Quadratic harnesses, q-commutations, and orthogonal martingale polynomials. Quadratic harnesses. Previous research. Notation and Motivation

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1 Quadratic harnesses, q-commutations, and orthogonal martingale polynomials Quadratic harnesses Slide 1 W lodek Bryc University of Cincinnati August 2005 Based on joint research with W Matysiak and J Wesolowski Slide 3 Covariance: Harness condition: EX t ) = 0, EX t X s ) = min{t, s} 1) EX t F s,u ) = a t,s,u X s + b t,s,u X u, 2) Quadratic harness condition: E Xt 2 ) F s,u = Qt,s,u X s, X u ), 3) mathucedu/~brycw/preprint/5-param/q-harnessespdf Q t,s,u x, y) = A t,s,u x 2 +B t,s,u xy +C t,s,u y 2 +D t,s,u x+e t,s,u y +F t,s,u 4) Note: Preserved by time-inversion X t ) tx 1/t ) Notation and Motivation Previous research i) Harnesses: [Hammersley, 1967] - linear regression based models of long-range misorientation [Williams, 1973] - discrete index, [1980] - continuous, [Mansuy and Yor, 2005] - integral repr Slide 2 Slide 4 ii) [Plucińska, 1983]: linear regressions and constant conditional variances Gaussian Discrete indexes [Bryc and Plucińska, 1985], L 2 -smooth processes [Szab lowski, 1989], Poisson [Bryc, 1987], Gamma [Weso lowski, 1989] F s,u = σ{x t : t 0, s] [u, )} F t = F t, = σ{x s : 0 < s t} iii) [Weso lowski, 1993]: conditional variance as a quadratic function of increments characterizes five Lévy processes: Wiener, Poisson, Pascal neg bin), Gamma, and Meixner hyp secant) 0 < r < s < t < u < v Assumption: EX t ) = 0 and EX s X t ) = min{s, t} iv) [Bryc, 2001b]: stationary quadratic harnesses are classical versions of non-commutative q-gaussian processes of [Frisch and Bourret, 1970] and [Bożejko et al, 1997]

2 Technical Assumptions: Suppose for each t > 0 the random variable X t has infinite support i) Q t,s,u 0, 0) = F t,s,u 0 ii) 1, X s, X t, X s X t, X 2 s, X 2 t are linearly independent for all 0 < s < t Theorem 2 If a quadratic harness X t ) with parameters q, η, θ, σ, τ has finite moments of all orders and martingale polynomials, then Ct) = tx + y, t > 0, 6) Slide 5 Theorem 1 [Bryc et al, 2005]) There exist η, θ R, σ, τ 0, and q στ such that Xu X s Var X t F s,u ) = F t,s,u K u s, ux ) s sx u 5) u s for all 0 < s < t < u, F t,s,u = u t)t s) u1+σs)+τ qs is the normalizing constant, and Kx, y) = 1 + θx + τx 2 + ηy + σy 2 xy qyx) Notation: X t ) is a quadratic harness with parameters q, η, θ, σ, τ Slide 7 and the infinite matrices x = C 1 C 0, y = C 0 satisfy the q- commuation equation Kx, y) = 0, ie [x, y] q = I + θx + ηy + τx 2 + σy 2, 7) [x, y] q = xy qyx, In [Frisch and Bourret, 1970], the equation [x, x ] q = I is the basis of a parastochastic model of Kraichnan s equation Guionnett-Mazza 2004) [Bryc and Weso lowski, 2005]: quadratic harnesses with η = θ = τ = σ = 0 are a classical version of the q-brownian motion Martingale polynomials Orthogonal martingale polynomials Slide 6 E p n X t ; t) F s ) = p n X s ; s), 0 < s < t, n = 0, 1, n+1 xp n x; t) = C k,n t)p k x; t) Take p 0 = 1, p 1 = x Then k=0 0 t C 02 t) C 0,3 t) 1 C 11 t) C 12 t) C 13 t) 0 C 21 t) C 22 t) C 23 t) Ct) := 0 0 C 32 t) C 33 t) C 43 t) Slide 8 0 t γ 1t + δ 1 ε 2t + ϕ α 2t + β 2 γ 2t + δ 2 ε 3t + ϕ 3 Ct) := 0 0 α 3t + β 3 γ 3t + δ α 4t + β 4

