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In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of I  -statistical convergence, which is a recently introduced summability method. The names of our new methods are A  I  -lacunary statistical convergence and strongly A  I  -lacunary convergence with respect to a sequence of modulus functions. These spaces are denoted by S  θ  A  I  ,  F  and N  θ  A  I  ,  F  ,  respectively. We give some inclusion relations between S  A  I  ,  F  ,  S  θ  A  I  ,  F  and N  θ  A  I  ,  F  . We also investigate Cesaro summability for A  I  and we obtain some basic results between A  I  -Cesaro summability, strongly A  I  -Cesaro summability and the spaces mentioned above.

New Definitions about A I -Statistical Convergence with Respect to a Sequence of Modulus Functions and Lacunary Sequences