The robbins-monro type stochastic differential equations. I. convergence of solutions

Rate of convergence of truncated stochastic approximation procedures with moving bounds

“…The proof can be found in Robbins and Siegmund (1985). Note also that this lemma is a special case of the theorem on the convergence sets of non-negative semi-martingales (see, e.g., Lazrieva et al (1997)). …”

“…Note that the idea of truncations is not new and goes back to Khasminskii and Nevelson (1972) and Fabian (1978) (see also Chen and Zhu (1986), Chen et al(1987), Andradóttir (1995), Sharia (1997), Tadic (1997Tadic ( ,1998, Lelong (2008). A comprehensive bibliography and some comparisons can be found in Sharia (2014)).…”

“…the components of θ). This estimator was introduced in Sakrison (1965) and studied by a number of authors (see e.g, Polyak and Tsypkin (1980), Campbell (1982), Ljung and Soderstrom (1987), Lazrieve and Toronjadze (1987), Englund et al (1989), Lazrieve et al (1997,2008), Sharia (1997). In particular, it has been shown that under certain conditions, the recursive estimatorθ t is asymptotically equivalent to the maximum likelihood estimator, i.e., it is consistent and asymptotically efficient.…”

Efficient on-line estimation of autoregressive parameters

“…Note also that this lemma is a special case of the theorem on the convergence sets nonnegative semimartingales (see, e.g., [11]). Remark B.5 The proof of this lemma is given [20] (see Lemma A2 in Appendix A).…”

Recursive parameter estimation: asymptotic expansion

“…In particular, Englund et al (1989) give an asymptotic representation results for certain type of X n processes. In Sharia (1998), theoretical results on convergence, rate of convergence and the asymptotic representation are given under certain regularity and ergodicity assumptions on the model, in the one-dimensional case with ψ n (x, θ) = ∂ ∂θ logf n (x, θ) (see also Campbell (1982), Sharia (1992), and Lazrieva et al (1997)). …”