3 Orthogonal martingale polynomials Worked out Example I: [x, y] q = I + θx + τx 2 Slide 9 xp n x; t) = a n t)p n+1 x; t) + b n t)p n x; t) + c n t)p n 1 x; t), 8) a n t) = α n+1 t + β n+1, b n t) = γ n t + δ n, c n t) = ε n t + ϕ n, 9) and the coefficients in 9) satisfy a system of 5 equations that result from the q-commutation equation 7) Theorem 3 [Bryc et al, 2005]) Suppose X t ) is a quadratic harness with parameters such that 0 στ < 1, 1 < q 1 2 στ Moreover, assume that for each t > 0 random variable X t has moments of all orders and infinite support Then 8) determines {p n x; t)} which are orthogonal martingale polynomials for X t ) Slide 11 Proof: D q f)z) = x = D q y = Z1 + θd q + τd 2 q) fz) fqz), Zf)z) = zfz), 1 q)z [x, y] q = [D q, Z] q + θ[d q, ZD q ] q + τ[d q, ZD 2 q] q = I + θd q + τd 2 q Note: D q z n ) = [n] q z n 1, [n] q = 1 + q + + q n 1 Calculation: tx + y)z n = z n+1 + θ[n] q z n + t + τ[n 1] q )[n] q z n 1 Slide 10 a n t) = σα n+1 t+β n+1, b n t) = γ n t+δ n, c n t) = β n t + τα n ) ω n, 10) i) α 1 = 0, β 1 = 1, γ 0 = δ 0 = 0, ω 1 = 1 ii) iii) α n+1 = q 1 α n, n 1 β n+1 στ 1 β n Moreover, λ n,k := β n β n+k στα n α n+k > 0 for all n 1, k 0 γ n+1 = q + στ λ n+2,0 λ n,2 γ n + α n+2 β n β n+2 α n )σδ n ) + σα n+2 λ n+2,0 ητα n+1 + θβ n+1 ) + β n+2 λ n+2,0 θσα n+1 + ηβ n+1 ), Similar equation for δ n+1 and ω n Slide 12 Identification of z n with p n x; t) gives Al-Salam-Chihara polynomials xp nx; t) = p n+1x; t) + θ[n] qp nx; t) + t + τ[n 1] q)[n] qp n 1x; t) [Feinsilver, 1990, Section 34], [Anshelevich, 2003, Remark 6] Theorem 4 [Bryc and Weso lowski, 2005]) If X t ) is a quadratic harness with parameters 1 < q 1, σ = η = 0 then X t ) is Markov with the transition probabilities P s,t x, dy) determined as the unique probability measure orthogonalizing polynomials {Q n y)} given by the recurrence yq n y) = Q n+1 y)+θ[n] q +xq n )Q n y)+t sq n 1 +τ[n 1] q )[n] q Q n 1 y) Conversely, each such Markov process is a quadratic harness

4 Classical versions [x, y] q = I + θx + τx 2 Slide 13 VarX t F s,u) 1+θ EX t1 X t2 X tk ) = τx t1 X t2 X tk ), t1 t 2 t k Xu Xs Xu Xs)2 Xu Xs)uXs sxu) +τ 1 q) u s u s) 2 u s) 2 i) Case τ = θ = 0 and 1 < q 1 corresponds to the q-brownian motion of [Bożejko et al, 1997] Slide 15 Theorem 5 work in progress) If X t ) is a quadratic harness with parameters 1 < q 1, σ = τ = 0, and 1+ηθ > max{q, 0} then X t ) is Markov with the transition probabilities P s,t x, dy) determined as the unique probability measure orthogonalizing polynomials {Q n y)} given by the recurrence yq ny) = Q n+1y) + A nx, t, s)q ny) + B nx, t, s)q n 1y), ii) Case θ 0, τ = 0 and 1 < q 1 correspond to the q-poisson process of [Anshelevich, 2001] It is also a classical version of the time-inverse of the q-poisson process of [Saitoh and Yoshida, 2000] iii) If τ 0, θ 0, σ = η = 0 and 1 < q < 1, q 0 then the the laws of q-meixner process X t ) are not closed