In this paper, the concepts of asymptotically \mathcal{I}_{\sigma}-equivalence, \sigma-asymptotically equivalence, st\-rong\-ly \sigma-asymptotically equivalence and strongly \sigma-asymptotically p-equivalence for real number sequences are de\-fi\-ned. Also, we give relationships among these new type equivalence concepts and the concept of \linebreak S_{\sigma}-asymptotically equivalence which is studied in [Savaş, E. and Patterson, R. F., \sigma-asymptotically lacunary statistical equivalent sequences, Cent. Eur. J. Math., 4 (2006), No. 4, 648–655].

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Ulusu, Uǧur