Online Appendix to “Strategic Trading in Informationally Complex Environments”

The proof of Theorem 2 in the main body of the paper applies to the special case in which the covariance matrix of random vector (θ; θM ;u) is full rank. In this Online Appendix, we prove Theorem 2 for the general case, imposing only Assumptions 1 and 2L: Cov(v, θ|θM ) 6= 0 and V ar(u|θ, θM ) > 0. Proof of Theorem 2 (General Case) Before proceeding to the proof, we first prove the following lemma. Lemma OA.1 Let φ be a random vector normally distributed with mean zero and variance-covariance matrix Σ. Let α be a vector of the same dimensionality, and let {αk} be a sequence of vectors such that lim k→∞ (αk) T φ̃→ α φ̃ for every realization φ̃ of random vector φ except possibly on a set of probability zero. Then the limit also holds in the L2 sense, that is, lim k→∞ E [( (αk) φ− αφ )2]→ 0. Proof. Let N be the rank of matrix V ar(φ), and take an orthogonal matrix Φ such that ΦV ar(φ)Φ = ( M 0 0 0 ) , where M is a symmetric positive definite matrix of size N (if matrix V ar(φ) is itself positive definite, and thus full rank, then we can simply take Φ to be the identity matrix and thus M = V ar(φ).) Let ψ := ΦTφ (and thus, since Φ is orthogonal, φ = Φψ). If, for almost all realizations φ̃ of φ, we have (αk) T φ̃ → αT φ̃, then for almost all realizations ψ̃ of ψ, we also have (αk)Φψ̃ → αTΦψ̃, which can be rewritten as (Φαk) T ψ̃ → (Φα) ψ̃. (OA.1) Since ψ is distributed normally with mean zero and variance ( M 0 0 0 ) , where M is a positive definite matrix of rank N , the fact that the convergence in equation (OA.1) holds for almost all