under q- convolution of [Nica, 1995], and are not the laws of the q-levy processes of [Anshelevich, 2004] A n x, t, s) = q n x + [n] q tη + θ [2] q q n 1 sη), B n x, t, s) = [n] q t sq n 1 ) 1 + ηxq n 1 + [n 1] q ηθ sηq n 1 ) ) Conversely, each such Markov process is a quadratic harness Worked Out Example II: [x, y] q = I + θx + ηy Free case: q = 0 Operator solution: Xu Xs uxs sxu Xu Xs)uXs sxu) VarX t F s,u) 1 + θ + η u s u s u s) 2 Slide 14 x = D q + ηzd q + θd 2 q), y = Z1 + θd q ) Calculation: tx + y)z n = z n+1 + θ + tη)[n] q z n + t1 + ηθ[n 1] q )[n] q z n 1, Slide 16 Denote by π t the law of X t Proposition 6 [Bryc and Weso lowski, 2004]) For every t 0, there exist a probability measure ν t such that the pairs π t, ν t ) form a semigroup with respect to the c-convolution, These are again Al-Salam-Chihara polynomials xp nx; t) = p n+1x; t)+θ+tη)[n] qp nx; t)+t1+ηθ[n 1] q)[n] qp n 1x; t), 11) π t+s, ν t+s ) = π t, ν t ) c π s, ν s ) Note: We must have 1 + ηθ q + [Bożejko et al, 1996], [Bożejko and Wysoczański, 2001], [Krystek and Wojakowski, 2004]

5 Classical case: q = 1 Birth-then-death process Slide 17 Xu Xs uxs sxu VarX t F s,u) 1 + θ + η u s u s VarX t F s,u) 1 + θ Xu Xs u s Poisson type VarX t F s,u) 1 + η uxs sxu u s time-inverse of Poisson X t = N t t X t = tn 1/t 1 Slide 19 θ ηt)z t t θ 0 < t < T, X t = Z T 1 η t = T ηt θ)z t 1 η t > T Here T = η/θ Fix η, θ > 0 Let T = θ/η Let Z t ) t>0 be a non-homogeneous Markov process with rates k k + 1 at rate k + 1 ηθ )/T s) on 0 < s < t < T k k 1 at rate k/s T ) on T < s < t Slide 18 Figure 1: Simulated sample trajectory when η = θ Slide 20 ) Z t Z s Z d 1 θ ηt s = nb + Zs, θη θ ηs Z T Z d 1 s = Gamma θη + Zs, 1 Z t Z T Z t Z s d = P Zθ/η ηt θ = d b Z ηs θ s, ηt θ, 0 s < t < θ/η, ) θ ηs, 0 s < T, ), t > T, ), T < s < t

6 [Bryc and Weso lowski, 2004] Bryc, W and Weso lowski, J 2004) Bi-Poisson process Submitted arxivorg/abs/mathpr/ [Bryc and Weso lowski, 2005] Bryc, W and Weso lowski, J 2005) Conditional moments of q- Meixner processes Probability Theory Related Fields, 131: arxivorg/abs/mathpr/ Concluding Remarks [Feinsilver, 1990] Feinsilver, P 1990) Lie algebras and recurrence relations III q-analogs and quantized algebras Acta Appl Math, 193): [Frisch and Bourret, 1970] Frisch, U and Bourret, R 1970) Parastochastics J Math Phys, 112): Slide 21 i) Little is known about the non-commutative processes corresponding to [x, y] q = I + θx + ηy ii) Little is known about which q-guassian processes have classical versions Markov, yes [Bożejko et al, 1997] There are q- Gaussian processes with no classical version [Bryc, 2001a] Slide 23 [Hammersley, 1967] Hammersley, J M 1967) Harnesses In Proc Fifth Berkeley Sympos Mathematical Statistics and Probability Berkeley, Calif, 1965/66), Vol III: Physical Sciences, pages Univ California Press, Berkeley, Calif [Krystek and Wojakowski, 2004] Krystek, A and Wojakowski, L 2004) Associative convolutions arising from conditionally free convolution Preprint [Mansuy and Yor, 2005] Mansuy, R and Yor, M 2005) Harnesses, Lévy bridges and Monsieur Jourdain