?url_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adc&rft.relation=https%3A%2F%2Fpure.iiasa.ac.at%2Fid%2Feprint%2F4573%2F&rft.title=Strong+Convergence+of+Stochastic+Approximation+Without+Lyapunov+Functions&rft.creator=Kaniovski%2C+Y.M.&rft.description=We+prove+convergence+with+probability+one+of+a+multivariate+Markov+stochastic+approximation+procedure+of+the+Robbins-Monro+type+with+several+roots.+The+argument+exploits+convergence+of+the+corresponding+system+of+ordinary+differential+equations+to+its+stationary+points.+If+the+points+are+either+linearly+stable+or+linearly+unstable%2C+we+prove+convergence+with+probability+1+of+the+procedure+to+a+random+vector+whose+distribution+concentrates+on+the+set+of+stable+stationary+points.+This+generalizes+for+procedures+with+several+roots+the+approach+suggested+by+L.+Ljung+for+processes+with+a+single+root.+%0D%0A%0D%0AAlong+with+stochastic+approximation+processes+as+such%2C+the+result+can+be+applied+to+generalized+urn+schemes+and+stochastic+models+of+technological+and+economic+dynamics+based+on+them%2C+in+particular%2C+evolutionary+games+with+incomplete+information.&rft.publisher=WP-95-019&rft.date=1995-02&rft.type=Monograph&rft.type=NonPeerReviewed&rft.format=text&rft.language=en&rft.identifier=https%3A%2F%2Fpure.iiasa.ac.at%2Fid%2Feprint%2F4573%2F1%2FWP-95-019.pdf&rft.identifier=++Kaniovski%2C+Y.M.+%3Chttps%3A%2F%2Fpure.iiasa.ac.at%2Fview%2Fiiasa%2F2009.html%3E++(1995).++Strong+Convergence+of+Stochastic+Approximation+Without+Lyapunov+Functions.+++IIASA+Working+Paper.+IIASA%2C+Laxenburg%2C+Austria%3A+WP-95-019+++++