: Construction 9.1 Definition and Heuristics The central limit theorem states that the standard Gaussian distribution arises as the weak limit of the rescaled partial sums S n / p n of
More informationL n = l n (π n ) = length of a longest increasing subsequence of π n.
Longest increasing subsequences π n : permutation of 1,2,...,n. L n = l n (π n ) = length of a longest increasing subsequence of π n. Example: π n = (π n (1),..., π n (n)) = (7, 2, 8, 1, 3, 4, 10, 6, 9,
More informationarxiv: v1 [math.pr] 15 Oct 2012
STIT Tessellations have trivial tail σ-algebra arxiv:121.3917v1 [math.pr] 15 Oct 212 S. Martínez Departamento Ingeniería Matemática and Centro Modelamiento Matemático, Universidad de Chile, UMI 287 CNRS,
More informationSTAT 200C: High-dimensional Statistics
STAT 200C: High-dimensional Statistics Arash A. Amini May 30, 2018 1 / 59 Classical case: n d. Asymptotic assumption: d is fixed and n. Basic tools: LLN and CLT. High-dimensional setting: n d, e.g. n/d
More informationStein s method and zero bias transformation: Application to CDO pricing
Stein s method and zero bias transformation: Application to CDO pricing ESILV and Ecole Polytechnique Joint work with N. El Karoui Introduction CDO a portfolio credit derivative containing 100 underlying
More informationPoisson line processes. C. Lantuéjoul MinesParisTech
Poisson line processes C. Lantuéjoul MinesParisTech christian.lantuejoul@mines-paristech.fr Bertrand paradox A problem of geometrical probability A line is thrown at random on a circle. What is the probability
More informationThe Wiener Itô Chaos Expansion
1 The Wiener Itô Chaos Expansion The celebrated Wiener Itô chaos expansion is fundamental in stochastic analysis. In particular, it plays a crucial role in the Malliavin calculus as it is presented in
More informationOn a class of stochastic differential equations in a financial network model
1 On a class of stochastic differential equations in a financial network model Tomoyuki Ichiba Department of Statistics & Applied Probability, Center for Financial Mathematics and Actuarial Research, University
More informationRecap. Probability, stochastic processes, Markov chains. ELEC-C7210 Modeling and analysis of communication networks
Recap Probability, stochastic processes, Markov chains ELEC-C7210 Modeling and analysis of communication networks 1 Recap: Probability theory important distributions Discrete distributions Geometric distribution
More informationA CLT FOR MULTI-DIMENSIONAL MARTINGALE DIFFERENCES IN A LEXICOGRAPHIC ORDER GUY COHEN. Dedicated to the memory of Mikhail Gordin
A CLT FOR MULTI-DIMENSIONAL MARTINGALE DIFFERENCES IN A LEXICOGRAPHIC ORDER GUY COHEN Dedicated to the memory of Mikhail Gordin Abstract. We prove a central limit theorem for a square-integrable ergodic
More informationPattern size in Gaussian fields from spinodal decomposition
Pattern size in Gaussian fields from spinodal decomposition Luigi Amedeo Bianchi (joint work with D. Blömker and P. Düren) 8th Random Dynamical Systems Workshop Bielefeld, 6th November 2015 Research partially
More informationMarch 25, 2010 CHAPTER 2: LIMITS AND CONTINUITY OF FUNCTIONS IN EUCLIDEAN SPACE
March 25, 2010 CHAPTER 2: LIMIT AND CONTINUITY OF FUNCTION IN EUCLIDEAN PACE 1. calar product in R n Definition 1.1. Given x = (x 1,..., x n ), y = (y 1,..., y n ) R n,we define their scalar product as
More information1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
More information(B(t i+1 ) B(t i )) 2
ltcc5.tex Week 5 29 October 213 Ch. V. ITÔ (STOCHASTIC) CALCULUS. WEAK CONVERGENCE. 1. Quadratic Variation. A partition π n of [, t] is a finite set of points t ni such that = t n < t n1
More informationarxiv: v3 [math.mg] 3 Nov 2017
arxiv:702.0069v3 [math.mg] 3 ov 207 Random polytopes: central limit theorems for intrinsic volumes Christoph Thäle, icola Turchi and Florian Wespi Abstract Short and transparent proofs of central limit
More informationExponential families also behave nicely under conditioning. Specifically, suppose we write η = (η 1, η 2 ) R k R p k so that
1 More examples 1.1 Exponential families under conditioning Exponential families also behave nicely under conditioning. Specifically, suppose we write η = η 1, η 2 R k R p k so that dp η dm 0 = e ηt 1
More informationConcentration of Measures by Bounded Couplings