Stochastic Processes and Their Applications, 115: [Meixner, 1934] Meixner, J 1934) Orthogonale polynomsysteme mit einer besonderen gestalt der erzeugenden funktion Journal of the London Mathematical Society, 9:6 13 [Nica, 1995] Nica, A 1995) A one-parameter family of transforms, linearizing convolution laws for probability distributions Comm Math Phys, 1681): iii) No general constructions of quadratic harnesses are known [Plucińska, 1983] Plucińska, A 1983) On a stochastic process determined by the conditional expectation and the conditional variance Stochastics, 10: [Saitoh and Yoshida, 2000] Saitoh, N and Yoshida, H 2000) q-deformed Poisson random variables on q-fock space J Math Phys, 418): [Szab lowski, 1989] Szab lowski, P J 1989) Can the first two conditional moments identify a mean square differentiable process? Comput Math Appl, 184): [Voiculescu, 2000] Voiculescu, D 2000) Lectures on free probability theory In Lectures on probability References [Anshelevich, 2001] Anshelevich, M 2001) Partition-dependent stochastic measures and q- deformed cumulants Doc Math, 6: electronic) [Anshelevich, 2003] Anshelevich, M 2003) Free martingale polynomials Journal of Functional Analysis, 201: ArXiv:mathCO/ [Anshelevich, 2004] Anshelevich, M 2004) q-lévy processes J Reine Angew Math, 576: ArXiv:mathOA/ [Biane, 1998] Biane, P 1998) Processes with free increments Math Z, 2271): theory and statistics Saint-Flour, 1998), volume 1738 of Lecture Notes in Math, pages Springer, Berlin [Weso lowski, 1989] Weso lowski, J 1989) A characterization of the gamma process by conditional moments Metrika, 365): [Weso lowski, 1993] Weso lowski, J 1993) Stochastic processes with linear conditional expectation and quadratic conditional variance Probab Math Statist, 14:33 44 [Williams, 1973] Williams, D 1973) Some basic theorems on harnesses In Stochastic analysis a tribute to the memory of Rollo Davidson), pages Wiley, London Slide 22 [Bożejko et al, 1997] Bożejko, M, Kümmerer, B, and Speicher, R 1997) q-gaussian processes: non-commutative and classical aspects Comm Math Phys, 1851): [Bożejko et al, 1996] Bożejko, M, Leinert, M, and Speicher, R 1996) Convolution and limit theorems for conditionally free random variables Pacific J Math, 1752): [Bożejko and Wysoczański, 2001] Bożejko, M and Wysoczański, J 2001) Remarks on t- transformations of measures and convolutions Ann Inst H Poincaré Probab Statist, 376): [Bryc, 1987] Bryc, W 1987) A characterization of the Poisson process by conditional moments Stochastics, 20:17 26 [Bryc, 2001a] Bryc, W 2001a) Classical versions of q-gaussian processes: conditional moments and Bell s inequality Comm Math Physics, 219: [Bryc, 2001b] Bryc, W 2001b) Stationary random fields with linear regressions Annals of Probability, 29: [Bryc et al, 2005] Bryc, W, Matysiak, W, and Weso lowski, J 2005) Quadratic harnesses, q- commuttions, and orthogonal martingale polynomials arxivorg/abs/mathpr/ [Bryc and Plucińska, 1985] Bryc, W and Plucińska, A 1985) A characterization of infinite gaussian sequences by conditional moments Sankhya A, 47: Slide 24