Fluctuations of the empirical quantiles of independent Brownian - PowerPoint PPT Presentation

Fluctuations of the empirical quantiles of independent Brownian motions Jason Swanson Department of Mathematics University of Central Florida October 9, 2008

α -quantiles ν – a probability measure ( ) Φ = ν −∞ ( ) ( , ] x x ν α ∈ (0,1) An α -quantile of ν is any number q such that Φ − ≤ α ≤ Φ ( ) ( ) q q ν ν { } α ≤ Φ is always an α -quantile of ν . ● inf : ( ) x x ν Φ is continuous, then Φ = α for any α -quantile, q . ● If ( ) q ν ν are α -quantiles, then ( ) < ν = ● If ( , ) 0 . q q q q 1 2 1 2

Order statistics , , … , – random variables X X X 1 2 n σ – a random permutation of { } 1,2, … , n such that ≤ ≤ ≤ � a.s. X X X σ (1) σ (2) σ ( ) n = = The j -th order statistic of ( , , … , ) is X X X X X X σ 1 2 : ( ) n j n j − = − ● ( ) X X : ( − + 1): j n n j n

The model B t – one-dimensional Brownian motion ( ) ( ) C ∞ ∈ R , with + n ( ) < ∞ (0) has a density ( ) sup 1 ( ) for all m n . , m B f x f x ∈ R x α ∈ Fix (0,1) . f x dx has a unique α -quantile, (0) > ( ) ( (0)) 0 Assume , such that . q f q { } – iid copies of B B n : ( ) t – j -th order statistic of ( ( ), ( ), … , ( )) B B t B t B t 1 2 j n n Note: is a continuous process B : j n ( ) { } ∞ ≤ ≤ = α + − 1/2 ( ) n = – integers such that 1 ( ) and ( ) / j n j n n j n n o n 1 = ( ) ( ) Q t B t ( ): n j n n

The model u x t – density of ( , ) B t ( ) q t – unique α -quantile of ( , ) ( ) u x t dx ∂ ( ( ), ) x u q t t ∈ ∞ ∩ C ∞ ∞ and ′ = − t > Lemma: [0, ) (0, ) ( ) for all 0 . q C q t 2 ( ( ), ) u q t t Proof sketch: ( ) ∫ q t α = ≤ = ( ( ) ( )) ( , ) P B t q t u x t dx −∞ ( ) ∫ q t ′ = + ∂ 0 ( ( ), ) ( ) ( , ) u q t t q t u x t dx t −∞ 1 ( ) ∫ q t ′ = + ∂ 2 0 ( ( ), ) ( ) ( , ) u q t t q t u x t dx x 2 −∞ 1 ′ = + ∂ 0 ( ( ), ) ( ) ( ( ), ) u q t t q t u q t t x 2

The model u x t – density of ( , ) B t ( ) q t – unique α -quantile of ( , ) ( ) u x t dx ∂ ( ( ), ) x u q t t ∈ ∞ ∩ C ∞ ∞ and ′ = − t > Lemma: [0, ) (0, ) ( ) for all 0 . q C q t 2 ( ( ), ) u q t t > . Then ∃ t ↓ ε > Proof sketch: Suppose (0) liminf ( ) 0 and 0 s.t. q q t n → 0 t − > ε (0) ( ) , which implies q q t n α = ≤ ≤ ≤ − ε ( ( ) ( )) ( ( ) (0) ) P B t q t P B t q n n n ⎯⎯⎯ →∞ → ≤ − ε ≤ ≤ = α n ( (0) (0) ) ( (0) (0)) . P B q P B q ≤ − ε = α , and the α -quantile is not unique, a Hence, ( (0) (0) ) P B q contradiction. �

The model ( ) = 1/2 − ( ) ( ) ( ) F t n Q t q t n n ⇒ ∞ , where F is a continuous, centered Gaussian Theorem: in [0, ) F F C n process with covariance ≤ ≤ − α 2 ( ( ) ( ), ( ) ( )) P B s q s B t q t ρ = ( , ) . s t ( ( ), ) ( ( ), ) u q s s u q t t

