www.ijcrt.org © 2020 IJCRT | Volume 8, Issue 8 August 2020 | ISSN: 2320-2882 IJCRT2008303 International Journal of Creative Research Thoughts (IJCRT) www.ijcrt.org 2702 Invariant means of sequences with statistical behaviour Abdul Hamid Ganie 1 Sameer Ahmad Gupkari 2 and Afroza Akhter 3 1,3 Department of Applied Sciences and Humanities, SSM College of Engineering and Technology Pattan, Jammu and Kahsmir. 2 J & K Institute of Mathematical Sciences, Kashmir-India. 3 AlJanoub Internationational School, Asir-KSA. Abstract: The current paper is thrusting to bring out some techniques of spaces by defining   and   ∅ of sigma strongly convergence in statistically nature and lacunary nature of convergent sequences of strong sigma statistically in nature. Some basic topological properties will be given. Mathematics Subject Classification [2010] : 46A45; 46CO5, 46B50. Keywords: Sequences; summable nature; convergence. 1. Introduction and historical background: We call a sequence to be function whose domain is the set of natural numbers. Let us represent  as set of all real or complex sequences, so the sequence space is any subspace of . With ,  and  we designate the set of non-negative integers, the set of real numbers and the set of complex numbers, respectively. Let  ∞ ,  and  0 , respectively, designates the set of bounded sequences, convergent sequences and those which has limit as zero [1], [8], 11-[15], [18]-[21]. Cesàro sums represents an “averaging” process. In 1890 the Italian mathematician Ernesto Cesàro used such sums while investigating products of infinite series. A expansion of the type ∑   ∞ =0 is said to be Cesaro summable to ∈  if and only if its Cesaro sum converges i.e.,   = ∑ [1 −   ]  −1 =0 =  0 + 1 +⋯+ −1  = 1  ∑  −1 =0 converges to  as → ∞. Thus, for a sequence   with ∑   ∞ =0 = converges if and only if its sequence of partial sums   converges to  i.e., given >0, we can find a natural number  0 ∈ in such a way that ≥ 0 implies |  − | <  or equivalently, if |  − | <  then ∑   ∞ =0 is Cesàro is summable [22]. Then, we can have |  − | = |  0 + 1 +⋯+ −1  − | = |∑ [1 −   ]  −1 =0 − |.