Concentration of Measures by Bounded Couplings Subhankar Ghosh, Larry Goldstein and Ümit Işlak University of Southern California [arxiv:0906.3886] [arxiv:1304.5001] May 2013 Concentration of Measure Distributional
More informationarxiv:math/ v4 [math.pr] 12 Apr 2007
arxiv:math/612224v4 [math.pr] 12 Apr 27 LARGE CLOSED QUEUEING NETWORKS IN SEMI-MARKOV ENVIRONMENT AND ITS APPLICATION VYACHESLAV M. ABRAMOV Abstract. The paper studies closed queueing networks containing
More informationNumerical Methods with Lévy Processes
Numerical Methods with Lévy Processes 1 Objective: i) Find models of asset returns, etc ii) Get numbers out of them. Why? VaR and risk management Valuing and hedging derivatives Why not? Usual assumption:
More informationExercises in stochastic analysis
Exercises in stochastic analysis Franco Flandoli, Mario Maurelli, Dario Trevisan The exercises with a P are those which have been done totally or partially) in the previous lectures; the exercises with
More informationCovariance function estimation in Gaussian process regression
Covariance function estimation in Gaussian process regression François Bachoc Department of Statistics and Operations Research, University of Vienna WU Research Seminar - May 2015 François Bachoc Gaussian
More informationLIMIT THEORY FOR PLANAR GILBERT TESSELLATIONS
PROBABILITY AND MATHEMATICAL STATISTICS Vol. 31, Fasc. 1 (2011), pp. 149 160 LIMIT THEORY FOR PLANAR GILBERT TESSELLATIONS BY TOMASZ S C H R E I B E R (TORUŃ) AND NATALIA S O JA (TORUŃ) Abstract. A Gilbert
More informationI forgot to mention last time: in the Ito formula for two standard processes, putting
I forgot to mention last time: in the Ito formula for two standard processes, putting dx t = a t dt + b t db t dy t = α t dt + β t db t, and taking f(x, y = xy, one has f x = y, f y = x, and f xx = f yy
More informationGARCH processes continuous counterparts (Part 2)
GARCH processes continuous counterparts (Part 2) Alexander Lindner Centre of Mathematical Sciences Technical University of Munich D 85747 Garching Germany lindner@ma.tum.de http://www-m1.ma.tum.de/m4/pers/lindner/
More informationThe number of Euler tours of random directed graphs
The number of Euler tours of random directed graphs Páidí Creed School of Mathematical Sciences Queen Mary, University of London United Kingdom P.Creed@qmul.ac.uk Mary Cryan School of Informatics University
More informationA Variational Analysis of a Gauged Nonlinear Schrödinger Equation
A Variational Analysis of a Gauged Nonlinear Schrödinger Equation Alessio Pomponio, joint work with David Ruiz Dipartimento di Meccanica, Matematica e Management, Politecnico di Bari Variational and Topological
More informationProperties of the Autocorrelation Function
Properties of the Autocorrelation Function I The autocorrelation function of a (real-valued) random process satisfies the following properties: 1. R X (t, t) 0 2. R X (t, u) =R X (u, t) (symmetry) 3. R
More informationIntroduction to Spatial Data and Models
Introduction to Spatial Data and Models Sudipto Banerjee 1 and Andrew O. Finley 2 1 Department of Forestry & Department of Geography, Michigan State University, Lansing Michigan, U.S.A. 2 Biostatistics,
More information25.1 Ergodicity and Metric Transitivity
Chapter 25 Ergodicity This lecture explains what it means for a process to be ergodic or metrically transitive, gives a few characterizes of these properties (especially for AMS processes), and deduces
More informationThe main results about probability measures are the following two facts:
Chapter 2 Probability measures The main results about probability measures are the following two facts: Theorem 2.1 (extension). If P is a (continuous) probability measure on a field F 0 then it has a
More informationTheory and Applications of Stochastic Systems Lecture Exponential Martingale for Random Walk
Instructor: Victor F. Araman December 4, 2003 Theory and Applications of Stochastic Systems Lecture 0 B60.432.0 Exponential Martingale for Random Walk Let (S n : n 0) be a random walk with i.i.d. increments
More informationStochastic Differential Equations
Chapter 5 Stochastic Differential Equations We would like to introduce stochastic ODE s without going first through the machinery of stochastic integrals. 5.1 Itô Integrals and Itô Differential Equations
More informationQ = (c) Assuming that Ricoh has been working continuously for 7 days, what is the probability that it will remain working at least 8 more days?
IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2005, Professor Whitt, Second Midterm Exam Chapters 5-6 in Ross, Thursday, March 31, 11:00am-1:00pm Open Book: but only the Ross
More informationLimit theorems for Hawkes processes
Limit theorems for nearly unstable and Mathieu Rosenbaum Séminaire des doctorants du CMAP Vendredi 18 Octobre 213 Limit theorems for 1 Denition and basic properties Cluster representation of Correlation
More informationAdditive functionals of infinite-variance moving averages. Wei Biao Wu The University of Chicago TECHNICAL REPORT NO. 535
Additive functionals of infinite-variance moving averages Wei Biao Wu The University of Chicago TECHNICAL REPORT NO. 535 Departments of Statistics The University of Chicago Chicago, Illinois 60637 June
More informationMalliavin-Stein Method in Stochastic Geometry
Malliavin-Stein Method in Stochastic Geometry Dissertation zur Erlangung des Doktorgrades (Dr. rer. nat.) des Fachbereichs Mathematik/Informatik der Universität Osnabrück vorgelegt von Matthias Schulte
More informationGeometry of log-concave Ensembles of random matrices
Geometry of log-concave Ensembles of random matrices Nicole Tomczak-Jaegermann Joint work with Radosław Adamczak, Rafał Latała, Alexander Litvak, Alain Pajor Cortona, June 2011 Nicole Tomczak-Jaegermann
More information11. Further Issues in Using OLS with TS Data
11. Further Issues in Using OLS with TS Data With TS, including lags of the dependent variable often allow us to fit much better the variation in y Exact distribution theory is rarely available in TS applications,
More informationExponential martingales: uniform integrability results and applications to point processes
Exponential martingales: uniform integrability results and applications to point processes Alexander Sokol Department of Mathematical Sciences, University of Copenhagen 26 September, 2012 1 / 39 Agenda
More informationAnisotropic cylinder processes
Anisotropic cylinder processes Evgeny Spodarev Joint work with A. Louis, M. Riplinger and M. Spiess Ulm University, Germany Evgeny Spodarev, 15 QIA, 8.05.2009 p.1 Modelling the structure of materials Gas
More informationCIMPA SCHOOL, 2007 Jump Processes and Applications to Finance Monique Jeanblanc
CIMPA SCHOOL, 27 Jump Processes and Applications to Finance Monique Jeanblanc 1 Jump Processes I. Poisson Processes II. Lévy Processes III. Jump-Diffusion Processes IV. Point Processes 2 I. Poisson Processes
More informationGeneralized Gaussian Bridges of Prediction-Invertible Processes
Generalized Gaussian Bridges of Prediction-Invertible Processes Tommi Sottinen 1 and Adil Yazigi University of Vaasa, Finland Modern Stochastics: Theory and Applications III September 1, 212, Kyiv, Ukraine
More information2. The averaging process
2. The averaging process David Aldous July 10, 2012 The two models in lecture 1 has only rather trivial interaction. In this lecture I discuss a simple FMIE process with less trivial interaction. It is
More informationPoint Process Control
Point Process Control The following note is based on Chapters I, II and VII in Brémaud s book Point Processes and Queues (1981). 1 Basic Definitions Consider some probability space (Ω, F, P). A real-valued
More informationStochastic Volatility and Correction to the Heat Equation
Stochastic Volatility and Correction to the Heat Equation Jean-Pierre Fouque, George Papanicolaou and Ronnie Sircar Abstract. From a probabilist s point of view the Twentieth Century has been a century
More informationRandom Processes. DS GA 1002 Probability and Statistics for Data Science.
