The follicles remaining in the ovaries of a peri menopausal woman get less sensitive to stimulation of FSH, than they were during the child bearing age. There will be occasionally abnormal cycles, in which the follicle does not develop fully.
When a follicle does not develop properly, it secretes less estrogen and the FSH levels rise. Sufficiently low estrogen levels may fail to trigger the LH surge, which is supposed to rupture the follicle. Consequently the egg isn't released and there will be a cycle without ovulation. This is termed an anovulatory cycle.
During this, progesterone cannot be released at the appropriate time and the lack of estrogen gives rise to all the familiar peri menopause symptoms.
In perimenopause the number of such abnormal cycles without a mature follicle increases. Finally there will be cycles in which no follicle develops at all.
The decline of progesterone and estrogen results in a general shortening of the length of the cycles. These irregular cycles are often the first signs that peri menopause has started.