Catherine matures considerably throughout the course of the play. When we're first introduced to her, she is more of an older child than a young adult. This is largely because of Eddie, her possessive, controlling uncle, who refuses to allow Catherine to develop into a woman, the better to continue to exert control over her life.
Eddie's efforts at keeping Catherine in a state of arrested development ultimately prove both fruitless and tragic. The arrival of Marco and Rodolpho into the household changes the whole nature of the relationship between uncle and niece. Right from the outset, Eddie senses his control is slipping away.
To make matters worse, in the shape of Rodolpho he now has a rival for Catherine's affections. Rodolpho has shown Catherine that there's a big old world outside the cramped confines of the family apartment. And once she sees this, the irreversible process from childhood into adulthood can begin in earnest.
Before long, Catherine becomes more independent, getting herself a job and spending more time outside the home. Though still far from the finished article in terms of maturity, Catherine shows that she's a young woman with her mind of her own.
We see this toward the end of the play in the way that she stands up to Eddie, something that would've been unthinkable at the start. Yet she also shows her independence from Rodolpho by justifying to him her close relationship to Eddie.
Catherine may still, to some considerable extent, be influenced in her behavior by others, but there seems little doubt by the end of the play that this is only a temporary phase and that Catherine will make her own way in life, whatever anyone else might say.