On ideal convergence of double sequences in probabilistic normed spaces

Lacunary ideal convergence of multiple sequences in probabilistic normed spaces

“…In 1993, Alsina et al [1] presented a new definition of probabilistic normed space which includes the definition of Sherstnev as a special case. For an extensive view on this subject, we refer ( [2,12,22,23,33,38,40,51,52]). Subsequently, Mursaleen and Mohiuddine [47] and Rahmat [48] studied the ideal convergence in probabilistic normed spaces and V. Kumar and K. Kumar [37] studied I-Cauchy and I * -Cauchy sequences in probabilistic normed spaces.…”

On Lacunary Mean Ideal Convergence in Generalized Randomn-Normed Spaces

“…Mursaleen and Alotaibi [17] studied the notion of ideal convergence for single and double sequences in random 2-normed spaces, respectively. For more details and linked concept, we refer to [18][19][20][21][22][23][24][25][26]. In [27,28], Gähler introduced a gorgeous theory of 2-normed and -normed spaces in the 1960s; we have studied these subjects and constructed some sequence spaces defined by ideal convergence in -normed spaces [29,30].…”

On Cluster Points, Continuity, and Boundedness Associated with the Generalized Statistical Convergence in Probabilistic Normed Spaces

“…The concepts of I and I * -convergence, two important generalizations of statistical convergence, were introduced and investigated by Kostyrko et al [6]. The ideas were based on the notion of ideal I of N. Subsequently, a lot of investigations have been done on ideal convergence (see [7][8][9][10][11][12][13][14][15][16][17] where many more references both on ideal as well as statistical convergence can be found). Very recently, ideals were used in a different way to generalize the notion of statistical convergence [18,19] and certain new and summability methods were introduced and their basic properties were investigated.…”

“…Şençimen and Pehlivan have very recently extended the notion of strong convergence to strong statistical convergence in probabilistic metric spaces [36] and carried out further investigations on statistical continuity and statistical -boundedness in PN spaces [37,38]. These were followed by the studies of strong ideal convergence in PM and PN spaces in [10,13,39], studies of lacunary statistical 2 Abstract and Applied Analysis convergence in PN spaces in [40]. As a natural extension, we had recently introduced the idea of strong I-statistical convergence in PM spaces [41] and as a followup in this paper we investigate the notion of strong I-statistical limit and cluster points in PN spaces.…”