Outline ● Convergence of finite-dimensional distributions In particular, ρ defines a Gaussian process ● Properties of the limit process 1/4 Comparison with fBm, B H ∈ ⋅ is a centered Gaussian process (for fixed (0,1) , ( ) H B 2 2 = − = − H with (0) 0 and ( ) ( ) .) H H H B E B t B s t s Related work: [Harris, 1965], [Dürr, Goldstein, Lebowitz, 1985] ● Tightness ♦ Connect quantile to iid sums ♦ Estimate iid sums in terms of their parameters ♦ Estimate those parameters in terms of our specific model

Outline ● Convergence of finite-dimensional distributions In particular, ρ defines a Gaussian process ● Properties of the limit process 1/4 Comparison with fBm, B H ∈ ⋅ is a centered Gaussian process (for fixed (0,1) , ( ) H B 2 2 = − = − H with (0) 0 and ( ) ( ) .) H H H B E B t B s t s Related work: [Harris, 1965], [Dürr, Goldstein, Lebowitz, 1985] ● Tightness ♦ Connect quantile to iid sums ♦ Estimate iid sums in terms of their parameters ♦ Estimate those parameters in terms of our specific model Done in generality, with future projects in mind.

Potential future projects ● quantiles of diffusions (with Tom Kurtz) ● quantiles of general Gaussian processes fBm: limiting fluctuations of iid copies of B H /2 should behave locally like H . B

Convergence of finite-dimensional distributions R -valued random variable = ( (1), (2), … , ( )) – d X X X X d α ∈ Φ = ≤ = ≤ ≤ ( ) ( ( ) ) , ( , ) ( ( ) , ( ) ) , fix (0,1) j x P X j x G x y P X i x X j y ij Assume ∃ = ∈ R such that Φ = α ( (1), (2), … , ( )) ( ( )) , d q q q q d j q j ′ Φ ( ( )) exists and is strictly positive, and G is continuous at ( ( ), ( )) . j q j q i q j ij { } – iid copies of X X n – component-wise order statistics of , , … , ; X X X X : 1 2 k n n i.e. : ( ) j is the k -th order statistic of ( ( ), ( ), … , ( )) X X j X j X j 1 2 k n n ( ) ( ) = α + − 1/2 1/2 − ⇒ Quantile CLT: If ( ) / , then , where k n n o n n X q N ( ): k n n N is mean zero, multi-normal, with covariance − α 2 ( ( ), ( )) G q i q j σ = = ( ) ( ) ij EN i N j ′ ′ Φ Φ ij ( ( )) ( ( )) q i q j i j

Convergence of finite-dimensional distributions Φ = ≤ = ≤ ≤ ( ) ( ( ) ) , ( , ) ( ( ) , ( ) ) j x P X j x G x y P X i x X j y ij Φ = α Φ ′ > ( ( )) , ( ( )) 0 , G continuous at ( ( ), ( )) . j q j j q j q i q j ij { } – iid copies, : ( ) j – k -th order statistic of ( ( ), … , ( )) X X X j X j 1 n k n n ( ) ( ) = α + − 1/2 1/2 − ⇒ Quantile CLT: If ( ) / , then , where k n n o n n X q N ( ): k n n − α 2 ( ( ), ( )) G q i q j σ = = ( ) ( ) ij EN i N j Φ ′ Φ ′ ij ( ( )) ( ( )) q i q j i j ● Convergence of finite-dimensional distributions is an immediate corollary.