Random Processes DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Aim Modeling quantities that evolve in time (or space)
More informationGaussian vectors and central limit theorem
Gaussian vectors and central limit theorem Samy Tindel Purdue University Probability Theory 2 - MA 539 Samy T. Gaussian vectors & CLT Probability Theory 1 / 86 Outline 1 Real Gaussian random variables
More informationPart IA Probability. Theorems. Based on lectures by R. Weber Notes taken by Dexter Chua. Lent 2015
Part IA Probability Theorems Based on lectures by R. Weber Notes taken by Dexter Chua Lent 2015 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
More informationApplications of random matrix theory to principal component analysis(pca)
Applications of random matrix theory to principal component analysis(pca) Jun Yin IAS, UW-Madison IAS, April-2014 Joint work with A. Knowles and H. T Yau. 1 Basic picture: Let H be a Wigner (symmetric)
More informationSTAT 331. Martingale Central Limit Theorem and Related Results
STAT 331 Martingale Central Limit Theorem and Related Results In this unit we discuss a version of the martingale central limit theorem, which states that under certain conditions, a sum of orthogonal
More informationIEOR 4701: Stochastic Models in Financial Engineering. Summer 2007, Professor Whitt. SOLUTIONS to Homework Assignment 9: Brownian motion
IEOR 471: Stochastic Models in Financial Engineering Summer 27, Professor Whitt SOLUTIONS to Homework Assignment 9: Brownian motion In Ross, read Sections 1.1-1.3 and 1.6. (The total required reading there
More informationIntroduction to Koecher Cones. Michael Orlitzky
Introduction to Koecher Cones Extreme Points Definition 1. Let S be a convex set in some vector space. We say that the point x S is an extreme point of the set S if x = λx 1 + (1 λ)x 2 for λ (0, 1) implies
More informationarxiv: v3 [math.pr] 17 May 2009
arxiv:95.1145v3 [math.pr] 17 May 29 Strong mixing property for STIT tessellations Raphaël Lachièze-Rey May 17, 29 abstract The so-called STIT tessellations form the class of homogeneous (spatially stationary)
More informationLiquidation in Limit Order Books. LOBs with Controlled Intensity
Limit Order Book Model Power-Law Intensity Law Exponential Decay Order Books Extensions Liquidation in Limit Order Books with Controlled Intensity Erhan and Mike Ludkovski University of Michigan and UCSB
More informationExtreme Value Analysis and Spatial Extremes
Extreme Value Analysis and Department of Statistics Purdue University 11/07/2013 Outline Motivation 1 Motivation 2 Extreme Value Theorem and 3 Bayesian Hierarchical Models Copula Models Max-stable Models
More informationRepresentation of Markov chains. Stochastic stability AIMS 2014
Markov chain model Joint with J. Jost and M. Kell Max Planck Institute for Mathematics in the Sciences Leipzig 0th July 204 AIMS 204 We consider f : M! M to be C r for r 0 and a small perturbation parameter
More informationHomogenization for chaotic dynamical systems
Homogenization for chaotic dynamical systems David Kelly Ian Melbourne Department of Mathematics / Renci UNC Chapel Hill Mathematics Institute University of Warwick November 3, 2013 Duke/UNC Probability
More informationConvergence to equilibrium for rough differential equations
Convergence to equilibrium for rough differential equations Samy Tindel Purdue University Barcelona GSE Summer Forum 2017 Joint work with Aurélien Deya (Nancy) and Fabien Panloup (Angers) Samy T. (Purdue)
More informationTopics in Theoretical Computer Science: An Algorithmist's Toolkit Fall 2007
MIT OpenCourseWare http://ocw.mit.edu 18.409 Topics in Theoretical Computer Science: An Algorithmist's Toolkit Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationLecture 21 Representations of Martingales
Lecture 21: Representations of Martingales 1 of 11 Course: Theory of Probability II Term: Spring 215 Instructor: Gordan Zitkovic Lecture 21 Representations of Martingales Right-continuous inverses Let
More informationFinite-Memory Universal Prediction of Individual Continuous Sequences
THE IBY AND ALADAR FLEISCHMAN FACULTY OF ENGINEERING THE ZANDMAN-SLANER SCHOOL OF GRADUATE STUDIES THE SCHOOL OF ELECTRICAL ENGINEERING Finite-Memory Universal Prediction of Individual Continuous Sequences
More informationConsistency of the maximum likelihood estimator for general hidden Markov models
Consistency of the maximum likelihood estimator for general hidden Markov models Jimmy Olsson Centre for Mathematical Sciences Lund University Nordstat 2012 Umeå, Sweden Collaborators Hidden Markov models
More informationCentral Limit Theorem for Non-stationary Markov Chains
Central Limit Theorem for Non-stationary Markov Chains Magda Peligrad University of Cincinnati April 2011 (Institute) April 2011 1 / 30 Plan of talk Markov processes with Nonhomogeneous transition probabilities
More informationExponential Tail Bounds
Exponential Tail Bounds Mathias Winther Madsen January 2, 205 Here s a warm-up problem to get you started: Problem You enter the casino with 00 chips and start playing a game in which you double your capital
More informationJump-type Levy Processes
Jump-type Levy Processes Ernst Eberlein Handbook of Financial Time Series Outline Table of contents Probabilistic Structure of Levy Processes Levy process Levy-Ito decomposition Jump part Probabilistic
More informationStochastic volatility models: tails and memory
: tails and memory Rafa l Kulik and Philippe Soulier Conference in honour of Prof. Murad Taqqu 19 April 2012 Rafa l Kulik and Philippe Soulier Plan Model assumptions; Limit theorems for partial sums and
More information