how to call the timing between 00.30 Am to 4.00 AM of day
11.50 PM of a day is called as mid night of a day. Similarly, the how to call the timing between 00.30 Am to 4.00 AM of day?

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For 11:50 pm, we say "Ten to midnight".  For 00:30, we say "Half past midnight".  Any other time,  we say for example 1 am or 1 O'Clock in the morning, 2 am or 2 O'clock in the morning etc..  Collectively you can call them "the early hours"

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For speakers of British English (outside of Scotland) the word 'wee' has other associations, especially when it comes to waking in the early hours...

I'd say the 'early hours of the morning'.

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Informally, I'd call it the wee hours of the morning.
http://forum.wordreference.com/showthread.php?t=2453409

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