Convergence of finite-dimensional distributions Φ = ≤ = ≤ ≤ ( ) ( ( ) ) , ( , ) ( ( ) , ( ) ) j x P X j x G x y P X i x X j y ij Φ = α Φ ′ > ( ( )) , ( ( )) 0 , G continuous at ( ( ), ( )) . j q j j q j q i q j ij { } – iid copies, : ( ) j – k -th order statistic of ( ( ), … , ( )) X X X j X j 1 n k n n ( ) ( ) = α + − 1/2 1/2 − ⇒ Quantile CLT: If ( ) / , then , where k n n o n n X q N ( ): k n n − α 2 ( ( ), ( )) G q i q j σ = = ( ) ( ) ij EN i N j Φ ′ Φ ′ ij ( ( )) ( ( )) q i q j i j x y ∈ R , x ≤ ≤ Proof sketch: For , d iff ( ) ( ) for all j . y x j y j ( ) ( ) ( ) − 1/2 1/2 − ≤ = ≤ + ∀ ( ) ( ) ( ), P n X q x P X j n x j q j j ( ): ( ): k n n k n n ⎛ ⎞ n ∑ = ≥ ∀ ⎜ ⎟ 1 ( ), P { } k n j ⎜ ⎟ ≤ − 1/2 + ( ) ( ) ( ) X j n x j q j ⎝ ⎠ m = 1 m ⎛ ⎞ n ∑ − ( ) 1/2 = ≥ − ∀ ⎜ ( ) ( ) ( ) , ⎟ P Y j n k n np j j ⎜ , ⎟ m n n ⎝ ⎠ = 1 m

Convergence of finite-dimensional distributions ⎛ ⎞ ( ) ( ) n ∑ ( ) − ≤ = ≥ − − ∀ 1/2 1/2 ( ) ( ) ( ) , , P n X q x P ⎜ Y j n k n np j j ⎟ ( ): , k n n m n n ⎝ ⎠ = 1 m where ( ) − 1/2 = ≤ + ( ) ( ) ( ) ( ) p j P X j n x j q j n ( ) − 1/2 = − ( ) 1 ( ) . { } Y j n p j , ≤ − + ( ) 1/2 ( ) ( ) m n n X j n x j q j m ∑ � , centered normal with n ⇒ Lindeberg-Feller: m Y N , = m n 1 � � = − α 2 ( ) ( ) ( ( ), ( )) . EN i N j G q i q j ij

Convergence of finite-dimensional distributions ⎛ ⎞ ( ) ( ) n ∑ ( ) − 1/2 − ≤ = ≥ 1/2 − ∀ ( ) ( ) ( ) , , P n X q x P ⎜ Y j n k n np j j ⎟ ( ): , k n n m n n ⎝ ⎠ = 1 m ∑ � , � � n ⇒ = − α 2 ( ) ( ) ( ( ), ( )) . m Y N EN i N j G q i q j , = 1 m n ij ( ) ( ) − − = − 1/2 1/2 ( ) ( ) ( ) / ( ) n k n np j n k n n p j n n ( ) = α − + 1/2 ( ) (1) n p j o n − Φ − Φ 1/2 + ( ( )) ( ( ) ( )) q j n x j q j = + j j (1) o − 1/2 n ′ → − Φ ( ) ( ( )) x j q j j Hence, ( ) ( ) ( ) ( ) � ′ 1/2 − ≤ → − Φ ≤ ∀ = ≤ ( ) / ( ( )) ( ), . � P n X q x P N j q j x j j P N x ( ): k n n j

Properties of the limit process ≤ ≤ − α 2 ( ( ) ( ), ( ) ( )) P B s q s B t q t ρ = ( , ) s t ( ( ), ) ( ( ), ) u q s s u q t t T > , ∃ δ > < < ≤ Theorem: For each 0 , , 0 such that for all 0 , C C s t T 0 1 2 − 1/2 − 1/2 − ≤ ∂ ρ ≤ − (i) ( , ) C t s s t C t s 1 2 s − − 1/2 1/2 − − ≤ ∂ ρ ≤ − − (ii) ( , ) C t s s t C t s 2 1 t − − 3/2 3/2 − − ≤ ∂ 2 ρ ≤ − − (iii) ( , ) C t s s t C t s 2 1 st − < δ whenever . t s 0 � = Heuristic: Define ( ) ( ( ), ) ( ) . F t u q t t F t ρ � = ≤ ≤ − α 2 ( , ) ( ( ) ( ), ( ) ( )) s t P B s q s B t q t