How do I put months in each array?

I'm making a program for expense statistics over a month; however, I don't know what I should do in the for loop in order to make it initialize each MonthlyExpensePeriods with the corresponding name of that month in the correct order?

for(int i = 0; i < 12; i++){ 
    monthlyExpensePeriods[i] = new MonthlyExpensePeriods(month's name); 
}

You could use DateFormatSymbols.getMonths():

String[] monthNames = new DateFormatSymbols().getMonths();
for(int i = 0; i < 12; i++){ 
    monthlyExpensePeriods[i] = new MonthlyExpensePeriods(monthNames[i]); 
}

tl;dr

Use Month enum objects rather than mere text of name of month.

for(int i = 1; i <= 12; i++){ 
monthlyExpensePeriods[i] = new MonthlyExpensePeriod( Month.of( i ) ); 
}

Use smart objects, not dumb strings

Using strings for special values, such as tracking each month of the year, is fragile and error-prone.

Month

Instead, use objects.

In Java 8 and later, we have the Month enum class built-in. This enum predefines a dozen objects, one for each month of the year, each assigned to a named constant.

Your class MonthlyExpensePeriod should hold a member variable of type Month.

public class MonthlyExpensePeriod {
// Member fields
public Month month ;
…
// Constructor
public MonthlyExpensePeriod( final Month monthArg ) {
this.month = monthArg ;
…
}
}

To use that class and its constructor, pass one of the named constants.

new MonthlyExpensePeriod ( Month.MARCH )

You can retrieve one of the Month objects by month number. Notice the sane numbering, 1-12 for January-December.

Month month = Month.of( 3 ) ;  // Month.MARCH

So your example code would look like the following. Change your for loop to count 1-12.

for(int i = 1; i <= 12; i++){ 
monthlyExpensePeriods[i] = new MonthlyExpensePeriod( Month.of( i ) ); 
}

Collections

You may want to use Java Collections rather than mere arrays.

List< MonthlyExpensePeriod > periods = new ArrayList<>( 12 ) ;
for(int i = 1; i <= 12; i++){ 
MonthlyExpensePeriod period = new MonthlyExpensePeriod( Month.of( i ) ) ;
periods.add( period ); 
}

Streams

I suppose we could get fancy and use streams. But in this case I do not see any added-value.

Calling Month.values() returns an array of all the objects defined on the enum, in the order in which they were defined. From that, we can generate a stream by calling the utility class method Arrays.stream.

List < MonthlyExpensePeriod > periods =
Arrays.stream( Month.values() )
.map( month -> new MonthlyExpensePeriod( month ) )
.collect( Collectors.toList() )
;

Or as a single-line of code.

List < MonthlyExpensePeriod > periods = Arrays.stream( Month.values() ).map( month -> new MonthlyExpensePeriod( month ) ).collect( Collectors.toList() );

You might want to make an unmodifiable copy of your list.

List < MonthlyExpensePeriod > periodsUnmod = List.copyOf( periods );

Or make the original list unmodifiable, by calling Collectors.toUnmodifiableList(), as discussed here.

List < MonthlyExpensePeriod > periods = 
Arrays
.stream( Month.values() )
.map( month -> new MonthlyExpensePeriod( month ) )
.collect( Collectors.toUnmodifiableList() )
;

Generating display text

You may want to display the name of the month.

Calling Month::toString generates text of the name in English in all caps. Instead you can automatically localize.

To localize, pass a TextStyle object to signal how long or abbreviated you want the name. Notice that TextStyle offers STANDALONE variations used in some languages (not English) where the month name appears outside the context of a date.

And pass a Locale to determine the human language and cultural norms to use in localizing.

Locale locale = Locale.CANADA_FRENCH ;  // Or Locale.US and so on.
String output = Month.MARCH.getDisplayName( TextStyle.FULL , locale ) ;